# 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.
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## References

SHOWING 1-10 OF 28 REFERENCES
Simple zeros of the Riemann zeta-function
• Mathematics, Philosophy
• 1993
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.
• Mathematics
Proceedings of the National Academy of Sciences of the United States of America
• 2005
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
• Mathematics
• 2006
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.