Pair Correlation Conjecture Research Articles

The variance of integers without small prime factors in short intervals

AbstractThe variance of primes in short intervals relates to the Riemann Hypothesis, Montgomery’s Pair Correlation Conjecture and the Hardy–Littlewood Conjecture. In regards to its asymptotics, very little is known unconditionally. We study the variance of integers without prime factors below y, in short intervals. We use complex analysis and sieve theory to prove an unconditional asymptotic result in a range for which we give evidence is qualitatively best possible. We find that this variance connects with statistics of y-smooth numbers, and, as with primes, is asymptotically smaller than the naive probabilistic prediction once the length of the interval is at least a power of y.

Quantitative upper bounds related to an isogeny criterion for elliptic curves

AbstractFor and elliptic curves defined over a number field , without complex multiplication, we consider the function counting nonzero prime ideals of the ring of integers of , of good reduction for and , of norm at most , and for which the Frobenius fields and are equal. Motivated by an isogeny criterion of Kulkarni, Patankar, and Rajan, which states that and are not potentially isogenous if and only if , we investigate the growth in of . We prove that if and are not potentially isogenous, then there exist positive constants , , and such that the following bounds hold: (i) ; (ii) under the Generalized Riemann Hypothesis for Dedekind zeta functions (GRH); (iii) under GRH, Artin's Holomorphy Conjecture for the Artin ‐functions of number field extensions, and a Pair Correlation Conjecture for the zeros of the Artin ‐functions of number field extensions.

Composite values of shifted exponentials

A well-known open problem asks to show that 2n+5 is composite for almost all values of n. This was proposed by Gil Kalai as a possible Polymath project, and was first posed by Christopher Hooley. We settle this problem assuming GRH and a form of the pair correlation conjecture. We in fact do not need the full power of the pair correlation conjecture, and it suffices to assume an inequality of Brun–Titchmarsh type in number fields that is implied by the pair correlation conjecture. Our methods apply in fact to any shifted exponential sequence of the form an−b and show that, under the same assumptions, such numbers are k-almost primes for a density 0 of natural numbers n. Furthermore, under the same assumptions we show that ap−b is composite for almost all primes p whenever (a,b)≠(2,1).

On the number variance of zeta zeros and a conjecture of Berry

AbstractAssuming the Riemann hypothesis, we prove estimates for the variance of the real and imaginary part of the logarithm of the Riemann zeta function in short intervals. We give three different formulations of these results. Assuming a conjecture of Chan for how often gaps between zeros can be close to a fixed non‐zero value, we prove a conjecture of Berry (1988) for the number variance of zeta zeros in the non‐universal regime. In this range, Gaussian unitary ensemble statistics do not describe the distribution of the zeros. We also calculate lower order terms in the second moment of the logarithm of the modulus of the Riemann zeta function on the critical line. Assuming Montgomery's pair correlation conjecture, this establishes a special case of a conjecture of Keating and Snaith (2000).

On the 𝑞-analogue of the Pair Correlation Conjecture via Fourier optimization

We study the q q -analogue of the average of Montgomery’s function F ( α , T ) F(\alpha ,\, T) over bounded intervals. Assuming the Generalized Riemann Hypothesis for Dirichlet L L -functions, we obtain upper and lower bounds for this average over an interval that are quite close to the pointwise conjectured value of 1 1 . To compute our bounds, we extend a Fourier analysis approach by Carneiro, Chandee, Chirre, and Milinovich, and apply computational methods of non-smooth programming.

A note on the mean values of the derivatives of 𝜁’/𝜁

Assuming the Riemann hypothesis, we obtain a formula for the mean value of the k k -derivative of ζ ′ / ζ \zeta ’/\zeta , depending on the pair correlation of zeros of the Riemann zeta-function. This formula allows us to obtain new equivalences to Montgomery’s pair correlation conjecture. This extends a result of Goldston, Gonek, and Montgomery where the mean value of ζ ′ / ζ \zeta ’/\zeta was considered.

On Montgomery’s pair correlation conjecture: A tale of three integrals

Abstract We study three integrals related to the celebrated pair correlation conjecture of H. L. Montgomery. The first is the integral of Montgomery’s function F ⁢ ( α , T ) {F(\alpha,T)} in bounded intervals, the second is an integral introduced by Selberg related to estimating the variance of primes in short intervals, and the last is the second moment of the logarithmic derivative of the Riemann zeta-function near the critical line. The conjectured asymptotic for any of these three integrals is equivalent to Montgomery’s pair correlation conjecture. Assuming the Riemann hypothesis, we substantially improve the known upper and lower bounds for these integrals by introducing new connections to certain extremal problems in Fourier analysis. In an appendix, we study the intriguing problem of establishing the sharp form of an embedding between two Hilbert spaces of entire functions naturally connected to Montgomery’s pair correlation conjecture.

Joint discrete universality for periodic zeta-functions. IV

In the paper, a theorem on the approximation of collections of a wide class of analytic functions by collections of shifts of zeta-functions with periodic co-efficients involving imaginary parts of non-trivial zeros of the Riemann zeta-function is obtained. For this, a version of the Montgomery pair correlation conjecture is required.

On the logarithm of the Riemann zeta-function near the nontrivial zeros

Assuming the Riemann hypothesis and Montgomery's Pair Correlation Conjecture, we investigate the distribution of the sequences $(\log|\zeta(\rho+z)|)$ and $(\arg\zeta(\rho+z)).$ Here $\rho=\frac12+i\gamma$ runs over the nontrivial zeros of the zeta-function, $0<\gamma \leq T,$ $T$ is a large real number, and $z=u+iv$ is a nonzero complex number of modulus $\ll 1/\log T.$ Our approach proceeds via a study of the integral moments of these sequences. If we let $z$ tend to $0$ and further assume that all the zeros $\rho$ are simple, we can replace the pair correlation conjecture with a weaker spacing hypothesis on the zeros and deduce that the sequence $(\log (|\zeta^\prime(\rho)|/\log T))$ has an approximate Gaussian distribution with mean $0$ and variance $\frac12\log\log T.$ This gives an alternative proof of an old result of Hejhal and improves it by providing a rate of convergence to the distribution.

Joint universality of periodic zeta-functions with multiplicative coefficients. II

In the paper, a joint discrete universality theorem for periodic zeta-functions with multiplicative coefficients on the approximation of analytic functions by shifts involving the sequence f kg of imaginary parts of nontrivial zeros of the Riemann zeta-function is obtained. For its proof, a weak form of the Montgomery pair correlation conjecture is used. The paper is a continuation of [A. Laurinčikas, M. Tekorė, Joint universality of periodic zeta-functions with multiplicative coefficients, Nonlinear Anal. Model. Control, 25(5):860–883, 2020] using nonlinear shifts for approximation of analytic functions.

UNIVERSALITY OF ZETA-FUNCTIONS OF CUSP FORMS AND NON-TRIVIAL ZEROS OF THE RIEMANN ZETA-FUNCTION

It is known that zeta-functions ζ(s,F) of normalized Hecke-eigen cusp forms F are universal in the Voronin sense, i.e., their shifts ζ(s + iτ,F), τ R, approximate a wide class of analytic functions. In the paper, under a weak form of the Montgomery pair correlation conjecture, it is proved that the shifts ζ(s+iγkh,F), where γ1 &lt; γ2 &lt; ... is a sequence of imaginary parts of non-trivial zeros of the Riemann zeta function and h &gt; 0, also approximate a wide class of analytic functions.

JOINT DISCRETE APPROXIMATION OF A PAIR OF ANALYTIC FUNCTIONS BY PERIODIC ZETA-FUNCTIONS

In the paper, the problem of simultaneous approximation of a pair of analytic functions by a pair of discrete shifts of the periodic and periodic Hurwitz zeta-function is considered. The above shifts are defined by using the sequence of imaginary parts of non-trivial zeros of the Riemann zeta-function. For the proof of approximation theorems, a weak form of the Montgomery pair correlation conjecture is applied.

Joint universality of Hurwitz zeta-functions and nontrivial zeros of the Riemann zeta-function

We prove that every collection of analytic functions (f1(s), . . . , fr(s)) defined on the right-hand side of the critical strip can be simultaneously approximated by shifts of Hurwitz zeta-functions (ζ(s + iγκh, α1), … , ζ(s + iγκh, αr)), h > 0, where 0 < γ1 ≤ γ2 ≤ … are the imaginary parts of nontrivial zeros of the Riemann zeta-function ζ(s). We use the weak form of the Montgomery pair correlation conjecture and the linear independence over ℚ of the set {log(m + αj) : m ∈ ℕ0, j = 1, … , r}.

Cyclic components of quotients of abelian varieties mod p

Let A an abelian variety of dimension r, defined over Q. For p a rational prime, we denote by Fp the finite field of cardinality p. If A has good reduction at p, let A¯p be the reduction of A at p. Let Γ be a free subgroup of the Mordell–Weil group A(Q), and let Γp be the reduction of Γ at p. In this paper for abelian varieties of type I, II, III, and IV, under Generalized Riemann Hypothesis, Artin's Holomorphy Conjecture, and Pair Correlation Conjecture, we obtain asymptotic formulas for the number of primes p, with p≤x, for which the quotient A¯p(Fp)Γp has at most 2r−1 cyclic components.

An extended pair-correlation conjecture and primes in short intervals

In this paper we extend the well-known investigations of Montgomery (1973) and Goldston and Montgomery (1987), concerning the pair-correlation function and its relations with the distribution of primes in short intervals, to a more general version of the pair-correlation function.

On the pair correlation conjecture and the alternative hypothesis

We prove the equivalence of certain asymptotic formulas for (a) averages over intervals for the 2-point form factor F(α,T) for the zeros of the Riemann zeta-function, ζ(s), (b) the mean square of the logarithmic derivative of ζ(s), (c) a variance for the number of primes in short intervals, and (d) the number of pairs of zeros of ζ(s) with small gaps. The main result is a generalization of the fusion of a theorem of Goldston and a theorem of Goldston, Gonek, and Montgomery. We apply our result to deduce several consequences of the Alternative Hypothesis.

An extension of the pair-correlation conjecture and applications

We study an extension of Montgomery's pair-correlation conjecture and its relevance in some problems on the distribution of prime numbers.

Explicit relations between primes in short intervals and exponential sums over primes

Under the assumption of the Riemann Hypothesis (RH), we prove explicit quantitative relations between hypothetical error terms in the asymptotic formulae for truncated mean-square average of exponential sums over primes and in the mean-square of primes in short intervals. We also remark that such relations are connected with a more precise form of Montgomery's pair-correlation conjecture.

Hilbert spaces and the pair correlation of zeros of the Riemann zeta-function

Abstract Montgomery’s pair correlation conjecture predicts the asymptotic behavior of the function N(T,β) defined to be the number of pairs γ and γ’ of ordinates of nontrivial zeros of the Riemann zeta-function satisfying 0 &lt; $&lt;$ γ,γ’ ≤ $\leq$ T and 0 &lt; $&lt;$ γ’ - $-$ γ ≤ $\leq$ (2πβ)/(log T) as T → ∞ ${T\to\infty}$ . In this paper, assuming the Riemann hypothesis, we prove upper and lower bounds for N(T,β), for all β &gt; $&gt;$ 0, using Montgomery’s formula and some extremal functions of exponential type. These functions are optimal in the sense that they majorize and minorize the characteristic function of the interval [ - $-$ β,β] in a way to minimize the L 1 ⁢ ( ℝ , { 1 - ( sin ⁡ π ⁢ x π ⁢ x ) 2 } ⁢ d ⁢ x ) ${L^{1}(\mathbb{R},\{1-(\frac{\sin\pi x}{\pi x})^{2}\}\,\text{\rm d}x)}$ -error. We give a complete solution for this extremal problem using the framework of reproducing kernel Hilbert spaces of entire functions. This extends previous work by P. X. Gallagher [J. reine angew. Math. 362 (1985), 72–86] in 1985, where the case β ∈ $\in$ (1/2) ℕ $\mathbb{N}$ was considered using non-extremal majorants and minorants.

Gaps between zeros of [formula omitted] and the distribution of zeros of [formula omitted

We assume the Riemann Hypothesis in this paper. We settle a conjecture of Farmer and Ki in a stronger form. Roughly speaking we show that there is a positive proportion of small gaps between consecutive zeros of the zeta-function ζ(s) if and only if there is a positive proportion of zeros of ζ′(s) lying very closely to the half-line. Our work has applications to the Siegel zero problem. We provide a criterion for the non-existence of the Siegel zero, solely in terms of the distribution of the zeros of ζ′(s). Finally on the Riemann Hypothesis and the Pair Correlation Conjecture we obtain near optimal bounds for the number of zeros of ζ′(s) lying very closely to the half-line. Such bounds are relevant to a deeper understanding of Levinson's method, allowing us to place one-third of the zeros of the Riemann zeta-function on the half-line.