The Values of the Periodic Zeta-Function at the Nontrivial Zeros of Riemann’s Zeta-Function

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 elementary combinatorial argument, we prove that assuming the Riemann Hypothesis at least 67.275% of the zeros are simple.

Recurrence for values of the zeta function

Using the Padé approximation of the exponential function, we obtain a general recurrence relation for values of the zeta function which contains, as particular cases, many of relations already proved. Applications to Bernoulli polynomials are given. At last, we derive some new recurrence relations with gap of length 4 for zeta numbers. MSC: 11B68 41A21

On the Multiplicity of Zeros of the Zeta-function

A b s t r a c t. Several results are obtained concerning multiplicities of zeros of the Riemann zeta-function ζ(s). They include upper bounds for multiplicities, showing that zeros with large multiplicities have to lie to the left of the line σ = 1. A zero-density counting function involving multiplicities is also discussed. AMS Subject Classification (1991): 11M06

Values of the Riemann zeta function at integers

Distribution of the zeros of the Riemann Zeta function

One of the most celebrated problem of mathematics is the Riemann hypothesis which states that all the non trivial zeros of the Zeta-function lie on the critical line <(s) = 1/2. Even if this famous problem is unsolved for so long, a lot of things are known about the zeros of ζ(s) and we expose here the most classical related results : all the non trivial zeros lie in the critical strip, the num...