Generations of Correlation Averages

Abstract

We give a general link between weighted Selberg integrals of any arithmetic function and averages of 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 , gauged by the Cesaro weight in the short interval and improved by these some recent results by Ivić. The same link provides, also, an unconditional improvement. Then, some remarkable conditional implications on the 2th moments of Riemann zeta function on the critical line are derived. We also give general requirements on that allow our treatment for weighted Selberg integrals.

1. Introduction and Statement of the Results

In the milestone paper [1], Selberg introduced an important tool in the study of the distribution of prime numbers in short intervals with as , that is, the integral where is the von Mangoldt function: if for some prime and ; otherwise, . Thus, is a weighted characteristic function of prime numbers generated by (hereafter, is the Riemann zeta function). The Selberg integral, being a quadratic mean, pertains precisely to the study of the distribution of primes in almost all short intervals , that is, with at most exceptional integers as . Here, we define the Selberg integral of any as where means and is the expected mean value of in short intervals (abbreviated as s.i. mean value). In order to avoid trivialities, we assume that goes to infinity with . In view of nontrivial bounds, the discrete version may be considered close enough to the original continuous integral; so, we are legitimate to use the same symbol for both. Similar considerations hold for the we work with, mostly the -divisor function for , where is the number of ways to write as a product of positive integers (see [2] and compare Section 4). Let us denote the Selberg integral of as with the s.i. mean value of given by where is the residual polynomial of degree such that .

The first author [2] has proved the lower bound for , where hereafter. We formulate, for the so-called modified Selberg integral of , where is the same s.i. mean value as in , the following conjecture. In Section 7, Propositions 16 and 18 justify the admissibility of such a choice for the s.i. mean value in modified Selberg integrals, even in the generalization for any divisor function (everything comes from Proposition 14; see Section 7).

Hereafter, we abbreviate whenever , for all .

Conjecture CL. If , then .

A first consequence of our conjecture is the following result.

Theorem 1. If Conjecture CL holds, then .

In Section 5, we deduce this from a general link (Lemma 10), between the Selberg integral and the corresponding modified Selberg integral, in the special case of an essentially bounded (see Section 4) real function such that vanishes identically.

Moreover, for the width of , that is, (more generally, is defined by for ), we will implicitly assume that any inequality (resp., ) means that fixed and absolute constant such that (resp., ). In particular, works as .

Noteworthy, Theorem 1 improves Ivić’s results [3] for , both in the bound and in the low range, while Ivić’s bound is nontrivial for , that is, for width ; ours is nontrivial for , that is, for (we think to be able to get a wider range). Remarkably, under Conjecture CL, the first author has recently derived the even better (see http://arxiv.org/).

As one may expect, our study applies to every divisor function , though the conjectured estimate for becomes less and less meaningful as grows: this is due to known bounds for in (compare Section 4)

We generalize Conjecture CL for the modified Selberg integral of with as follows.

General Conjecture CL. Fix and assume that in . If , then Consequently, setting , for every width , there exists an such that

From the last inequality, our second main result follows, proved together with Theorem 1 in Section 5.

Theorem 2. If General Conjecture CL holds, then for .

As a consequence, one would get an improvement in the low range of with respect to the results of Ivić for the mean-square of in short intervals [3]. In fact, Theorem 1 in [3] holds for with defined in terms of Carlson’s abscissa (see Section 6). In particular, it holds for , , and , whereas from bounds [4], we get instead with , , .

More generally, is a kind of smoothing (see Section 2) for the Selberg integral , both in the arithmetic and in the harmonic analysis aspects. The arithmetic one goes back to Ernesto Cesaro: This arithmetic mean of the inner sum in justifies the choice of the same in . The analytic aspects of such a smoothing process will be clear after the introduction of the correlation in Section 3, where it is shown that Selberg integrals are averages of in short intervals . In fact (see Section 3 title), a kind of elementary dispersion method proves this link.

The next corollaries, applying such an intimate link, improve recent results [5, 6] on an additive divisor problem for (first two conditionally, third unconditionally). They concern the deviation of ; that is, where is the correlation (see Section 3) of and is the so-called logarithmic polynomial of . The following consequence of Theorem 1 is proved in Section 6.

Corollary 3. Let be such that for a fixed . If Conjecture CL is true, then

In a completely analogous way, from Theorem 2, we deduce the following result on for .

Corollary 4. Under the same hypotheses of Theorem 2, one has, in the same ranges and for the same ,

The aforementioned link to [5, 6] results relies on the identity (see Section 4) where one has to be acquainted that one’s notations and are not consistent with those in [5, 6]. In particular, from (3.8) in [6], the main term in the [5, 6] formulas for sums of correlations is Since it is easily seen that, for every , one has then our is comparable with the average for correlations, estimated in [5] when and in [6] for every . Thus, Corollary 3 and the bound (see [4]) imply for , improving [5] in the low range of short intervals (while [6] bounds are better for if ); in fact, their remainders (worse than ours when ) are nontrivial only for . Similarly, since , Corollary 4 improves [6] bounds for the low range (but not for ).

Our Lemma 12 (see Remark 13, Section 5) and Ivić’s Theorem 1 [3] prove the following unconditional result.

Corollary 5. For a fixed integer , let be defined as in [6] and for a fixed , where is Carlson’s abscissa, let as . Then where might depend also on given in [3].

The novelty of our approach is that, though conditionally, it improves the analogous achievements obtained via the classical moments of the Riemann zeta function on the critical line (a major example being the known upper bounds for Carlson’s abscissa in Ivić [3] results). We think that our conjectures might be approachable by elementary arguments, starting with the so-called -folding method (Section 7, Proposition 14).

On the other hand, estimates for have nontrivial consequences on the th moments of the Riemann function on the critical line (see [7]): At present, we content ourselves with having found an alternative way to pursue possible improvements on the th moments of , for at least relatively low values of . Indeed, in Section 8, we take a glance at the effect of hypothetical estimates for Selberg integrals on the th moments, through Theorem 1.1 of [7], whereas the first author (see http://arxiv.org/) provides the best known estimate for the th moment (under CL).

In Section 8, we prove next link, between bounds and bounds (compare [8] conditional bounds).

Theorem 6. Let be fixed. If holds for and for some constants , then (beside the dependence, implicit constant may depend on ).

This result encourages us to seek nontrivial bounds for the Selberg integrals, in the future.

As already said, we will give some Propositions, besides above results; we will not prove them here. We state Proposition 9 (about proximity of two different forms of expected value in short intervals) in Section 4, while Propositions 14, 16, and 18 (about, resp., the -folding, the nontrivial arithmetic bound of weighted Selberg integrals, and the mean-square proximity of expected value in s.i., for all ) are stated in Section 7.

We will provide the necessary Lemmas for the proof of our results in due course, in next sections. In particular, we prove Lemma 7 (our elementary version of Linnik’s method) in Section 3 and Lemma 10 (linking 2nd and 3rd generation) and Lemma 12 (linking 1st and 2nd generation) in Section 5.

1.1. Some Notation and Conventions

If the implicit and constants depend on parameters like , mostly, we write and , but we omit subscripts for the symbol. As usual, is arbitrarily small, changing from statement to statement. The relation means that as the main variable tends to infinity typically. There is no confusion with the dyadic notation, , that means . The Möbius function is ,   if is the product of distinct primes, and otherwise. Symbol denotes constant function and is the characteristic function. The Dirichlet convolution product of the arithmetic functions and is . The -fold Dirichlet product of is , so , for all . For any , we call the Eratosthenes transform of if or equivalently (Möbius’ inversion formula): is the Eratosthenes transform of , the divisor function. We omit in sums like . The distance of from is and is the fractional part of . As usual, , for all , and , for all , for all .

2. Introducing Weighted Selberg Integrals

Given positive integers and , the -Selberg integral of an arithmetic function is the weighted quadratic mean where the complex valued weight has support in for some fixed real number , so that the inner sum is genuinely finite. The term is the expected mean value of weighted with in the short interval of length and depends on , when has logarithmic polynomial ; see Section 4; set

Here, , with the convention that constant has degree , while has .

Clearly, these include the most celebrated Selberg integral , with being the characteristic function of ; more generally, is the -Selberg integral of , while the modified Selberg integral (introduced in [9]) is recognizable by taking the Cesaro weight; say Since where is the correlation of the weight (see next Section 3), it follows that the Cesaro weight is the normalized correlation of . We say that we smooth when we get the modified -Selberg integral of : where the new weight is the normalized correlation of ; that is, As a general strategy, from a nonpositive definite weight (with support of length ), we smooth it; the new weight has a nonnegative exponential sum (see next Section 3).

Another important instance, studied by the first author (see [10–13]), is the symmetry integral where , for , and vanishes identically for every . Such a study has been motivated by the link found by Kaczorowski and Perelli [14], between the classical Selberg integral and the symmetry properties of the prime numbers.

We will exploit the links between the Selberg integral , the symmetry integral , and the modified Selberg integral (also, within other weighted integrals) in the future (see http://arxiv.org/).

Finally, the considerations in Section 4 suggest that a satisfactory general theory on the weighted Selberg integrals may be built within the Selberg Class (see [15] and http://arxiv.org/abs/1205.1706 in which v3 refers to the arxiv paper with same title of present paper).

3. Weighted Selberg Integrals Are Correlation Averages

The correlation of an arithmetic function is a shifted convolution sum of the form where is the shift. Observe that one may restrict to . Further, a correlation of shift is strictly related to a weighted count of such that ; namely, We use the last formula to define the correlation of a weight by neglecting the -term conveniently in the present context; namely, The reason of such a different definition will be clarified after the next lemma, where we prove a strict connection between correlations and weighted Selberg integrals by applying an elementary dispersion method.

Lemma 7. Let be positive integers such that and as . For every uniformly bounded weight with support in and every arithmetic function , one has where .

Proof. It is readily seen that, after expanding the square and exchanging sums, it suffices to show, say, Since we may clearly assume that and , we write By noting that the condition is implied by , one has

Remark 8. Essentially the remainder term comes from the estimate for short segments of length within long sums of length . We refer to these short segments as the tails in the summations. To simplify our exposition, the symbol () within some of the next formulas will warn the reader of some tails discarded to abbreviate the formulas themselves.

Thus, by using the exponential sum (hereafter, we will not specify that is a finite sum) we write An appealing aspect is that the exponential sums, whose coefficients are correlations of a weight , are nonnegative. More precisely, In particular, for the correlations of , one gets the well-known Fejér kernel More generally, the Fejér-Riesz theorem [16] states that every nonnegative exponential sum is the square modulus of another exponential sum. A particularly easy instance of this theorem follows by recalling that the Cesaro weight is the normalized correlation of ; that is, (see Section 2). Hence, again Fejér’s kernel makes its appearance in We also say that the Cesaro weights are positive definite and think that this is an advantage, whereas the Selberg integral and the symmetry integral have weights and that are far from being positive definite.

We expect to be able to exploit such a positivity condition, in the future.

4. Essentially Bounded, Balanced, Quasi-Constant, and Stable Arithmetic Functions

The wide class of arithmetic functions under our consideration consists of functions bounded asymptotically by every arbitrarily small power of the variable, in agreement with the definition (i.e., satisfies one of the Selberg Class axioms, the so-called Ramanujan hypothesis (see especially [15])), that we denote briefly by . A well-known prototype of an essentially bounded function is the divisor function , whose Dirichlet series is . Similarly, the Dirichlet series is defined in at least the right half-plane , whenever the generating function is essentially bounded. Assuming that is meromorphic, recall that the expansion of at leads to an asymptotic formula, defining as the residual polynomial of , which either has degree (the polar order is the order of the pole of at . In particular, is the polar order of )    or vanishes identically when (compare [15] for hypotheses). For the remainder term , a good estimate would be with a suitable (negative values of are discarded as meaningless); see [15] formulas.

This is the case for any divisor function . Indeed, from of Section 1, one has where the degree of the residual polynomial (see Section 1) is , because the polar order of is . Here, comes, inductively, from the elementary Dirichlet hyperbola method applied to (precisely, ). We find, by partial summation, the logarithmic polynomial such that Thus, the balanced part of is obtained by subtracting the very slowly increasing function : Note that the product of and the Dirichlet series of has zero residue at .

Moreover, is equivalent to This invites us to formulate the following definitions (although with a different meaning, such a terminology has been coined by Ben Green and Terence Tao. Mainly, Green [17] calls a function balanced when is a characteristic function of a set with density ): (i.e., has an analytic continuation in ), Of course, well-balanced implies balanced (because, from previous bound, is regular at ). However, the converse needs not to be true; particularly, is a balanced function (by the prime number Theorem), but the existence of an exponent is far from being proved (compare [18]).

An essentially bounded arithmetic function is said to be quasi-constant if there exists such that (the condition on the derivative implies that is essentially bounded, provided that depends only on . However, we do not exclude the possibility that and might depend on auxiliary parameters)     and on . Clearly, the logarithmic polynomial is quasi-constant with respect to . This, together with the fact that is a well-balanced arithmetic function of exponent , suggests the following further definition.

An arithmetic function is stable of exponent if there exist a quasi-constant function and a well-balanced function of exponent such that , whereas the amplitude of is defined as In what follows, we will assume that stable have a logarithmic polynomial (above is the general definition).

Recall that the Dirichlet divisor problem requires to prove the conjectured amplitude , while one infers by the Dirichlet hyperbola method and the best known bound at the present moment is (Huxley, 2003). In the sequel, is the best possible exponent in and .

According to Ivić [3], the mean value in the Selberg integral of an arithmetic function , with Dirichlet series converging absolutely (at least) in and meromorphic in , has analytic form given by where is the derivative of the residual polynomial . We remark that is linear in and is separable; that is, the variables are separated. The logarithmic polynomial has the property For , this identity permits us to compare [5, 6] results with ours (see Section 1).

In particular, the analytic form of the mean value in the Selberg integral of is explicitly given by with , where is the Euler-Mascheroni constant and Regarding the (short) sum of weighted with Cesaro weights, that is, Proposition 16 in Section 7 (see Remark 17 too) suggests (from Proposition 14 in Section 7) that the expected mean value is where . Next, Proposition 9 justifies the interchange of analytic form and arithmetic form of the mean value , inside ; its proof is designed for the specific case .

Proposition 9. Uniformly, for every , one has

The idea of the proof (details on arxiv: v3 of this paper) is to apply Amitsur’s formula [19] with Tull’s error term [20] (Amitsur derived a symbolic method to calculate main terms of asymptotic formulas. Tull gave a refined partial summation that allows us to transfer the error terms from the formula for the sum , like Dirichlet’s classical , to this formula); that is,

Finally, by Lemma 7, the Selberg integrals and the modified one for a real and essentially bounded function are related to the correlation averages (see Remark 8), respectively, as In particular, when is also balanced, that is, vanishes identically, from these formulas, we get

5. Averages of Correlations: Smoothing Correlations’ Formulas by Arithmetic Means

Recalling that with , one easily infers the formulas In particular, for every balanced real function , from the last formulas of the previous section, we get Formulas , , and correspond, respectively, to the following iterations of correlations’ averages. 1st generation:  2nd generation:  3rd generation: Such an obstinate process of averaging is motivated by the fact that it is rarely possible to prove an asymptotic formula for the single correlation . This counts the number of -twins (see Section 3) not only when is a pure characteristic function (the von Mangoldt function is a typical case). In general, the underlying Diophantine equation is a binary problem still out of reach; see Section 3 of [7] (and also http://arxiv.org/: v3 of present paper). On the other hand, for higher generations of correlations’ averages, we get smoother averages; for which, consequently, there is more hope of proving nontrivial estimates. However, even for the nd generation, this hope is quite frustrated by the lack of efficient elementary methods to bound the Selberg integral directly. Indeed, Ivić [3] applies the th moments of , since has a strong connection with them (see Section 8).

Further, it is interesting to analyze the cost of the loss when nontrivial information on the correlations’ averages at some th generation level is transferred to the th generation (see Lemmas 10 and 12).

In this regard, if is real, essentially bounded, and balanced, then the trivial inequality gives

Then, in order to obtain an inequality in the opposite direction, we prove the following lemma.

Lemma 10. For every essentially bounded, balanced, and real arithmetic function and for every ,

Proof. By applying Cauchy’s inequality and Parseval’s identity, we see that The Lemma is proved (using ), because the formulas for and from and yield

Now, let us prove Theorems 1 and 2.

Proof of Theorems 1 and 2. First, we note that , where is the logarithmic polynomial of and the implicit remainders give a negligible contribution in view of the stated estimate for . Then, from Lemma 10, it follows that if, for some , a nontrivial estimate of the form holds with some gain , then, for the same range of , one has Thus, using these inequalities for the balanced part of , that is, , Theorems 1 and 2 follow by taking, respectively, in the Conjecture CL for and in the General Conjecture CL for .

Remark 11. Lemma 10 reveals that, applying only Cauchy inequality, a third generation gain leads to the gain for the second generation. We say that the exponent gain has halved.

What about the trade of information from the second generation to the first one?

Let us define the deviation of a stable arithmetic function , with logarithmic polynomial , as

Lemma 12 estimates in terms of the Selberg integral of (when is real and stable).

Lemma 12. Let be an essentially bounded, stable, and real arithmetic function with amplitude . Then, for every , one has

Proof. Let where as above and let be well-balanced. Then, we write By the Cauchy inequality, one has where the mean value is (see Section 2) , since is quasi-constant. Hence, we get and the lemma follows applying partial summation to , since is stable with amplitude .

Remark 13. From previous Lemma, if, for a suitable , , one has then In particular, for , we get (the essence of Corollary 5; see above)

In conclusion, Lemmas 10 and 12 provide the following chain of implications of nontrivial bounds: The exponent gains halves at each step. If it is a neat positive one, we say that is stable through generations.

In the future, we will explore the hard case of stable (possibly) having no logarithmic polynomial.

6. Asymptotic Formulas for in Almost All Short Intervals

We turn our attention to nontrivial bounds, postponing those on to the next section. Ivić [3] proved that for the width , with a neat exponent gain . In other words, if is the width range for such an inequality to hold, Ivić has proved .

This comes from , where is the Carlson’s abscissa for the Riemann zeta th moment: (see [4] to get some of the known upper bounds for ).

Conjecture CL provides improvements on Ivić’s result, since, for every width , it yields the best possible estimate, that is, the square-root cancellation (compare with the abovementioned lower bound holding for ; see [2]), which in turn implies the optimal through the arguments of Section 5 (compare Lemma 10 and Theorems 1 and 2 proofs).

Our estimate for leads to improvements on [5, 6] for deviation. That is what we prove now.

Proof of Corollary 3. From the decomposition introduced in Section 4, one gets where we recall that , are essentially bounded and is a quasi-constant function of . Therefore, By applying partial summation and , one has Since holds for at least width (see [2] and recall that ), the conclusion from Theorem 1 follows, because Cauchy’s inequality implies where (compare Section 2)

7. Main Theme: From All Long Intervals to Almost All Short Intervals

The inductive identity invites us to explore the possibility to infer formulas for in almost all short intervals, from suitable information on in long intervals.

However, the known bounds on the amplitudes seem to be a first serious bottle-neck. Further, while it might be comparatively easy to attack Conjecture CL, the path climbs up drastically when . Although a general -folding method is available (Proposition 14, the core of our elementary approach), we want to prove the following mean-square proximity of analytic and arithmetic forms of s.i. mean value: for a suitable arithmetic mean value . The right form is suggested by Proposition 14 that also implies (via the Large Sieve inequality) a nontrivial bound for in short intervals of width (for any satisfying a property shared by our weights , , and ), indeed, Proposition 16.

On the other hand, the above mean-square proximity of and , at the present moment, is obtained with a gain weaker than ; see next Proposition 18, where we apply the nontrivial estimates for given by Ivić [3] in the common range where Proposition 16 also gives nontrivial bounds.

Now, for the so-called -folding method, we need some ad hoc notation. For all , let us consider where is the -fold Dirichlet product of a fixed arithmetic function and the weight is uniformly bounded with respect to , independent of . Set and denote where

Proposition 14. If is essentially bounded, then uniformly, for , one has

The proof runs on the lines of (initial part of) the proof of Corollary 1 in [2] and we will omit it.

Remark 15. The uncovered range gives in mean-square a tail contribution .
We keep hereafter the notation even for and set .

Let us state the following consequence of Proposition 14, denoting .

Proposition 16. Assume that is uniformly bounded in and Then, for every integer , one has

The proof applies the Large Sieve inequality in the form given by Lemma 3 of [13] and will appear elsewhere.

Remark 17. The Cesaro weight satisfies previous assumptions: recalling Section 3 last formula and the well-known , if we set , then (for ; otherwise, the following vanishes)

Similar calculations prove that and satisfy previous Proposition assumptions, too.

The replacement, in general, of analytic by arithmetic lies at the heart of our arithmetic approach.

We see that the analytic form is close to the arithmetic form in Proposition 16:

Proposition 18. For every integer , there exists such that

The proof follows by comparing (above quoted) Ivić’s bounds with Proposition 16 and will appear elsewhere.

Notice that, for , this Proposition yields a much weaker inequality than Proposition 9.

Remark 19. As for the case , implications in Section 5 and Propositions 16 and 18 immediately prove Corollary 4.

8. Conditional Bounds for the Moments of on the Critical Line

We do not use explicitly any deep property of , relying upon known bounds, but these, actually, follow from nontrivial estimates (for a clear digression on this; see the wonderful book [4] by Ivić).

On the other hand, bounds for have nontrivial consequences on , as applied in next proof.

Proof of Theorem 6. For and for every , Theorem 1.1 of [7] yields ( saves here) Indeed, in Theorem 1.1 of [7], one finds the same inequality, with replaced by the Selberg integral However, an easy dyadic argument allows one to replace by an interval like , while the substitution of the integral on with generates only negligible remainder terms.
Since holds for , by hypothesis, the previous inequality implies

Remark 20. Define the excess to be such that ; Theorem 6 yields . Known values are , , followed by Hölder’s inequality from and (see Section 2 in [7]).

According to Section 5, our belong to a 2nd generation approach, while Ivić’s [8] is a 1st generation one. Under Conjecture CL, our new approach, based on a modified version of Gallagher’s Lemma [21], gives .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are very grateful to Alberto Perelli for interesting discussions and invaluable suggestions. The authors wish to give Corollary 5 as a gift to Professor Aleksandar Ivić.