We give a method for explicitly constructing an elementary cubic extension <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.0498209pt" id="M1" height="8.68572pt" version="1.1" viewBox="-0.0498162 -8.6359 7.31558 8.68572" width="7.31558pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-77" d="M541 160L512 170C485 116 462 88 439 67C411 41 376 35 318 35C272 35 238 37 224 48C207 61 205 85 212 121L290 533C305 611 308 615 391 622L397 650H139L133 622C217 615 221 612 206 533L126 118C111 41 103 34 23 26L17 0H474C489 31 528 124 541 160Z"/><glyph.data ascent="3473" descent="-2876" horiz-adv-x="553" vert-adv-y="553"/></g></svg> over which an elliptic curve <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.429399pt" id="M2" height="15.0208pt" version="1.1" viewBox="-0.0498162 -11.5914 149.372 15.0208" width="149.372pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path 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In our method, the extension degree <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.15067pt" id="M4" height="11.439pt" version="1.1" viewBox="-0.0498162 -9.28833 36.5404 11.439" width="36.5404pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-92" d="M290 -163V-135C183 -126 181 -122 181 -44V583C181 662 184 666 290 675V703H120V-163H290Z"/><glyph.data ascent="3473" descent="-2876" horiz-adv-x="345" vert-adv-y="345"/></g><g transform="matrix(.013,0,0,-0.013,4.485,0)"><path id="g113-77" d="M541 160L512 170C485 116 462 88 439 67C411 41 376 35 318 35C272 35 238 37 224 48C207 61 205 85 212 121L290 533C305 611 308 615 391 622L397 650H139L133 622C217 615 221 612 206 533L126 118C111 41 103 34 23 26L17 0H474C489 31 528 124 541 160Z"/><glyph.data ascent="3473" descent="-2876" horiz-adv-x="553" vert-adv-y="553"/></g><g transform="matrix(.013,0,0,-0.013,15.302,0)"><path id="g113-59" d="M116 412C83 412 59 388 59 355C59 321 83 297 115 297S170 321 170 355C170 388 147 412 116 412ZM114 -12C147 -12 171 11 171 46C171 78 147 103 115 103C84 103 60 78 60 46C60 11 84 -12 114 -12Z"/><glyph.data ascent="3473" descent="-2876" horiz-adv-x="228" vert-adv-y="228"/></g><g transform="matrix(.013,0,0,-0.013,21.898,0)"><path id="g197-28" d="M703 -12C678 -16 653 -21 626 -21C591 -21 567 -1 542 24C650 81 715 195 715 332C715 529 588 676 385 676C179 676 45 536 45 332C45 139 185 -14 381 -14C422 -14 460 -8 494 4C525 -31 561 -65 626 -65C654 -65 681 -60 709 -56L703 -12ZM591 548C648 488 671 412 671 332C671 255 647 175 591 119V548ZM547 79C536 70 523 63 510 58C478 91 441 122 384 122C329 122 290 96 257 56C240 63 226 70 213 79V584C251 616 334 632 381 632C434 632 504 619 547 584V79ZM169 119C116 175 89 255 89 332C89 412 112 488 169 548V119ZM460 41C432 34 406 30 381 30C358 30 333 34 305 41C326 64 348 78 381 78C416 78 439 62 460 41Z"/><glyph.data ascent="693" descent="-218" horiz-adv-x="765" vert-adv-y="765"/></g><g transform="matrix(.013,0,0,-0.013,31.843,0)"><path id="g113-94" d="M226 -163V703H56V676C162 667 165 662 165 584V-43C165 -122 162 -126 56 -136V-163H226Z"/><glyph.data ascent="3473" descent="-2876" horiz-adv-x="345" vert-adv-y="345"/></g></svg> often becomes small.

Rintaro Kozuma

On the Rank of Elliptic Curves in Elementary Cubic Extensions

Aleksandar Ivi&#x107;

Hybrid Moments of the Riemann Zeta-Function

We prove general theorems for the explicit evaluations of the level 13
analogue of Rogers-Ramanujan continued fraction and find some new explicit values. This work is
a sequel to some recent works of S. Cooper and D. Ye.

Nipen Saikia

New General Theorems and Explicit Values of the Level 13 Analogue of Rogers-Ramanujan Continued Fraction

Jordan Schettler

Using Continued Fractions to Compute Iwasawa Lambda Invariants of Imaginary Quadratic Number Fields

 We deduce a simple representation and the invariant factor decompositions of the subgroups of the group <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42776pt" id="M2" height="12.4801pt" version="1.1" viewBox="-0.0498162 -9.05234 47.4443 12.4801" width="47.4443pt"><g transform="matrix(.0136,0,0,-0.0136,0,0)"><use xlink:href="#g197-37"/></g><g transform="matrix(.0095,0,0,-0.0095,9.85,3.264)"><use xlink:href="#g50-110"/></g><g transform="matrix(.0135,0,0,-0.0135,21.039,0)"><use xlink:href="#g117-42"/></g><g transform="matrix(.0136,0,0,-0.0136,32.02,0)"><use xlink:href="#g197-37"/></g><g transform="matrix(.0095,0,0,-0.0095,41.869,3.264)"><use xlink:href="#g50-111"/></g></svg>, where <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2129903pt" id="M3" height="6.42315pt" version="1.1" viewBox="-0.0498162 -6.21016 10.8292 6.42315" width="10.8292pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-110" d="M766 88L752 113C719 83 690 64 681 64C674 64 672 74 679 103L724 292C758 436 724 448 701 448C680 448 666 442 639 429C594 407 514 350 441 252H439L447 289C476 423 450 448 419 448C398 448 379 441 355 427C307 400 234 344 162 249H160L180 324C203 409 197 448 170 448C144 448 82 413 23 349L35 321C57 343 96 374 108 374C115 374 117 371 111 341C87 227 57 112 24 -6L32 -12C53 -4 81 4 108 6C119 68 134 128 149 171C177 229 309 383 364 383C387 383 388 355 373 282C354 190 330 92 303 -6L309 -12C332 -4 356 3 386 6C396 63 411 122 424 171C458 236 590 383 642 383C658 383 664 369 652 315L603 91C587 20 593 -12 619 -12C642 -12 708 23 766 88Z"/></g></svg> and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2129903pt" id="M4" height="6.42315pt" version="1.1" viewBox="-0.0498162 -6.21016 6.92631 6.42315" width="6.92631pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-111" d="M495 86L479 114C446 82 419 66 409 66C401 66 401 72 406 97C420 166 436 231 453 297C489 435 454 448 428 448C406 448 384 439 354 422C305 394 222 327 161 247H159L183 345C200 415 194 448 173 448C143 448 82 410 23 351L38 325C64 349 95 371 105 371C111 371 116 365 109 336L25 -4L31 -12C50 -4 77 3 107 9C119 69 132 122 145 168C197 254 321 381 370 381C387 381 393 374 378 305L329 95C309 17 320 -12 345 -12C372 -12 430 19 495 86Z"/></g></svg> are arbitrary positive integers. We obtain formulas for the total number of subgroups and the number of subgroups of a given order.

Mario Hampejs

Nicki Holighaus

L&#xe1;szl&#xf3; T&#xf3;th

Christoph Wiesmeyr

Representing and Counting the Subgroups of the Group <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42776pt" height="12.4801pt" version="1.1" viewBox="-0.0498162 -9.05234 47.4443 12.4801" width="47.4443pt"><g transform="matrix(.0136,0,0,-0.0136,0,0)"><path id="g197-37" d="M677 618V662H100V618H469L50 44V0H668V44H252L677 618ZM617 618L200 44H103L523 618H617Z" /><glyph.data ascent="693" descent="-218" horiz-adv-x="727" vert-adv-y="727" /></g><g transform="matrix(.0095,0,0,-0.0095,9.85,3.264)"><path id="g50-110" d="M781 91L765 118C733 86 702 70 693 70C686 70 685 77 692 106L738 297C772 439 736 451 712 451S675 445 647 431C605 410 526 356 452 257H450L459 294C491 426 459 451 429 451C403 451 386 445 361 431C312 404 239 343 170 253H168L188 333C208 411 205 451 175 451C148 451 85 417 24 346L39 318C57 338 100 374 112 374C120 374 122 372 115 339C91 226 62 112 28 -6L37 -12C59 -4 87 3 114 6C125 68 142 127 154 168C184 227 316 383 372 383C397 383 396 366 381 289C362 192 338 92 312 -6L317 -12C339 -5 367 3 396 6L434 170C468 234 598 383 650 383C668 383 676 369 664 319L609 90C593 23 600 -12 629 -12C654 -12 722 23 781 91Z" /><glyph.data ascent="3443" descent="-2856" horiz-adv-x="805" vert-adv-y="805" /></g><g transform="matrix(.0135,0,0,-0.0135,21.039,0)"><path id="g117-42" d="M528 54L331 254L528 455L492 493L294 291L96 493L60 455L257 254L60 54L96 16L294 217L492 16L528 54Z" /><glyph.data ascent="3473" descent="-2876" horiz-adv-x="587" vert-adv-y="587" /></g><g transform="matrix(.0136,0,0,-0.0136,32.02,0)"><use xlink:href="#g197-37"/><glyph.data ascent="693" descent="-218" horiz-adv-x="727" vert-adv-y="727" /></g><g transform="matrix(.0095,0,0,-0.0095,41.869,3.264)"><path id="g50-111" d="M504 89L488 119C453 85 428 69 416 69C408 69 405 75 412 102L462 304C492 438 460 451 436 451S388 441 356 423C311 397 229 336 170 256H168L189 340C208 418 200 451 177 451C144 451 82 413 24 352L40 322C65 345 98 369 108 369C114 369 119 363 112 335L28 -3L36 -12C54 -3 83 4 112 10C125 73 138 122 153 174C205 258 328 382 376 382C395 382 399 369 384 305L330 93C312 25 323 -12 351 -12C378 -12 439 21 504 89Z" /><glyph.data ascent="3443" descent="-2856" horiz-adv-x="512" vert-adv-y="512" /></g></svg>

We consider the Diophantine equation <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.552pt" id="M1" height="15.3998pt" version="1.1" viewBox="-0.0498162 -12.8478 322.212 15.3998" width="322.212pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-50" d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"/></g><g transform="matrix(.0095,0,0,-0.0095,6.503,-5.984)"><path id="g50-108" d="M485 418C485 434 469 451 441 451C389 451 312 387 255 335C221 304 194 278 159 242H157L259 678C264 698 264 710 255 710C242 710 183 681 96 673L90 643L128 642C165 641 171 640 162 602L24 -5L31 -12C51 -4 81 4 110 9L145 181C154 195 180 220 205 242C233 170 263 106 291 53C318 1 335 -12 360 -12C383 -12 425 0 483 82L463 105C437 79 411 57 400 57C388 57 378 68 355 110C329 156 288 244 268 298C299 333 355 376 405 376C426 376 440 373 448 368C453 365 465 368 468 373C477 386 485 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transform="matrix(.0135,0,0,-0.0135,183.958,0)"><use xlink:href="#g117-36"/></g><g transform="matrix(.0135,0,0,-0.0135,194.939,.19)"><use xlink:href="#g113-41"/></g><g transform="matrix(.0135,0,0,-0.0135,199.644,0)"><use xlink:href="#g113-121"/></g><g transform="matrix(.0135,0,0,-0.0135,210.246,0)"><use xlink:href="#g117-36"/></g><g transform="matrix(.0135,0,0,-0.0135,221.226,0)"><path id="g113-52" d="M285 378C315 398 338 416 353 432C373 451 384 474 384 503C384 579 325 635 236 635H235C182 635 136 610 108 579L65 516L85 496C110 533 150 575 205 575C258 575 300 543 300 481C300 407 232 369 141 339L147 310C163 315 188 321 211 321C268 321 338 284 338 192C338 94 288 40 217 40C160 40 119 68 93 91C85 98 77 97 69 91C60 84 47 71 46 58C44 46 48 35 62 22C75 10 116 -12 162 -12C234 -12 424 62 424 224C424 297 373 359 285 376V378Z"/></g><g transform="matrix(.0135,0,0,-0.0135,227.73,.19)"><use xlink:href="#g113-42"/></g><g transform="matrix(.0095,0,0,-0.0095,232.517,-6.058)"><use xlink:href="#g50-108"/></g><g transform="matrix(.0135,0,0,-0.0135,240.888,0)"><use xlink:href="#g117-36"/></g><g transform="matrix(.0135,0,0,-0.0135,251.869,0)"><use xlink:href="#g113-46"/></g><g transform="matrix(.0135,0,0,-0.0135,257.229,0)"><use xlink:href="#g113-46"/></g><g transform="matrix(.0135,0,0,-0.0135,262.589,0)"><use xlink:href="#g113-46"/></g><g transform="matrix(.0135,0,0,-0.0135,268.741,0)"><use xlink:href="#g117-36"/></g><g transform="matrix(.0135,0,0,-0.0135,279.722,.19)"><use xlink:href="#g113-41"/></g><g transform="matrix(.0135,0,0,-0.0135,284.427,0)"><use xlink:href="#g113-121"/></g><g transform="matrix(.0135,0,0,-0.0135,295.029,0)"><use xlink:href="#g117-36"/></g><g transform="matrix(.0135,0,0,-0.0135,306.01,0)"><path id="g113-115" d="M393 379C402 394 400 411 393 422C384 437 365 448 348 448C301 448 237 372 186 285H182L193 335C210 408 205 448 178 448C150 448 80 402 29 344L45 321C80 355 114 373 122 373C128 373 130 365 124 330C106 228 76 98 50 -5L57 -12C82 -5 112 3 132 6L172 203C196 256 234 304 254 329C275 355 293 367 306 367C318 367 330 360 342 348C347 343 355 343 365 350S386 367 393 379Z"/></g><g transform="matrix(.0135,0,0,-0.0135,311.729,.19)"><use xlink:href="#g113-42"/></g><g transform="matrix(.0095,0,0,-0.0095,316.511,-6.058)"><use xlink:href="#g50-108"/></g></svg> for some natural numbers <i>x, k</i>, and <i>r</i>, and we call <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.3625803pt" id="M2" height="9.06134pt" version="1.1" viewBox="-0.0498162 -8.69876 34.7951 9.06134" width="34.7951pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-51"/></g><g transform="matrix(.0135,0,0,-0.0135,6.503,0)"><use xlink:href="#g113-121"/></g><g transform="matrix(.0135,0,0,-0.0135,17.105,0)"><use xlink:href="#g117-36"/></g><g transform="matrix(.0135,0,0,-0.0135,28.086,0)"><use xlink:href="#g113-50"/></g></svg> as <i>k</i>th order 2-gap balancing number. It was also proved that there are infinitely many first order 2-gap balancing numbers. In this paper, we show that the only second order 2-gap balancing number is 1.

S. S. Rout

On Second Order Gap Balancing Numbers

It is proved in this paper that, for any point on an elliptic curve, the mean value of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.16258pt" id="M1" height="6.23217pt" version="1.1" viewBox="0 -6.06959 7.57344 6.23217" width="7.57344pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-121" d="M536 404C536 423 520 448 491 448C445 448 398 404 308 283L286 338C255 416 242 448 217 448C182 448 138 402 92 341L111 321C149 368 169 378 178 378C188 378 198 363 214 320L254 212C185 117 138 65 107 65C98 65 85 69 82 75C79 82 73 86 65 86C44 86 23 60 23 39C23 7 44 -12 71 -12C119 -12 168 33 265 177L306 61C321 17 347 -12 373 -12C413 -12 465 33 507 96L491 119C459 84 432 60 413 60C395 60 378 92 358 148L321 250C341 279 369 310 389 332C417 363 439 382 456 382C466 382 475 376 481 368C486 361 492 358 496 358C513 358 536 381 536 404Z"></path></g></svg>-coordinates of its <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.16258pt" id="M2" height="6.23217pt" version="1.1" viewBox="0 -6.06959 6.80119 6.23217" width="6.80119pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-111" d="M495 86L479 114C446 82 419 66 409 66C401 66 401 72 406 97C420 166 436 231 453 297C489 435 454 448 428 448C406 448 384 439 354 422C305 394 222 327 161 247H159L183 345C200 415 194 448 173 448C143 448 82 410 23 351L38 325C64 349 95 371 105 371C111 371 116 365 109 336L25 -4L31 -12C50 -4 77 3 107 9C119 69 132 122 145 168C197 254 321 381 370 381C387 381 393 374 378 305L329 95C309 17 320 -12 345 -12C372 -12 430 19 495 86Z"></path></g></svg>-division points is the same as its <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.16258pt" id="M3" height="6.23217pt" version="1.1" viewBox="0 -6.06959 7.57344 6.23217" width="7.57344pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-121"></use></g></svg>-coordinate and that of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.53608pt" id="M4" height="9.60567pt" version="1.1" viewBox="0 -6.06959 7.8444 9.60567" width="7.8444pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-122" d="M556 393C556 426 537 448 514 448C496 448 478 435 471 420C468 414 462 401 466 394C473 382 480 368 480 346C480 268 392 143 338 67H336C329 163 319 253 309 330S286 448 254 448C215 448 160 383 127 337L143 311C167 344 200 373 208 373S222 365 229 320C246 214 264 66 268 -20C219 -83 131 -171 17 -239L25 -261L137 -235C248 -110 273 -79 335 8C384 77 481 215 520 287C544 332 556 363 556 393Z"></path></g></svg>-coordinates of its <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.16258pt" id="M5" height="6.23217pt" version="1.1" viewBox="0 -6.06959 6.80119 6.23217" width="6.80119pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-111"></use></g></svg>-division points is <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.16258pt" id="M6" height="6.23217pt" version="1.1" viewBox="0 -6.06959 6.80119 6.23217" width="6.80119pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-111"></use></g></svg> times that of its <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.53608pt" id="M7" height="9.60567pt" version="1.1" viewBox="0 -6.06959 7.8444 9.60567" width="7.8444pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-122"></use></g></svg>-coordinate.

Rongquan Feng

Hongfeng Wu

A Mean Value Formula for Elliptic Curves

We consider certain quartic twists of an elliptic curve. We establish the rank of these curves under the Birch and Swinnerton-Dyer conjecture and obtain bounds on the size of Shafarevich-Tate group of these curves. We also establish a reduction between the problem of factoring integers of a certain form and the problem of computing rational points on these twists.

Iftikhar A. Burhanuddin

Ming-Deh A. Huang

On the Equation <svg style="vertical-align:-2.29482pt" height="20.275" version="1.1" viewBox="0 0 97.025002 20.275" width="97.025002" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,15.775)"><path id="x1D466" d="M556 393q0 -39 -36 -106q-42 -78 -185 -279q-47 -66 -81 -108t-117 -135l-112 -26l-8 22q150 90 251 219q-6 136 -39 340q-8 53 -21 53q-6 0 -27 -19.5t-38 -42.5l-16 26q80 111 127 111q23 0 35 -28t20 -90q18 -137 27 -263h2q142 200 142 279q0 24 -14 48q-4 7 5 26&#xA;q13 28 43 28q18 0 30 -15.5t12 -39.5z" /></g><g transform="matrix(.012,-0,0,-.012,9.9,7.613)"><path id="x32" d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" /></g><g transform="matrix(.017,-0,0,-.017,20.963,15.775)"><path id="x3D" d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" /></g><g transform="matrix(.017,-0,0,-.017,35.649,15.775)"><path id="x1D465" d="M536 404q0 -17 -13.5 -31.5t-26.5 -14.5q-8 0 -15 10q-11 14 -25 14q-22 0 -67 -50q-47 -52 -68 -82l37 -102q31 -88 55 -88t78 59l16 -23q-32 -48 -68.5 -78t-65.5 -30q-19 0 -37.5 20t-29.5 53l-41 116q-72 -106 -114.5 -147.5t-79.5 -41.5q-21 0 -34.5 14t-13.5 37&#xA;q0 16 13.5 31.5t28.5 15.5q12 0 17 -11q5 -10 25 -10q22 0 57.5 36t89.5 111l-40 108q-22 58 -36 58q-21 0 -67 -57l-19 20q81 107 125 107q17 0 30 -22t39 -88l22 -55q68 92 108.5 128.5t74.5 36.5q20 0 32.5 -14t12.5 -30z" /></g><g transform="matrix(.012,-0,0,-.012,45.162,7.613)"><path id="x33" d="M285 378v-2q65 -13 102 -54.5t37 -97.5q0 -57 -30.5 -104.5t-74 -75t-85.5 -42t-72 -14.5q-31 0 -59.5 11t-40.5 23q-19 18 -16 36q1 16 23 33q13 10 24 0q58 -51 124 -51q55 0 88 40t33 112q0 64 -39 96.5t-88 32.5q-29 0 -64 -11l-6 29q77 25 118 57.5t41 84.5&#xA;q0 45 -26.5 69.5t-68.5 24.5q-67 0 -120 -79l-20 20l43 63q51 56 127 56h1q66 0 107 -37t41 -95q0 -42 -31 -71q-22 -23 -68 -54z" /></g><g transform="matrix(.017,-0,0,-.017,55.275,15.775)"><path id="x2212" d="M535 230h-483v50h483v-50z" /></g><g transform="matrix(.017,-0,0,-.017,69.027,15.775)"><path id="x1D45D" d="M570 304q0 -108 -87 -199q-40 -42 -94.5 -74t-105.5 -43q-41 0 -65 11l-29 -141q-9 -45 -1.5 -58t45.5 -16l26 -2l-5 -29l-241 -10l4 26q51 10 67.5 24t26.5 60l113 520q-54 -20 -89 -41l-7 26q38 28 105 53l11 49q20 25 77 58l8 -7l-17 -77q39 14 102 14q82 0 119 -36&#xA;t37 -108zM482 289q0 114 -113 114q-26 0 -66 -7l-70 -327q12 -14 32 -25t39 -11q59 0 118.5 81.5t59.5 174.5z" /></g><g transform="matrix(.017,-0,0,-.017,79.107,15.775)"><path id="x1D45E" d="M474 429q-19 -69 -47 -222l-65 -347q-8 -49 0 -60t49 -16l26 -3l-4 -26l-246 -12l4 25l17 3q37 6 50 20.5t23 60.5l67 321h-2q-65 -79 -150 -138q-69 -47 -104 -47q-28 0 -48.5 32t-20.5 81q0 78 35 148t90 117q67 57 161 77q23 5 58 5q43 0 90 -15zM387 387&#xA;q-30 16 -75 16q-58 0 -92 -27q-49 -39 -78.5 -112t-29.5 -136q0 -34 8.5 -52.5t21.5 -18.5q40 0 109.5 63.5t103.5 115.5q16 53 32 151z" /></g><g transform="matrix(.017,-0,0,-.017,87.454,15.775)"><use xlink:href="#x1D465"/></g></svg>

A theorem of Dubickas, affirming a conjecture of Kuba, states that a nonzero algebraic number is a root of a polynomial <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.34499pt;width:11.0375px;" id="M1" height="16.6625" version="1.1" viewBox="0 0 11.0375 16.6625" width="11.0375">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path id="x1D453" d="M619 670q0 -13 -9 -26t-18 -19q-13 -10 -25 2q-36 38 -66 38q-31 0 -54.5 -50t-45.5 -185h120l-20 -31l-107 -12q-23 -138 -57 -293q-27 -122 -55 -184.5t-75 -109.5q-60 -61 -114 -61q-25 0 -47.5 15t-22.5 31q0 17 31 44q11 8 20 -1q10 -11 31 -19t35 -8q26 0 47 19&#xA;q34 34 71 253l54 315h-90l-3 12l31 30h70q28 138 90 204q35 37 75 57.5t70 20.5q26 0 45 -14t19 -28z"></path></g>

		
	
</svg> with positive rational coefficients if and only if none of its conjugates is a positive real number. A certain quantitative version of this result, yielding a growth factor for the coefficients of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.34499pt;width:11.0375px;" id="M2" height="16.6625" version="1.1" viewBox="0 0 11.0375 16.6625" width="11.0375">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x1D453"></use></g>

		
	
</svg> similar to the condition of the classical Eneström-Kakeya theorem of such polynomial, is derived. The bound for the growth factor so obtained is shown to be sharp for some particular classes of algebraic numbers.

Vichian Laohakosol

Suton Tadee

Algebraic Numbers Satisfying Polynomials with Positive Rational Coefficients

We give a general link between weighted Selberg integrals of any arithmetic function <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.34499pt;width:12.1625px;" id="M1" height="17.7875" version="1.1" viewBox="0 0 12.1625 17.7875" width="12.1625">
	
		
			<g transform="matrix(.017,-0,0,-.017,.625,12.725)"><path id="x1D453" d="M619 670q0 -13 -9 -26t-18 -19q-13 -10 -25 2q-36 38 -66 38q-31 0 -54.5 -50t-45.5 -185h120l-20 -31l-107 -12q-23 -138 -57 -293q-27 -122 -55 -184.5t-75 -109.5q-60 -61 -114 -61q-25 0 -47.5 15t-22.5 31q0 17 31 44q11 8 20 -1q10 -11 31 -19t35 -8q26 0 47 19&#xA;q34 34 71 253l54 315h-90l-3 12l31 30h70q28 138 90 204q35 37 75 57.5t70 20.5q26 0 45 -14t19 -28z"></path></g>

		
	
</svg> and averages of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.34499pt;width:12.1625px;" id="M2" height="17.7875" version="1.1" viewBox="0 0 12.1625 17.7875" width="12.1625">
	
		
			<g transform="matrix(.017,-0,0,-.017,.625,12.725)"><use xlink:href="#x1D453"></use></g>

		
	
</svg> correlations in short intervals, proved by the elementary dispersion method (our version of Linnik’s method). We formulate conjectural bounds for the so-called modified Selberg integral of the divisor functions <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.2316pt;width:37.587502px;" id="M3" height="17.575001" version="1.1" viewBox="0 0 37.587502 17.575001" width="37.587502">
	
		
			<g transform="matrix(.017,-0,0,-.017,.625,12.725)"><path id="x1D451" d="M530 686q-79 -330 -126 -595q-5 -24 4 -24q22 0 77 53l16 -24q-42 -49 -88 -78.5t-72 -29.5q-37 0 -19 83l21 99h-2q-62 -83 -153 -141q-65 -41 -97 -41q-28 0 -48 31.5t-20 91.5q0 71 32.5 144t88.5 118q36 30 94 52.5t94 22.5q38 0 68 -13l29 158q7 40 2 47.5t-38 7.5&#xA;h-35l1 26q39 4 76 13.5t58.5 17t27.5 7.5q16 0 9 -26zM387 375q-11 11 -38.5 20t-51.5 9q-45 0 -79 -23q-46 -32 -77.5 -107.5t-31.5 -143.5q0 -40 9.5 -58.5t23.5 -18.5q40 0 113.5 70t99.5 118z"></path></g>

			<g transform="matrix(.012,-0,0,-.012,10.062,16.8)"><path id="x1D458" d="M480 416q0 -21 -18 -41q-9 -11 -17 -7q-20 9 -42 9q-62 0 -140 -78q23 -69 88 -192q17 -31 27 -42t20 -11q16 0 62 46l17 -20q-64 -92 -119 -92q-35 0 -70 66q-41 73 -84 187q-36 -30 -62 -61q-27 -115 -35 -172q-41 -8 -78 -20l-6 6l140 612q7 28 0.5 34t-37.5 7l-34 1&#xA;l5 26q38 4 74 13.5t57 17t25 7.5q12 0 4 -32l-104 -443h2q35 38 97 93q39 35 65.5 56t62 41.5t58.5 20.5q19 0 30.5 -10t11.5 -22z"></path></g>

			<g transform="matrix(.017,-0,0,-.017,16.662,12.725)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z"></path></g><g transform="matrix(.017,-0,0,-.017,22.544,12.725)"><path id="x1D45B" d="M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2&#xA;q42 56 97 103.5t96 71.5q46 26 74 26q9 0 16 -2.5t14 -11.5t9.5 -24.5t-1 -44t-13.5 -68.5q-30 -117 -47 -200q-4 -19 -3.5 -25t6.5 -6q21 0 70 48z"></path></g><g transform="matrix(.017,-0,0,-.017,31.077,12.725)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z"></path></g>

		
	
</svg>, gauged by the Cesaro weight in the short interval <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-1.95624pt;width:130.52499px;" id="M4" height="15.9625" version="1.1" viewBox="0 0 130.52499 15.9625" width="130.52499">
	
		
			<g transform="matrix(.017,-0,0,-.017,.625,12.575)"><use xlink:href="#x1D45B"></use></g><g transform="matrix(.017,-0,0,-.017,13.867,12.575)"><path id="x2208" d="M448 1h-83q-118 0 -201.5 74.5t-83.5 179.5t83.5 179.5t201.5 74.5h83v-50h-84q-87 0 -150.5 -51.5t-73.5 -127.5h308v-50h-308q10 -76 73.5 -127.5t150.5 -51.5h84v-50z"></path></g><g transform="matrix(.017,-0,0,-.017,27.568,12.575)"><path id="x5B" d="M290 -163h-170v866h170v-28q-79 -7 -94 -19.5t-15 -72.5v-627q0 -59 14.5 -71.5t94.5 -19.5v-28z"></path></g><g transform="matrix(.017,-0,0,-.017,33.433,12.575)"><path id="x1D465" d="M536 404q0 -17 -13.5 -31.5t-26.5 -14.5q-8 0 -15 10q-11 14 -25 14q-22 0 -67 -50q-47 -52 -68 -82l37 -102q31 -88 55 -88t78 59l16 -23q-32 -48 -68.5 -78t-65.5 -30q-19 0 -37.5 20t-29.5 53l-41 116q-72 -106 -114.5 -147.5t-79.5 -41.5q-21 0 -34.5 14t-13.5 37&#xA;q0 16 13.5 31.5t28.5 15.5q12 0 17 -11q5 -10 25 -10q22 0 57.5 36t89.5 111l-40 108q-22 58 -36 58q-21 0 -67 -57l-19 20q81 107 125 107q17 0 30 -22t39 -88l22 -55q68 92 108.5 128.5t74.5 36.5q20 0 32.5 -14t12.5 -30z"></path></g><g transform="matrix(.017,-0,0,-.017,46.709,12.575)"><path id="x2212" d="M535 230h-483v50h483v-50z"></path></g><g transform="matrix(.017,-0,0,-.017,60.461,12.575)"><path id="x1D43B" d="M865 650q-1 -4 -4 -14t-4 -14q-62 -5 -77 -19.5t-29 -82.5l-74 -394q-12 -61 -0.5 -77t75.5 -21l-6 -28h-273l8 28q64 5 82 21t29 76l36 198h-380l-37 -197q-11 -64 0.5 -78.5t79.5 -19.5l-6 -28h-268l6 28q60 6 75.5 21.5t26.5 76.5l75 394q13 66 2 81.5t-77 20.5l8 28&#xA;h263l-6 -28q-58 -5 -75.5 -21t-30.5 -81l-26 -153h377l29 153q12 67 2 81t-74 21l5 28h268z"></path></g><g transform="matrix(.017,-0,0,-.017,75.369,12.575)"><path id="x2C" d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z"></path></g><g transform="matrix(.017,-0,0,-.017,82.083,12.575)"><use xlink:href="#x1D465"></use></g><g transform="matrix(.017,-0,0,-.017,95.359,12.575)"><path id="x2B" d="M535 230h-212v-233h-58v233h-213v50h213v210h58v-210h212v-50z"></path></g><g transform="matrix(.017,-0,0,-.017,109.111,12.575)"><use xlink:href="#x1D43B"></use></g><g transform="matrix(.017,-0,0,-.017,124.019,12.575)"><path id="x5D" d="M226 -163h-170v27q79 7 94 20t15 73v627q0 59 -15 72t-94 20v27h170v-866z"></path></g>

		
	
</svg> and improved by these some recent results by Ivić. The same link provides, also, an unconditional improvement. Then, some remarkable conditional implications on the 2<svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.13794pt;width:9.8000002px;" id="M5" height="13.55" version="1.1" viewBox="0 0 9.8000002 13.55" width="9.8000002">
	
		
			<g transform="matrix(.017,-0,0,-.017,.625,12.725)"><use xlink:href="#x1D458"></use></g>

		
	
</svg>th moments of Riemann zeta function on the critical line are derived. We also give general requirements on <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.34499pt;width:12.1625px;" id="M6" height="17.7875" version="1.1" viewBox="0 0 12.1625 17.7875" width="12.1625">
	
		
			<g transform="matrix(.017,-0,0,-.017,.625,12.725)"><use xlink:href="#x1D453"></use></g>

		
	
</svg> that allow our treatment for <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.34499pt;width:12.1625px;" id="M7" height="17.7875" version="1.1" viewBox="0 0 12.1625 17.7875" width="12.1625">
	
		
			<g transform="matrix(.017,-0,0,-.017,.625,12.725)"><use xlink:href="#x1D453"></use></g>

		
	
</svg> weighted Selberg integrals.

Giovanni Coppola

Maurizio Laporta

Generations of Correlation Averages

Given an arbitrary prime number <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.29482pt;width:8.4750004px;" id="M1" height="12.1125" version="1.1" viewBox="0 0 8.4750004 12.1125" width="8.4750004">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path id="x1D45E" d="M474 429q-19 -69 -47 -222l-65 -347q-8 -49 0 -60t49 -16l26 -3l-4 -26l-246 -12l4 25l17 3q37 6 50 20.5t23 60.5l67 321h-2q-65 -79 -150 -138q-69 -47 -104 -47q-28 0 -48.5 32t-20.5 81q0 78 35 148t90 117q67 57 161 77q23 5 58 5q43 0 90 -15zM387 387&#xA;q-30 16 -75 16q-58 0 -92 -27q-49 -39 -78.5 -112t-29.5 -136q0 -34 8.5 -52.5t21.5 -18.5q40 0 109.5 63.5t103.5 115.5q16 53 32 151z"></path></g>

		
	
</svg>, set <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.29482pt;width:61.5625px;" id="M2" height="19.862499" version="1.1" viewBox="0 0 61.5625 19.862499" width="61.5625">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,16.588)"><path id="x1D709" d="M425 682q0 -40 -49 -59q-36 -13 -110 -13q-31 -18 -52.5 -49.5t-21.5 -67.5q0 -34 17.5 -57.5t45.5 -28.5q63 15 97 15q18 0 18 -16q0 -13 -22 -36t-34 -23q-50 0 -93 8q-47 -20 -80.5 -66.5t-33.5 -96.5q0 -76 84 -103q33 -11 75 -21q101 -26 101 -93q0 -51 -39 -95.5&#xA;t-76 -68.5l-22 20q60 69 60 111q0 21 -14 34t-55 22q-21 4 -63 16q-68 19 -101.5 52.5t-33.5 96.5q0 125 157 206q-27 10 -46 36t-19 58q0 84 103 152q-43 3 -71.5 21t-28.5 45q0 37 65 59l17 -6q-9 -14 -9 -33q0 -21 17.5 -36t51.5 -19q82 53 147 53q18 0 18 -17z"></path></g><g transform="matrix(.017,-0,0,-.017,12.387,16.588)"><path id="x3D" d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z"></path></g><g transform="matrix(.017,-0,0,-.017,27.091,16.588)"><path id="x1D452" d="M391 364q0 -28 -20.5 -53.5t-50 -43.5t-70 -34t-73.5 -25t-68 -17v-29q0 -48 20 -81t65 -33q73 0 157 76l16 -23q-107 -113 -221 -113q-21 0 -40.5 6.5t-39 22t-31.5 47t-12 75.5q0 69 33.5 139.5t95.5 114.5q77 55 143 55q42 0 69 -24.5t27 -59.5zM313 350&#xA;q0 26 -14.5 40.5t-37.5 14.5q-32 0 -44 -7q-82 -46 -104 -175q200 49 200 127z"></path></g>

			<g transform="matrix(.012,-0,0,-.012,34.138,8.425)"><path id="x32" d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z"></path></g><g transform="matrix(.012,-0,0,-.012,39.849,8.425)"><path id="x1D70B" d="M574 449q-23 -58 -38 -79q-16 -4 -79 -4q-27 -100 -46 -219q-14 -87 7 -87t74 36l13 -27q-74 -81 -139 -81q-48 0 -48 62q0 22 8 59l60 258l-154 4q-21 -103 -54.5 -209t-64.5 -151q-38 -23 -83 -23l-7 15q46 36 94 152.5t64 216.5q-81 0 -138 -54l-18 23q22 27 39.5 43&#xA;t61.5 33.5t100 17.5q43 0 131.5 -2.5t129.5 -2.5q23 0 33.5 5t24.5 25z"></path></g><g transform="matrix(.012,-0,0,-.012,46.881,8.425)"><path id="x1D456" d="M244 607q0 -25 -15.5 -43t-37.5 -18q-19 0 -32 13t-13 35q0 21 15 41t39 20q20 0 32 -14t12 -34zM222 91q-29 -33 -79 -68t-75 -35q-13 0 -19 7.5t-6 31t10 65.5l62 253q5 26 -1 26q-21 0 -72 -43l-13 24q43 40 91 68t71 28q30 0 10 -78l-71 -274q-8 -30 3 -30&#xA;q16 0 76 48z"></path></g><g transform="matrix(.012,-0,0,-.012,50.129,8.425)"><path id="x2F" d="M368 703l-264 -866h-60l265 866h59z"></path></g><g transform="matrix(.012,-0,0,-.012,55.032,8.425)"><use xlink:href="#x1D45E"></use></g>

		
	
</svg>. We use a clever selection of the values of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.29482pt;width:15.125px;" id="M3" height="16.9625" version="1.1" viewBox="0 0 15.125 16.9625" width="15.125">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,13.688)"><use xlink:href="#x1D709"></use></g>

			<g transform="matrix(.012,-0,0,-.012,7.675,5.388)"><path id="x1D6FC" d="M545 106q-67 -118 -134 -118q-24 0 -40 37.5t-30 129.5h-2q-47 -72 -103 -119.5t-108 -47.5q-47 0 -76 45.5t-29 119.5q0 113 85 204t174 91q47 0 70 -33.5t43 -119.5h3q32 47 80 140l55 13l10 -9q-47 -80 -138 -201q17 -99 27.5 -136t22.5 -37q23 0 69 61zM333 204&#xA;q-14 98 -31 149.5t-50 51.5q-49 0 -94 -70t-45 -164q0 -55 15.5 -86t40.5 -31q70 0 164 150z"></path></g>

		
	
</svg>, <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-1.76814pt;width:82.137497px;" id="M4" height="12.9125" version="1.1" viewBox="0 0 82.137497 12.9125" width="82.137497">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,10.862)"><use xlink:href="#x1D6FC"></use></g><g transform="matrix(.017,-0,0,-.017,14.426,10.862)"><use xlink:href="#x3D"></use></g><g transform="matrix(.017,-0,0,-.017,29.13,10.862)"><path id="x31" d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z"></path></g><g transform="matrix(.017,-0,0,-.017,37.29,10.862)"><path id="x2C" d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z"></path></g><g transform="matrix(.017,-0,0,-.017,44.004,10.862)"><use xlink:href="#x32"></use></g><g transform="matrix(.017,-0,0,-.017,52.164,10.862)"><use xlink:href="#x2C"></use></g><g transform="matrix(.017,-0,0,-.017,58.861,10.862)"><path id="x2026" d="M160 -12q-24 0 -39.5 16t-15.5 42q0 24 16 40.5t40 16.5t40 -16.5t16 -40.5q0 -26 -16 -42t-41 -16zM485 -12q-24 0 -39.5 16t-15.5 42q0 24 16 40.5t40 16.5t40 -16.5t16 -40.5q0 -26 -16 -42t-41 -16zM806 -12q-24 0 -39.5 16t-15.5 42q0 24 16 40.5t40 16.5t40 -16.5&#xA;t16 -40.5q0 -26 -15.5 -42t-41.5 -16z"></path></g><g transform="matrix(.017,-0,0,-.017,78.189,10.862)"><use xlink:href="#x2C"></use></g>

		
	
</svg> in order to create normal numbers. We also use a famous result of André Weil concerning Dirichlet characters  to construct a family of normal numbers.

Jean Marie De Koninck

Imre K&#xe1;tai

Complex Roots of Unity and Normal Numbers

We prove two identities for Ramanujan&#x2019;s cubic continued fraction and
a continued fraction of Ramanujan, which are analogues of Ramanujan&#x2019;s
identities for the Rogers-Ramanujan continued fraction. We further derive
Eisenstein series identities associated with Ramanujan&#x2019;s cubic continued
fraction and Ramanujan&#x2019;s continued fraction of order six.

Chandrashekar Adiga

A. Vanitha

M. S. Surekha

Continued Fractions of Order Six and New Eisenstein Series Identities

Given two unitary automorphic cuspidal representations <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.16302pt;width:10.175px;" id="M2" height="8.1499996" version="1.1" viewBox="0 0 10.175 8.1499996" width="10.175">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,7.888)"><path id="x1D70B" d="M574 449q-23 -58 -38 -79q-16 -4 -79 -4q-27 -100 -46 -219q-14 -87 7 -87t74 36l13 -27q-74 -81 -139 -81q-48 0 -48 62q0 22 8 59l60 258l-154 4q-21 -103 -54.5 -209t-64.5 -151q-38 -23 -83 -23l-7 15q46 36 94 152.5t64 216.5q-81 0 -138 -54l-18 23q22 27 39.5 43&#xA;t61.5 33.5t100 17.5q43 0 131.5 -2.5t129.5 -2.5q23 0 33.5 5t24.5 25z"></path></g>

		
	
</svg> and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.16302pt;width:14.325px;" id="M3" height="17.9125" version="1.1" viewBox="0 0 14.325 17.9125" width="14.325">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,17.65)"><use xlink:href="#x1D70B"></use></g>

			<g transform="matrix(.012,-0,0,-.012,10.113,9.488)"><path id="x2032" d="M227 744l-123 -338l-31 15l73 368q12 3 41.5 -8t36.5 -20z"></path></g>

		
	
</svg> defined on <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.20526pt;width:67.175003px;" id="M4" height="16.4125" version="1.1" viewBox="0 0 67.175003 16.4125" width="67.175003">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,12.138)"><path id="x1D43A" d="M713 296l-5 -25q-47 -7 -59 -20t-23 -72l-15 -79q-9 -48 -3 -74q-15 -3 -55 -13t-63 -15t-59.5 -10t-70.5 -5q-149 0 -243 80t-94 220q0 169 127.5 276.5t336.5 107.5q91 0 206 -36l-10 -165l-29 -1q1 85 -47.5 126t-146.5 41q-153 0 -243.5 -97t-90.5 -242&#xA;q0 -122 68.5 -198.5t188.5 -76.5q121 0 139 75l20 86q13 58 -1.5 70.5t-99.5 20.5l5 26h267z"></path></g><g transform="matrix(.017,-0,0,-.017,12.37,12.138)"><use xlink:href="#x1D43F"></use></g>

			<g transform="matrix(.012,-0,0,-.012,22.075,16.212)"><path id="x1D45A" d="M766 88q-40 -45 -83.5 -72.5t-63.5 -27.5q-39 0 -16 103l49 224q15 68 -10 68q-36 0 -112.5 -78.5t-105.5 -133.5q-27 -101 -38 -165q-35 -4 -77 -18l-6 6q42 153 70 288q12 55 10 78t-19 23q-38 0 -114 -80.5t-101 -131.5q-24 -68 -41 -165q-36 -3 -76 -18l-8 6&#xA;q56 202 87 347q7 33 -3 33q-8 0 -31.5 -17.5t-41.5 -35.5l-12 28q42 45 84 72t63 27q44 0 10 -124l-20 -75h2q92 121 193 178q36 21 64 21q62 0 28 -159l-8 -37h2q50 67 103.5 112t94.5 65q39 19 62 19q60 0 23 -156l-45 -189q-9 -39 2 -39q17 0 71 49z"></path></g>

			<g transform="matrix(.017,-0,0,-.017,32.088,12.138)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z"></path></g><g transform="matrix(.017,-0,0,-.017,37.969,12.138)"><path id="x1D538" d="M751 0h-370v28q47 6 58 16t1 39l-55 154h-166l-44 -128q-16 -48 -5 -61.5t68 -19.5v-28h-218v28q50 7 68.5 23.5t43.5 80.5l205 518l111 15l193 -535q23 -63 41.5 -80t68.5 -22v-28zM371 280l-65 189h-2l-70 -189h137zM614 60q0 20 -18 70l-180 499q-55 -8 -67 -37&#xA;l167 -462q11 -31 18 -46.5t22.5 -32.5t34.5 -17q23 0 23 26z"></path></g>

			<g transform="matrix(.012,-0,0,-.012,53.088,16.212)"><path id="x1D438" d="M609 650l-19 -162l-30 -2q2 69 -14 94q-9 18 -28.5 26t-74.5 8h-88q-30 0 -37 -6.5t-12 -36.5l-41 -212h103q48 0 69 5.5t32 19t26 50.5h29l-40 -198h-30q2 58 -11.5 70.5t-85.5 12.5h-99l-30 -167q-17 -87 3 -101q18 -15 111 -15q59 0 89 7t54 27q8 7 15.5 16t16 21.5&#xA;l14 20.5t15 23.5t13.5 21.5l29 -10q-52 -129 -71 -163h-500l6 28q66 4 83 17.5t28 73.5l77 409q11 61 -0.5 75.5t-78.5 18.5l10 28h467z"></path></g>

			<g transform="matrix(.017,-0,0,-.017,61.225,12.138)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z"></path></g>

		
	
</svg> and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.94505pt;width:70.824997px;" id="M5" height="19.487499" version="1.1" viewBox="0 0 70.824997 19.487499" width="70.824997">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,12.137)"><use xlink:href="#x1D43A"></use></g><g transform="matrix(.017,-0,0,-.017,12.37,12.137)"><use xlink:href="#x1D43F"></use></g>

			<g transform="matrix(.012,-0,0,-.012,22.075,19.287)"><use xlink:href="#x1D45A"></use></g>

			<g transform="matrix(.008,-0,0,-.008,31.463,12.762)"><use xlink:href="#x2032"></use></g>

			<g transform="matrix(.017,-0,0,-.017,35.862,12.137)"><use xlink:href="#x28"></use></g><g transform="matrix(.017,-0,0,-.017,41.744,12.137)"><use xlink:href="#x1D538"></use></g>

			<g transform="matrix(.012,-0,0,-.012,56.85,16.212)"><path id="x1D439" d="M599 650l-18 -160l-30 -2q-2 70 -11 94q-5 16 -26 24t-75 8h-94q-27 0 -33.5 -6.5t-12.5 -32.5l-43 -226h108q52 0 72.5 5t31 18t28.5 52h28l-40 -200h-28q-1 42 -6 56.5t-24.5 21t-70.5 6.5h-108l-32 -177q-13 -65 1 -80.5t88 -22.5l-8 -28h-279l6 28q61 5 77.5 20.5&#xA;t29.5 81.5l72 387l7.5 40.5t1 26.5t-4 18t-15.5 10t-24 6t-37 4l8 28h461z"></path></g>

			<g transform="matrix(.017,-0,0,-.017,64.875,12.137)"><use xlink:href="#x29"></use></g>

		
	
</svg>, respectively, with <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.0pt;width:10.8625px;" id="M6" height="11.175" version="1.1" viewBox="0 0 10.8625 11.175" width="10.8625">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D438"></use></g>

		
	
</svg> and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.0pt;width:10.7px;" id="M7" height="11.175" version="1.1" viewBox="0 0 10.7 11.175" width="10.7">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D439"></use></g>

		
	
</svg> being Galois extensions of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.8151pt;width:14.0125px;" id="M8" height="14.15" version="1.1" viewBox="0 0 14.0125 14.15" width="14.0125">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.363)"><path id="x211A" d="M794 -138l-9 -21q-74 -1 -142 15q-79 19 -254 130q-139 7 -231 87q-52 44 -83 107t-31 135q0 145 103 247.5t274 102.5q145 0 249.5 -93.5t104.5 -240.5q0 -126 -73 -212.5t-179 -116.5q137 -85 271 -140zM673 302q0 152 -78 239.5t-201 87.5q-50 0 -84.5 -74t-34.5 -209&#xA;q0 -265 118 -317l66 18l31 -19q76 19 129.5 90t53.5 184zM298 54q-66 94 -66 292q0 178 50 256q-62 -32 -99 -98t-37 -162q0 -98 41 -174t111 -114z"></path></g>

		
	
</svg>, we consider two generalized Rankin-Selberg <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.0pt;width:9.8249998px;" id="M9" height="11.175" version="1.1" viewBox="0 0 9.8249998 11.175" width="9.8249998">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D43F"></use></g>

		
	
</svg>-functions obtained by forcefully factoring <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.21957pt;width:44.724998px;" id="M10" height="15.0875" version="1.1" viewBox="0 0 44.724998 15.0875" width="44.724998">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x1D43F"></use></g><g transform="matrix(.017,-0,0,-.017,9.769,12.138)"><use xlink:href="#x28"></use></g><g transform="matrix(.017,-0,0,-.017,15.65,12.138)"><path id="x1D460" d="M352 391q0 -31 -27 -44q-14 -7 -24 6q-39 48 -84 48q-23 0 -39.5 -15t-16.5 -40q0 -43 73 -90q49 -32 70 -58t21 -57q0 -58 -62 -105.5t-129 -47.5q-40 0 -75.5 25t-35.5 52q0 28 32 46q7 4 15 3t11 -6q19 -31 48.5 -50.5t54.5 -19.5q34 0 54 19.5t20 42.5q0 43 -65 81&#xA;q-97 56 -97 123q0 50 51 96q19 17 58 32.5t62 15.5q37 0 61 -18t24 -39z"></path></g><g transform="matrix(.017,-0,0,-.017,22.025,12.138)"><path id="x2C" d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z"></path></g><g transform="matrix(.017,-0,0,-.017,28.722,12.138)"><use xlink:href="#x1D70B"></use></g><g transform="matrix(.017,-0,0,-.017,38.769,12.138)"><use xlink:href="#x29"></use></g>

		
	
</svg>  and  <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.21957pt;width:48.487499px;" id="M11" height="23.612499" version="1.1" viewBox="0 0 48.487499 23.612499" width="48.487499">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,20.662)"><use xlink:href="#x1D43F"></use></g><g transform="matrix(.017,-0,0,-.017,9.769,20.662)"><use xlink:href="#x28"></use></g><g transform="matrix(.017,-0,0,-.017,15.65,20.662)"><use xlink:href="#x1D460"></use></g><g transform="matrix(.017,-0,0,-.017,22.025,20.662)"><use xlink:href="#x2C"></use></g><g transform="matrix(.017,-0,0,-.017,28.722,20.662)"><use xlink:href="#x1D70B"></use></g>

			<g transform="matrix(.008,-0,0,-.008,38.775,6.8)"><use xlink:href="#x2032"></use></g>

			<g transform="matrix(.017,-0,0,-.017,42.55,20.662)"><use xlink:href="#x29"></use></g>

		
	
</svg>. We prove the absolute convergence of these <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.0pt;width:9.8249998px;" id="M12" height="11.175" version="1.1" viewBox="0 0 9.8249998 11.175" width="9.8249998">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D43F"></use></g>

		
	
</svg>-functions for <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.21957pt;width:63.674999px;" id="M13" height="15.0875" version="1.1" viewBox="0 0 63.674999 15.0875" width="63.674999">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,12.138)"><path id="x52" d="M631 18l-4 -26q-24 0 -35 1q-57 3 -92.5 23t-69.5 68q-21 30 -101 160q-13 22 -30.5 31t-54.5 9h-32v-159q0 -63 14 -77.5t74 -19.5v-28h-262v28q62 5 76 19.5t14 77.5v402q0 62 -13 76t-73 19v28h248q109 0 159 -34q67 -42 67 -131q0 -115 -127 -172q27 -52 89 -144&#xA;q49 -71 81 -104q31 -36 72 -47zM212 316h48q73 0 108 29q55 42 55 124q0 76 -42 112t-104 36q-41 0 -54 -10q-11 -7 -11 -44v-247z"></path></g><g transform="matrix(.017,-0,0,-.017,10.653,12.138)"><path id="x65" d="M380 106l15 -22q-42 -52 -87 -74t-76 -22q-89 0 -142 62t-53 151q0 103 66 175q67 73 146 73h1q61 0 105.5 -46.5t44.5 -105.5q0 -16 -4.5 -22.5t-18.5 -9.5q-54 -10 -262 -17q1 -95 46 -143.5t104 -48.5q61 0 115 50zM225 412q-35 0 -64.5 -32.5t-40.5 -90.5q83 0 170 3&#xA;q14 0 19 4.5t5 15.5q1 41 -23.5 70.5t-65.5 29.5z"></path></g>

			<g transform="matrix(.017,-0,0,-.017,17.9,12.138)"><use xlink:href="#x28"></use></g><g transform="matrix(.017,-0,0,-.017,23.782,12.138)"><use xlink:href="#x1D460"></use></g><g transform="matrix(.017,-0,0,-.017,30.156,12.138)"><use xlink:href="#x29"></use></g><g transform="matrix(.017,-0,0,-.017,40.746,12.138)"><path id="x3E" d="M512 230l-437 -233v58l378 199v2l-378 200v58l437 -233v-51z"></path></g><g transform="matrix(.017,-0,0,-.017,55.45,12.138)"><path id="x31" d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z"></path></g>

		
	
</svg>. The main difficulty  in our case is that the two extension fields may be completely unrelated, so  we are forced to work either “downstairs” in some intermediate extension between <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.38873pt;width:38.974998px;" id="M14" height="11.175" version="1.1" viewBox="0 0 38.974998 11.175" width="38.974998">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D438"></use></g><g transform="matrix(.017,-0,0,-.017,14.579,11.113)"><path id="x2229" d="M535 0h-58v354q0 68 -54.5 113.5t-128.5 45.5t-128.5 -45.5t-54.5 -113.5v-354h-58v354q0 90 71.5 149.5t169.5 59.5t169.5 -59.5t71.5 -149.5v-354z"></path></g><g transform="matrix(.017,-0,0,-.017,28.331,11.113)"><use xlink:href="#x1D439"></use></g>

		
	
</svg> and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.8151pt;width:14.0125px;" id="M15" height="14.15" version="1.1" viewBox="0 0 14.0125 14.15" width="14.0125">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.363)"><use xlink:href="#x211A"></use></g>

		
	
</svg>, or “upstairs” in some extension field containing the composite extension <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.0pt;width:21.4375px;" id="M16" height="11.175" version="1.1" viewBox="0 0 21.4375 11.175" width="21.4375">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D438"></use></g><g transform="matrix(.017,-0,0,-.017,10.806,11.113)"><use xlink:href="#x1D439"></use></g>

		
	
</svg>. We close by investigating some special cases when analytic continuation is possible and show that when the degrees of the extension fields <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.0pt;width:10.8625px;" id="M17" height="11.175" version="1.1" viewBox="0 0 10.8625 11.175" width="10.8625">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D438"></use></g>

		
	
</svg> and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.0pt;width:10.7px;" id="M18" height="11.175" version="1.1" viewBox="0 0 10.7 11.175" width="10.7">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D439"></use></g>

		
	
</svg> are relatively prime, the two different definitions give the same generating function.

Tim Gillespie

On a Rankin-Selberg <svg style="vertical-align:-0.0pt" height="11.175" version="1.1" viewBox="0 0 9.8249998 11.175" width="9.8249998" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x1D43F" d="M559 163q-23 -66 -68 -163h-474l6 26q62 4 79.5 19.5t28.5 75.5l78 409q7 35 8.5 49t-8 25t-24 13t-51.5 5l5 28h266l-6 -28q-65 -5 -79.5 -18t-25.5 -74l-76 -406q-10 -57 14 -75q12 -13 96 -13q93 0 126 29q41 40 76 109z" /></g></svg>-Function over Different Fields

For <i>k</i>, being a fixed integer &#x2265;2, a positive integer <i>n</i> is called <i>k</i>-free number if <i>n</i> is not divisible by the <i>k</i>th power of any integer &#x3e;1. In this paper, we studied the distribution of <i>r</i>-tuples of <i>k</i>-free numbers and derived an asymptotic formula.

Ting Zhang

Huaning Liu

On the Distribution of <svg style="vertical-align:-0.0pt" height="7.9499998" version="1.1" viewBox="0 0 7.3000002 7.9499998" width="7.3000002" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path id="x1D45F" d="M393 379q-9 -16 -28 -29q-15 -10 -23 -2q-19 19 -36 19q-21 0 -52 -38q-57 -72 -82 -126l-40 -197q-23 -3 -75 -18l-7 7q49 196 74 335q7 43 -2 43q-7 0 -30 -14.5t-47 -37.5l-16 23q37 42 82 73t67 31q41 0 15 -113l-11 -50h4q41 71 85 117t77 46q29 0 45 -26&#xA;q13 -21 0 -43z" /></g></svg>-Tuples of <svg style="vertical-align:-0.13794pt" height="12.4375" version="1.1" viewBox="0 0 8.6750002 12.4375" width="8.6750002" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path id="x1D458" d="M480 416q0 -21 -18 -41q-9 -11 -17 -7q-20 9 -42 9q-62 0 -140 -78q23 -69 88 -192q17 -31 27 -42t20 -11q16 0 62 46l17 -20q-64 -92 -119 -92q-35 0 -70 66q-41 73 -84 187q-36 -30 -62 -61q-27 -115 -35 -172q-41 -8 -78 -20l-6 6l140 612q7 28 0.5 34t-37.5 7l-34 1&#xA;l5 26q38 4 74 13.5t57 17t25 7.5q12 0 4 -32l-104 -443h2q35 38 97 93q39 35 65.5 56t62 41.5t58.5 20.5q19 0 30.5 -10t11.5 -22z" /></g></svg>-Free Numbers

A sequence of rational numbers as a generalization of the sequence of Bernoulli numbers is introduced. Sums of products involving the terms of this generalized sequence are then obtained using an application of Fa&#xe0; di Bruno's formula. These sums of products are analogous to the higher order Bernoulli numbers and are used to develop the closed form expressions for the sums of products involving the power sums <svg     style="vertical-align:-5.15417pt;width:271.35001px;" id="M2" height="23.174999" version="1.1" viewBox="0 0 271.35001 23.174999" width="271.35001"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,16.7)"><path id="x3A8" d="M699 659l10 -27q-31 -17 -42 -39t-21 -74l-14 -69q-10 -53 -26 -82t-47 -53q-58 -45 -156 -48v-145q0 -58 14.5 -74t71.5 -20v-28h-256v28q57 4 71 20t14 74v145q-104 3 -163 49q-29 23 -45 54t-25 82l-12 72q-11 62 -20 77q-11 19 -38 31l10 27q74 -11 106 -50&#xA;q20 -24 30 -78l11 -60q16 -92 48.5 -128t97.5 -41v227q0 57 -13.5 73t-66.5 20v28h245v-28q-53 -4 -66.5 -20t-13.5 -73v-227q75 4 103.5 37.5t45.5 132.5l10 58q8 51 33 80q35 38 104 49z" /></g>

			<g transform="matrix(.012,-0,0,-.012,12.387,20.775)"><path id="x1D458" d="M480 416q0 -21 -18 -41q-9 -11 -17 -7q-20 9 -42 9q-62 0 -140 -78q23 -69 88 -192q17 -31 27 -42t20 -11q16 0 62 46l17 -20q-64 -92 -119 -92q-35 0 -70 66q-41 73 -84 187q-36 -30 -62 -61q-27 -115 -35 -172q-41 -8 -78 -20l-6 6l140 612q7 28 0.5 34t-37.5 7l-34 1&#xA;l5 26q38 4 74 13.5t57 17t25 7.5q12 0 4 -32l-104 -443h2q35 38 97 93q39 35 65.5 56t62 41.5t58.5 20.5q19 0 30.5 -10t11.5 -22z" /></g>

			<g transform="matrix(.017,-0,0,-.017,19,16.7)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,24.882,16.7)"><path id="x1D465" d="M536 404q0 -17 -13.5 -31.5t-26.5 -14.5q-8 0 -15 10q-11 14 -25 14q-22 0 -67 -50q-47 -52 -68 -82l37 -102q31 -88 55 -88t78 59l16 -23q-32 -48 -68.5 -78t-65.5 -30q-19 0 -37.5 20t-29.5 53l-41 116q-72 -106 -114.5 -147.5t-79.5 -41.5q-21 0 -34.5 14t-13.5 37&#xA;q0 16 13.5 31.5t28.5 15.5q12 0 17 -11q5 -10 25 -10q22 0 57.5 36t89.5 111l-40 108q-22 58 -36 58q-21 0 -67 -57l-19 20q81 107 125 107q17 0 30 -22t39 -88l22 -55q68 92 108.5 128.5t74.5 36.5q20 0 32.5 -14t12.5 -30z" /></g><g transform="matrix(.017,-0,0,-.017,34.384,16.7)"><path id="x2C" d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" /></g><g transform="matrix(.017,-0,0,-.017,41.081,16.7)"><use xlink:href="#x1D45B"/></g><g transform="matrix(.017,-0,0,-.017,49.615,16.7)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,60.222,16.7)"><path id="x2236" d="M223 355q0 -25 -15.5 -41.5t-39.5 -16.5t-40 16.5t-16 41.5t16 41t40 16t39.5 -16t15.5 -41zM223 46q0 -26 -16 -42t-40 -16t-39.5 16t-15.5 42q0 24 16 40.5t39 16.5q24 0 40 -16.5t16 -40.5z" /></g><g transform="matrix(.017,-0,0,-.017,65.9,16.7)"><path id="x3D" d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" /></g><g transform="matrix(.017,-0,0,-.017,80.586,16.7)"><path id="x2211" d="M697 -46l-42 -182h-602v25l325 457l-308 460v25h585l12 -154l-28 -5q-17 63 -39 88q-12 15 -36.5 20.5t-80.5 5.5h-277l268 -396l-311 -440h382q51 0 75.5 19.5t47.5 83.5z" /></g>

			<g transform="matrix(.012,-0,0,-.012,93.35,21.175)"><path id="x1D451" d="M530 686q-79 -330 -126 -595q-5 -24 4 -24q22 0 77 53l16 -24q-42 -49 -88 -78.5t-72 -29.5q-37 0 -19 83l21 99h-2q-62 -83 -153 -141q-65 -41 -97 -41q-28 0 -48 31.5t-20 91.5q0 71 32.5 144t88.5 118q36 30 94 52.5t94 22.5q38 0 68 -13l29 158q7 40 2 47.5t-38 7.5&#xA;h-35l1 26q39 4 76 13.5t58.5 17t27.5 7.5q16 0 9 -26zM387 375q-11 11 -38.5 20t-51.5 9q-45 0 -79 -23q-46 -32 -77.5 -107.5t-31.5 -143.5q0 -40 9.5 -58.5t23.5 -18.5q40 0 113.5 70t99.5 118z" /></g><g transform="matrix(.012,-0,0,-.012,99.954,21.175)"><path id="x7C" d="M162 -163h-61v866h61v-866z" /></g><g transform="matrix(.012,-0,0,-.012,103.083,21.175)"><use xlink:href="#x1D45B"/></g>

			<g transform="matrix(.017,-0,0,-.017,112.512,16.7)"><path id="x1D707" d="M538 96q-38 -45 -84 -76.5t-71 -31.5q-44 0 -15 117q12 49 18 89h-2q-99 -128 -163 -178q-36 -28 -70 -28q-26 0 -44 38h-2q-16 -69 -16 -129q0 -43 11 -75.5t24 -42.5l1 -6q-7 -12 -24.5 -23t-37.5 -11q-40 0 -40 76q0 52 35 202l94 407l78 24l5 -5q-32 -122 -72 -310&#xA;q-7 -33 0 -55t25 -22q36 0 105.5 75t107.5 145l32 146l75 26h10q-50 -197 -78 -347q-7 -38 6 -38q7 0 31 17t47 39z" /></g><g transform="matrix(.017,-0,0,-.017,121.947,16.7)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,127.828,16.7)"><use xlink:href="#x1D451"/></g><g transform="matrix(.017,-0,0,-.017,137.263,16.7)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,143.144,16.7)"><use xlink:href="#x1D451"/></g>

			<g transform="matrix(.012,-0,0,-.012,152.575,8.538)"><use xlink:href="#x1D458"/></g>

			<g transform="matrix(.017,-0,0,-.017,159.188,16.7)"><path id="x1D446" d="M457 488l-30 -3q-17 148 -131 148q-53 0 -84.5 -34.5t-31.5 -82.5q0 -42 25.5 -72t74.5 -62l33 -22q63 -42 95 -85t32 -102q0 -84 -67 -137t-163 -53q-58 0 -113 22t-70 43l-4 152l27 4q4 -32 15 -62.5t31 -59.5t53.5 -47t76.5 -18q56 0 92 35t36 96q0 39 -25 70t-78 68&#xA;l-31 22q-32 23 -53.5 41.5t-45 57t-23.5 77.5q0 82 58 132.5t156 50.5q46 0 101 -17l18.5 -6t17 -6t8.5 -3q-4 -55 0 -147z" /></g>

			<g transform="matrix(.012,-0,0,-.012,167.35,20.775)"><use xlink:href="#x1D458"/></g>

			<g transform="matrix(.017,-0,0,-.017,173.95,16.7)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,179.832,16.7)"><use xlink:href="#x1D465"/></g><g transform="matrix(.017,-0,0,-.017,189.334,16.7)"><path id="x2F" d="M368 703l-264 -866h-60l265 866h59z" /></g><g transform="matrix(.017,-0,0,-.017,196.337,16.7)"><use xlink:href="#x1D451"/></g><g transform="matrix(.017,-0,0,-.017,205.772,16.7)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,211.653,16.7)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,224.028,16.7)"><use xlink:href="#x1D45B"/></g><g transform="matrix(.017,-0,0,-.017,237.287,16.7)"><path id="x2208" d="M448 1h-83q-118 0 -201.5 74.5t-83.5 179.5t83.5 179.5t201.5 74.5h83v-50h-84q-87 0 -150.5 -51.5t-73.5 -127.5h308v-50h-308q10 -76 73.5 -127.5t150.5 -51.5h84v-50z" /></g><g transform="matrix(.017,-0,0,-.017,250.971,16.7)"><path id="x2124" d="M734 170q-25 -130 -36 -170h-661l-8 15l420 599h-173q-77 0 -102 -7t-39 -29q-19 -29 -35 -87l-29 1l12 183h24q9 -16 19 -20.5t34 -4.5h543l7 -14l-416 -598l166 -2q94 0 131.5 7.5t58.5 29.5q28 29 55 103zM639 614h-89l-397 -567l88 -7z" /></g>

			<g transform="matrix(.012,-0,0,-.012,263.688,8.538)"><path id="x2B" d="M535 230h-212v-233h-58v233h-213v50h213v210h58v-210h212v-50z" /></g>

		
	
</svg> which are defined via the M&#xf6;bius function <i  >&#x3bc;</i> and the usual power sum <svg     style="vertical-align:-3.3907pt;width:36.162498px;" id="M3" height="16.4125" version="1.1" viewBox="0 0 36.162498 16.4125" width="36.162498"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x1D446"/></g>

			<g transform="matrix(.012,-0,0,-.012,8.225,16.212)"><use xlink:href="#x1D458"/></g>

			<g transform="matrix(.017,-0,0,-.017,14.825,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,20.707,12.138)"><use xlink:href="#x1D465"/></g><g transform="matrix(.017,-0,0,-.017,30.209,12.138)"><use xlink:href="#x29"/></g>

		
	
</svg> of a real or complex variable <svg     style="vertical-align:-0.1638pt;width:9.625px;" id="M4" height="7.9499998" version="1.1" viewBox="0 0 9.625 7.9499998" width="9.625"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><use xlink:href="#x1D465"/></g>

		
	
</svg>. The power sum <svg     style="vertical-align:-3.3907pt;width:36.162498px;" id="M5" height="16.4125" version="1.1" viewBox="0 0 36.162498 16.4125" width="36.162498"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x1D446"/></g>

			<g transform="matrix(.012,-0,0,-.012,8.225,16.212)"><use xlink:href="#x1D458"/></g>

			<g transform="matrix(.017,-0,0,-.017,14.825,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,20.707,12.138)"><use xlink:href="#x1D465"/></g><g transform="matrix(.017,-0,0,-.017,30.209,12.138)"><use xlink:href="#x29"/></g>

		
	
</svg> is expressible in terms of the well-known Bernoulli polynomials by <svg     style="vertical-align:-3.3907pt;width:270.02499px;" id="M6" height="16.450001" version="1.1" viewBox="0 0 270.02499 16.450001" width="270.02499"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x1D446"/></g>

			<g transform="matrix(.012,-0,0,-.012,8.225,16.25)"><use xlink:href="#x1D458"/></g>

			<g transform="matrix(.017,-0,0,-.017,14.825,12.162)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,20.707,12.162)"><use xlink:href="#x1D465"/></g><g transform="matrix(.017,-0,0,-.017,30.209,12.162)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,40.816,12.162)"><use xlink:href="#x2236"/></g><g transform="matrix(.017,-0,0,-.017,46.494,12.162)"><use xlink:href="#x3D"/></g><g transform="matrix(.017,-0,0,-.017,61.181,12.162)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,67.062,12.162)"><path id="x1D435" d="M594 511q0 -122 -171 -157l1 -2q158 -33 158 -159q0 -52 -34.5 -95t-90.5 -65q-76 -33 -217 -33h-223l8 28q63 5 79.5 19t26.5 72l83 426q9 48 -2.5 60t-77.5 17l6 28h259q195 0 195 -139zM499 509q0 59 -37 83t-91 24q-36 0 -51 -9q-17 -9 -22 -44l-35 -195h62&#xA;q82 0 128 37t46 104zM481 199q0 71 -48 102.5t-121 31.5h-56l-37 -201q-11 -58 7.5 -77t80.5 -19q76 0 125 44.5t49 118.5z" /></g>

			<g transform="matrix(.012,-0,0,-.012,77.562,16.25)"><use xlink:href="#x1D458"/></g><g transform="matrix(.012,-0,0,-.012,83.548,16.25)"><use xlink:href="#x2B"/></g><g transform="matrix(.012,-0,0,-.012,90.532,16.25)"><path id="x31" d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" /></g>

			<g transform="matrix(.017,-0,0,-.017,96.862,12.162)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,102.744,12.162)"><use xlink:href="#x1D465"/></g><g transform="matrix(.017,-0,0,-.017,116.02,12.162)"><use xlink:href="#x2B"/></g><g transform="matrix(.017,-0,0,-.017,129.772,12.162)"><use xlink:href="#x31"/></g><g transform="matrix(.017,-0,0,-.017,137.931,12.162)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,147.604,12.162)"><path id="x2212" d="M535 230h-483v50h483v-50z" /></g><g transform="matrix(.017,-0,0,-.017,161.356,12.162)"><use xlink:href="#x1D435"/></g>

			<g transform="matrix(.012,-0,0,-.012,171.85,16.25)"><use xlink:href="#x1D458"/></g><g transform="matrix(.012,-0,0,-.012,177.835,16.25)"><use xlink:href="#x2B"/></g><g transform="matrix(.012,-0,0,-.012,184.82,16.25)"><use xlink:href="#x31"/></g>

			<g transform="matrix(.017,-0,0,-.017,191.15,12.162)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,197.032,12.162)"><use xlink:href="#x31"/></g><g transform="matrix(.017,-0,0,-.017,205.191,12.162)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,211.073,12.162)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,216.954,12.162)"><use xlink:href="#x2F"/></g><g transform="matrix(.017,-0,0,-.017,223.958,12.162)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,229.839,12.162)"><use xlink:href="#x1D458"/></g><g transform="matrix(.017,-0,0,-.017,242.163,12.162)"><use xlink:href="#x2B"/></g><g transform="matrix(.017,-0,0,-.017,255.915,12.162)"><use xlink:href="#x31"/></g><g transform="matrix(.017,-0,0,-.017,264.075,12.162)"><use xlink:href="#x29"/></g>

		
	
</svg>.

Jitender Singh

Sums of Products Involving Power Sums of <svg style="vertical-align:-3.56265pt" height="16.625" version="1.1" viewBox="0 0 30.3125 16.625" width="30.3125" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.138)"><path id="x1D711" d="M559 271q0 -109 -83 -187t-189 -89l-36 -213q-16 -30 -86 -43l-7 9q10 36 53 242q-91 11 -139.5 63.5t-48.5 132.5q0 186 242 271l11 -28q-76 -34 -119.5 -91.5t-43.5 -136.5q0 -66 29 -108.5t77 -55.5q39 205 56 260q21 71 65 111t93 40q50 0 88 -45.5t38 -131.5z&#xA;M484 273q0 65 -26.5 99t-52.5 34q-37 0 -53 -75l-59 -298q80 5 135.5 74.5t55.5 165.5z" /></g><g transform="matrix(.017,-0,0,-.017,9.956,12.138)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,15.837,12.138)"><path id="x1D45B" d="M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2&#xA;q42 56 97 103.5t96 71.5q46 26 74 26q9 0 16 -2.5t14 -11.5t9.5 -24.5t-1 -44t-13.5 -68.5q-30 -117 -47 -200q-4 -19 -3.5 -25t6.5 -6q21 0 70 48z" /></g><g transform="matrix(.017,-0,0,-.017,24.371,12.138)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g></svg> Integers

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Journal of Numbers