A discrete mean value of the derivative of the Riemann zeta function

@article{Ng2007ADM,
  title={A discrete mean value of the derivative of the Riemann zeta function},
  author={Nathan Ng},
  journal={arXiv: Number Theory},
  year={2007}
}
  • N. Ng
  • Published 12 June 2007
  • Mathematics
  • arXiv: Number Theory
In this article we compute a discrete mean value of the derivative of the Riemann zeta function. This mean value will be important for several applications concerning the size of $\zeta'(\rho)$ where $\zeta(s)$ is the Riemann zeta function and $\rho$ is a non-trivial zero of the Riemann zeta function. 
Lower bounds for discrete negative moments of the Riemann zeta function
We prove lower bounds for the discrete negative $2k$th moment of the derivative of the Riemann zeta function for all fractional $k\geqslant 0$. The bounds are in line with a conjecture of Gonek and
Negative values of the Riemann zeta function on the critical line
We investigate the intersections of the curve $\mathbb{R}\ni t\mapsto \zeta({1\over 2}+it)$ with the real axis. We show unconditionally that the zeta-function takes arbitrarily large positive and
Lower bounds for moments of zeta prime rho
Assuming the Riemann Hypothesis, we establish lower bounds for moments of the derivative of the Riemann zeta-function averaged over the non-trivial zeros of $\zeta(s)$. Our proof is based upon a
Value-distribution of the Riemann zeta-function and related functions near the critical line
We study the value-distribution of the Riemann zeta-function and related functions on and near the critical line. Amongst others, we focus on the following: The critical line is a natural boundary
Large values of Dirichlet L-functions at zeros of a class of L-functions
  • Junxian Li
  • Mathematics
    Canadian Journal of Mathematics
  • 2020
Abstract In this paper, we are interested in obtaining large values of Dirichlet L-functions evaluated at zeros of a class of L-functions, that is, $$ \begin{align*}\max_{\substack{F(\rho)=0\\ T\leq
Lower bounds of discrete moments of the derivatives of the Riemann zeta-function on the critical line
L’accès aux articles de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://jtnb.cedram.

References

SHOWING 1-10 OF 28 REFERENCES
Simple zeros of the Riemann zeta-function
Assuming the Riemann Hypothesis, Montgomery and Taylor showed that at least 67.25% of the zeros of the Riemann zeta-function are simple. Using Montgomery and Taylor's argument together with an
The fourth moment of ζ′(ρ)
Discrete moments of the Riemann zeta function were studied by Gonek and Hejhal in the 1980's. They independently formulated a conjecture concerning the size of these moments. In 1999, Hughes,
The Distribution of the Summatory Function of the Möbius Function
The summatory function of the Möbius function is denoted M(x). In this article we deduce conditional results concerning M(x) assuming the Riemann hypothesis and a conjecture of Gonek and Hejhal on
Extreme values of ζ′(ρ)
Assuming the Riemann hypothesis, we exhibit large and small values of the derivative of the zeta function evaluated at the non‐trivial zeros of the zeta function. These results are proved by applying
Lower bounds for moments of L-functions.
TLDR
A simple method is developed to establish lower bounds of the conjectured order of magnitude for several such families of L-functions, including the case of the family of all Dirichlet L-Functions to a prime modulus.
Topics in Multiplicative Number Theory
Three basic principles.- The large sieve.- Arithmetic formulations of the large sieve.- A weighted sieve and its application.- A lower bound of Roth.- Classical mean value theorems.- New mean value
Lower bounds for moments of L-functions: symplectic and orthogonal examples
We give lower bounds of the conjectured order of magnitude for an orthogonal and a symplectic family of L-functions.
Introduction to Analytic and Probabilistic Number Theory
Foreword Notation Part I. Elementary Methods: Some tools from real analysis 1. Prime numbers 2. Arithmetic functions 3. Average orders 4. Sieve methods 5. Extremal orders 6. The method of van der
Bombieri's mean value theorem
The purpose of this paper is to give a short proof of an important recent theorem of Bombieri [2] on the mean value of the remainder term in the prime number theorem for arithmetic progressions.
...
...