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Jeffrey Stopple

Publications and source records attributed to Jeffrey Stopple.

14 recordsLinked to original sources
arXiv2025

Level curves for Zhang's Eta Function

Study of the level curve for the real part of $\eta(s)=0$ with $\eta(s)=\pi^{-s/2}\Gamma(s/2)\zeta^\prime(s)$ gives a new classification of the zeros of $\zeta(s)$ and of $\zeta^\prime(s)$. We conjecture that for type 2 zeros, $\liminf (\beta^\prime -1/2)\log\gamma^\prime = 0$ if and only if $\liminf (\gamma^+-\gamma^-)\log \gamma^\prime=0$, and reduce the conjecture to a lower bound on the curvature of the level curve. We compute and classify $10^6$ zeros of $\zeta^\prime(s)$ near $T=10^{10}$. The Riemann Hypothesis is assumed throughout. An appendix develops the analogous classification for characteristic polynomials of unitary matrices.

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arXiv2023

X-Ray of Zhang's eta function

Study of the level curves the real part of $η(s)=0$ and imaginary part of $η(s)=0$, for $η(s)=π^{-s/2}Γ(s/2)ζ^\prime(s)$ gives a new classification of the zeros of $ζ(s)$ and of $ζ^\prime(s)$. Numerical evidence indicates that the statistics of the gaps (between zeros of $ζ$), or distance from the critical line (for zeros of $ζ^\prime$) is related to the classification. Theorem 6 gives the full conjecture of Soundararajan for the zeros we classify as type 2. We assume the Riemann Hypothesis throughout.

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arXiv2020

Notes on the Phase Statistics of the Riemann Zeros

We numerically investigate, for zeros $ρ=1/2+iγ$, the statistics of the imaginary part of $\log(ζ^\prime(1/2+iγ))$, computed by continuous variation along a vertical line from $σ=4$ to $4+iγ$ and then along a horizontal line to $1/2+iγ$.

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arXiv2015

Lehmer pairs revisited

We seek to understand how the technical definition of Lehmer pair can be related to more analytic properties of the Riemann zeta function, particularly the location of the zeros of $ζ^\prime(s)$. Because we are interested in the connection between Lehmer pairs and the de Bruijn-Newman constant $Λ$, we assume the Riemann Hypothesis throughout. We define strong Lehmer pairs via an inequality on the derivative of the pre-Schwarzian of Riemann's function $Ξ(t)$, evaluated at consecutive zeros. Theorem 1 shows that strong Lehmer pairs are Lehmer pairs. Theorem 2 describes the derivative of the pre-Schwarzian in terms of $ζ^\prime(ρ)$. Theorem 3 expresses the criteria for strong Lehmer pairs in terms of nearby zeros $ρ^\prime$ of $ζ^\prime(s)$. We examine 114661 pairs of zeros of $ζ(s)$ around height t=10^6, finding 855 strong Lehmer pairs. These are compared to the corresponding zeros of $ζ^\prime(s)$ in the same range.

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arXiv2014

Notes on $\log(ζ(s))^{\prime\prime}$

Motivated by the connection to the pair correlation of the Riemann zeros, we investigate the second derivative of the logarithm of the Riemann zeta function, in particular the zeros of this function. Theorem 1 gives a zero-free region. Theorem 2 gives an asymptotic estimate for the number of nontrivial zeros to height T. Theorem 3 is a zero density estimate.

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arXiv2013

Notes on Low discriminants and the generalized Newman conjecture

Generalizing work of Polya, de Bruijn and Newman, we allow the backward heat equation to deform the zeros of quadratic Dirichlet L-functions. There is a real constant Λ_Kr (generalizing the de Bruijn-Newman constant Λ) such that for time t>=Λ_Kr all such L-functions have all their zeros on the critical line; for time t<Λ_Kr there exist zeros off the line. Under GRH, Λ_Kr<=0; we make the complementary conjecture 0<=Λ_Kr. Following the work of Csordas et. al. on Lehmer pairs of Riemann zeros, we use low-lying zeros of quadratic Dirichlet L-functions to show that -1.13* 10^{-7}<Λ_Kr. In the last section we develop a precise definition of a Low discriminant which is motivated by considerations of random matrix theory. The existence of infinitely many Low discriminants would imply 0<=Λ_Kr.

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arXiv2012

Elementary Deuring-Heilbronn Phenomenon

Adapting a technique of Pintz, we give an elementary demonstration of the Deuring phenomenon: a zero of ζ(s) off the critical line gives a lower bound on L(1,χ). The necessary tools are Dirichlet's 'method of the hyperbola', Euler summation, summation by parts, and the Polya-Vinogradov inequality.

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arXiv2012

On Zagier's conjecture for $L(E,2)$: a number field example

We work out an example, for a CM elliptic curve E defined over a real quadratic field F, of Zagier's conjecture. This relates L(E,2) to values of the elliptic dilogarithm function at a divisor in the Jacobian of E which arises from K-theory.

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arXiv2012

Repulsive behavior in an exceptional family

The existence of a Landau-Siegel zero leads to the Deuring-Heilbronn phenomenon, here appearing in the 1-level density in a family of quadratic twists of a fixed genus character L-function. We obtain explicit lower order terms describing the vertical distribution of the zeros, and realize the influence of the Landau-Siegel zero as a resonance phenomenon.

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arXiv2010

The quadratic character experiment

A fast new algorithm is used compute the zeros of the quadratic character L-functions for all negative fundamental discriminants with absolute value 10^12<d<10^12+10^7. These are compared to the 1-level density, including various lower order terms. These terms come from, on the one hand the Explicit Formula, and on the other the L-functions Ratios Conjecture. The latter give a much better fit to the data, providing numerical evidence for the conjecture.

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arXiv2006

Computing $L$-functions with large conductor

An algorithm is given to efficiently compute $L$-functions with large conductor in a restricted range of the critical strip. Examples are included for about 21000 dihedral Galois representations with conductor near $10^7$. The data shows good agreement with a symplectic random matrix model. Also included are an elliptic curve $L$-function and one for a quadratic Dirichlet character.

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arXiv2001

Stark conjectures for CM curves over number fields

We present an elliptic curve analog of the Stark conjecture for the value of the $L$-function at $s=0$. Although implied by the general Beilinson conjectures, the approach here is very concrete. Several cases are proved.

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