Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fourier optimization and pair correlation problems

Mithun Kumar Das, Tolibjon Ismoilov, Antonio Pedro Ramos

TL;DR

This work develops an axiomatic, Fourier-analytic framework to bound pair-correlation-type form factors for general sequences, linking long-term averages to two extremal Fourier optimization problems and reproducing-kernel Hilbert space methods. By solving EP1 and EP2, and leveraging RKHS structure, the authors derive explicit average bounds applicable to zeros of primitive L-functions in the Selberg class, Dedekind zeta functions, and the real/imaginary parts of zeta along vertical lines. The analysis yields concrete bounds in terms of limiting measures and reproducing kernels, and provides new insight into conjectures on pair correlations, including a counterexample to a conjecture of Gonek and Ki in certain regimes. The results unify several prior averages for L-function zeros under a common framework and supply explicit kernel-based constants for practical bounds. The approach blends harmonic analysis, reproducing-kernel theory, and analytic number theory to quantify how zero patterns constrain average correlations across diverse arithmetic objects.

Abstract

We introduce a generic framework to provide bounds related to the pair correlation of sequences belonging to a wide class. We consider analogues of Montgomery's form factor for zeros of the Riemann zeta function in the case of arbitrary sequences satisfying some basic assumptions, and connect their estimation to two extremal problems in Fourier analysis, which are promptly studied. As applications, we provide average bounds of form factors related to some sequences of number theoretic interest, such as the zeros of primitive elements of the Selberg class, Dedekind zeta functions, and the real and imaginary parts of the Riemann zeta function. In the last case, our results bear an implication to a conjecture of Gonek and Ki (2018), showing it cannot hold in some situations.

Fourier optimization and pair correlation problems

TL;DR

This work develops an axiomatic, Fourier-analytic framework to bound pair-correlation-type form factors for general sequences, linking long-term averages to two extremal Fourier optimization problems and reproducing-kernel Hilbert space methods. By solving EP1 and EP2, and leveraging RKHS structure, the authors derive explicit average bounds applicable to zeros of primitive L-functions in the Selberg class, Dedekind zeta functions, and the real/imaginary parts of zeta along vertical lines. The analysis yields concrete bounds in terms of limiting measures and reproducing kernels, and provides new insight into conjectures on pair correlations, including a counterexample to a conjecture of Gonek and Ki in certain regimes. The results unify several prior averages for L-function zeros under a common framework and supply explicit kernel-based constants for practical bounds. The approach blends harmonic analysis, reproducing-kernel theory, and analytic number theory to quantify how zero patterns constrain average correlations across diverse arithmetic objects.

Abstract

We introduce a generic framework to provide bounds related to the pair correlation of sequences belonging to a wide class. We consider analogues of Montgomery's form factor for zeros of the Riemann zeta function in the case of arbitrary sequences satisfying some basic assumptions, and connect their estimation to two extremal problems in Fourier analysis, which are promptly studied. As applications, we provide average bounds of form factors related to some sequences of number theoretic interest, such as the zeros of primitive elements of the Selberg class, Dedekind zeta functions, and the real and imaginary parts of the Riemann zeta function. In the last case, our results bear an implication to a conjecture of Gonek and Ki (2018), showing it cannot hold in some situations.

Paper Structure

This paper contains 21 sections, 17 theorems, 169 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. Montgomery's pair correlation conjecture
  4. Axiomatic formulation
  5. Bounds via Fourier optimization
  6. Reproducing kernel Hilbert spaces
  7. Explicit bounds for averages
  8. Applications
  9. Primitive $L$-functions in $\mathcal{S}$ of degree $m$
  10. Dedekind zeta functions of number fields
  11. The real and imaginary parts of the Riemann zeta function
  12. Fourier Optimization Bounds
  13. Proof of Theorem \ref{['thm:boundswithextremalproblem']}
  14. Proof of Theorem \ref{['thm:indptlowerbound']}
  15. The Reproducing Kernel Hilbert Space Hnu
  16. ...and 6 more sections

Key Result

Theorem 1

Let $\Gamma$ be a sequence of real numbers satisfying (A1) and (A2). Let $\nu$ be the limiting measure for $\mathfrak{F}_\Gamma(\alpha, T)$ in the interval $[-\Delta, \Delta]$. Given $\varepsilon>0$, for any $b\geq 1$ and sufficiently large $\ell$, one has as $T \to \infty$, with $\ell \geq \ell_0(\varepsilon)$ for the upper bound and $\ell \geq \ell_0(b,\varepsilon)$ for the lower bound.

Figures (1)

  • Figure 1: Upper and lower bounds for $\frac{1}{\ell}\int_{b}^{b+\ell}\mathfrak{F}_\Gamma(\alpha, T)\,\text{\rm d}\alpha$ as a function of $c$, where $c_1=1$, $c_2=1$, $c_3 = 4c$, and $\Delta = 0.5$.

Theorems & Definitions (27)

  • Remark
  • Theorem 1
  • Remark
  • Theorem 2
  • Remark
  • Theorem 3
  • Lemma 4
  • Theorem 5
  • Theorem 6
  • Remark
  • ...and 17 more