About

Jeffrey Stopple is a professor whose area of research is Analytic Number Theory. His work involves the study of the properties and behaviors of functions such as the Riemann Zeta Function, which is a central object in number theory. He has authored a book titled 'A Primer of Analytic Number Theory: from Pythagoras to Riemann,' which provides insights into the field, and offers supplementary materials including graphics, videos, and audio to enhance understanding of the topics covered, particularly in Chapter 8 on the Riemann Zeta Function and Chapter 10 on the Explicit Formula. His research contributions are also reflected in recent papers available on platforms such as arXiv, the Notices of the AMS, and Mathematics of Computation.

Research topics

• History
• Mathematics
• Mathematical physics
• Pure mathematics
• Combinatorics
• Physics

Selected publications

• Level Curves for Zhang’s Eta Function

  Experimental Mathematics · 2025-04-11

  articleOpen access1st authorCorresponding

  Study of the level curve Re((s)) = 0 for (s) = -s/2 (s/2) (s) gives a new classification of the zeros of (s) and of (s).We conjecture that for type 2 zeros, lim inf( -1/2) log = 0 lim inf( + -) log = 0, and reduce the conjecture to a lower bound on the curvature of the level curve.We compute and classify 10 6 zeros of (s) near T = 10 10 .The Riemann Hypothesis is assumed throughout.An appendix develops the analogous classification for characteristic polynomials of unitary matrices.

  PublisherOA PDFDOI

• Level curves for Zhang's Eta Function

  ArXiv.org · 2025-03-10

  preprintOpen access1st authorCorresponding

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

  PublisherOA PDFDOI

• Lynne Heather Walling (1958–2021)

  Notices of the American Mathematical Society · 2023

  • History
  DOI

• Notes on the Phase Statistics of the Riemann Zeros

  arXiv (Cornell University) · 2020-07-15

  preprintOpen access1st authorCorresponding

  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 $1/2+iγ$.

  PublisherOA PDFDOI

• X-Ray of Zhang's eta function

  arXiv (Cornell University) · 2020

  1st authorCorresponding
  • Mathematics
  • Mathematical physics
  • Pure mathematics

  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.

  PublisherOA PDF

• Lehmer Pairs Revisited

  Experimental Mathematics · 2016-07-07 · 7 citations

  article1st authorCorresponding

  We seek to understand how the technical definition of a Lehmer pair can be related to more analytic properties of the Riemann zeta function, particularly the location of the zeros of ζ′(s). Because we are interested in the connection [Csordas et al. 94 [Csordas et al. 94] G. Csordas, W. Smith, R. Varga. “Lehmer Pairs of Zeros, the de Bruijn-Newman Constant, and the Riemann Hypothesis.” Constr. Approx. 10 (1994), 107–129.[Crossref], [Web of Science ®] , [Google Scholar]] 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 PΞ′(γ) in terms of ζ′(ρ) where ρ = 1/2 + iγ. Theorem 3 expresses PΞ′(γ+) + PΞ′(γ−) in terms of nearby zeros ρ′ of ζ′(s). We examine 114, 661 pairs of zeros of ζ(s) around height t = 106, finding 855 strong Lehmer pairs. These are compared to the corresponding zeros of ζ′(s) in the same range.

  PublisherDOI

• Notes on $\log (\zeta (s))^\prime \prime $

  Rocky Mountain Journal of Mathematics · 2016-10-01 · 2 citations

  preprintOpen access1st authorCorresponding

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

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• Lehmer pairs revisited

  arXiv (Cornell University) · 2015-08-24

  preprintOpen access1st authorCorresponding

  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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• Notes on low discriminants and the generalized Newman conjecture

  Functiones et Approximatio Commentarii Mathematici · 2014-09-01 · 10 citations

  articleOpen access1st authorCorresponding

  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 $\Lambda_{Kr}$ (generalizing the de Bruijn-Newman constant $\Lambda$) such that for time $t\ge\Lambda_{Kr}$ all such $L$-functions have all their zeros on the critical line; for time $t<\Lambda_{Kr}$ there exist zeros off the line. Under GRH, $\Lambda_{Kr}\le 0$; we make the complementary conjecture $0\le \Lambda_{Kr}$. Following the work of Csordas \emph{et. al}. on Lehmer pairs of Riemann zeros, we use low-lying zeros of quadratic Dirichlet $L$-functions to show that $-1.13\cdot 10^{-7}<\Lambda_{Kr}$. 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 $0\le \Lambda_{Kr}$.

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• On the theorem of Conrey and Iwaniec

  arXiv (Cornell University) · 2013-07-02

  preprintOpen access1st authorCorresponding

  An exposition on "Spacing of zeros of Hecke L-functions and the class number problem" by Conrey and Iwaniec; any errors are my own.

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Frequent coauthors

• Kimberly Hopkins

  Towson University

  5 shared

• Nancy Childress

  Arizona State University

  2 shared

• Jeffrey Hoffstein

  Providence College

  1 shared

• Eric Stade

  University of Colorado Boulder

  1 shared

• Rainer Schulze‐Pillot

  Saarland University

  1 shared

• Dorothy Wallace

  Dartmouth College

  1 shared

• Jim Brown

  1 shared

• Solomon Friedberg

  Boston College

  1 shared