Landau-Siegel zeros and zeros of the derivative of the Riemann zeta function

@article{Farmer2010LandauSiegelZA,
  title={Landau-Siegel zeros and zeros of the derivative of the Riemann zeta function},
  author={David W. Farmer and Haseo Ki},
  journal={Advances in Mathematics},
  year={2010},
  volume={230},
  pages={2048-2064}
}
  • D. Farmer, Haseo Ki
  • Published 8 February 2010
  • Mathematics, Philosophy
  • Advances in Mathematics

Figures from this paper

Roots of the derivative of the Riemann-zeta function and of characteristic polynomials
We investigate the horizontal distribution of zeros of the derivative of the Riemann-zeta function and compare this with the radial distribution of zeros of the derivative of the characteristic
On the distribution of the zeros of the derivative of the Riemann zeta-function
  • S. Lester
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 2014
Abstract We establish an asymptotic formula describing the horizontal distribution of the zeros of the derivative of the Riemann zeta-function. For ℜ(s) = σ satisfying (log T)−1/3+ε ⩽ (2σ − 1) ⩽ (log
Arithmetic consequences of the GUE conjecture for zeta zeros
Conditioned on the Riemann hypothesis, we show that the conjecture that the zeros of the Riemann zeta function resemble the eigenvalues of a random matrix is logically equivalent to a statement about
ON THE DISTRIBUTION OF ZEROS OF THE DERIVATIVE OF SELBERG’S ZETA FUNCTION ASSOCIATED TO FINITE VOLUME RIEMANN SURFACES
We study the distribution of zeros of the derivative of the Selberg zeta function associated to a noncompact, finite volume hyperbolic Riemann surface $M$ . Actually, we study the zeros of
Gaps between zeros of and the distribution of zeros of C1 (8)
Gaps between zeros of $\zeta(s)$ and the distribution of zeros of $\zeta'(s)$
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 $\zeta(s)$ if and
Lehmer Pairs Revisited
TLDR
This work seeks 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), finding 855 strong Lehmer pairs.
The distribution of zeros of $\zeta'(s)$ and gaps between zeros of $\zeta(s)$
Assume the Riemann Hypothesis, and let $\gamma^+>\gamma>0$ be ordinates of two consecutive zeros of $\zeta(s)$. It is shown that if $\gamma^+-\gamma < v/ \log \gamma $ with $v<c$ for some absolute

References

SHOWING 1-10 OF 32 REFERENCES
Roots of the derivative of the Riemann-zeta function and of characteristic polynomials
We investigate the horizontal distribution of zeros of the derivative of the Riemann-zeta function and compare this with the radial distribution of zeros of the derivative of the characteristic
Moments of the Riemann zeta function
Assuming the Riemann hypothesis, we obtain an upper bound for the moments of the Riemann zeta function on the critical line. Our bound is nearly as sharp as the conjectured asymptotic formulae for
Spacing of zeros of Hecke L-functions and the class number problem
We derive strong and effective lower bounds for the class number h(q) of the imaginary quadratic field Q(\sqrt{-q}), conditionally subject to the existence of many small (subnormal) gaps between
The Theory of the Riemann Zeta-Function
The Riemann zeta-function embodies both additive and multiplicative structures in a single function, making it our most important tool in the study of prime numbers. This volume studies all aspects
Random matrix theory and the zeros of ζ′(s)
We study the density of the roots of the derivative of the characteristic polynomial Z(U, z) of an N × N random unitary matrix with distribution given by Haar measure on the unitary group. Based on
Pair correlation of zeros of the zeta function.
s T— »oo and U-+Q in such a way that UL = A; here L = ̂ —logT is the average 2n density of zeros up to T and A is an arbitrary positive constant. If the same number of points were distributed at
On the zeros of ?'(s near the critical line
Let ρ = β ′ + i γ ′ denote the zeros of ζ (s), s = σ + i t . It is shown that there is a positive proportion of the zeros of ζ (s) in 0 < t < T satisfyingβ ′ − 1/2 (logT)−1. Further results relying
The Zeros of the Derivative of the Riemann Zeta Function Near the Critical Line
We study the horizontal distribution of zeros of ζ ′ (s) which are denoted as ρ ′ =β ′ +iγ ′ . We assume the Riemann hypothesis which implies β ′ ≥ 1/2 for any nonreal zero ρ ′ , equality being
Differentiation evens out zero spacings
If f is a polynomial with all of its roots on the real line, then the roots of the derivative f' are more evenly spaced than the roots of f. The same holds for a real entire function of order 1 with
On Small Distances Between Ordinates of Zeros of ζ(s) and ζ′(s)
In this paper s = σ + it will denote a complex variable, where σ and t are real, and T will denote a large parameter. The relations between the zeros of a function and the zeros of its derivatives
...
...