On the multiplicity of zeros of the zeta-function

We investigate bounds for the multiplicities $m(\beta+i\gamma)$, where $\beta+i\gamma\,$ ($\beta\ge \1/2, \gamma>0)$ denotes complex zeros of $\zeta(s)$. It is seen that the problem can be reduced to the estimation of the integrals of the zeta-function over "very short" intervals. A new, explicit bound for $m(\beta+i\gamma)$ is also derived, which is relevant when $\beta$ is close to unity. The related Karatsuba conjectures are also discussed.

In this article, we prove an explicit bound for $N(\sigma,T)$, the number of zeros of the Riemann zeta function satisfying $\sigma < \Re s <1 $ and $0 < \Im s < T$. This result provides a significant improvement over Rosser's bound for $N(T)$ when used for estimating prime counting functions. For instance this is applied to obtain new bounds for $\psi(x)$ (arXiv:1310.6374).

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

Let $N(\sigma,T)$ denote the number of nontrivial zeros of the Riemann zeta function with real part greater than $\sigma$ and imaginary part between $0$ and $T$. We provide explicit upper bounds for $N(\sigma,T)$ commonly referred to as a zero density result. In 1937, Ingham showed the following asymptotic result $N(\sigma,T)=\mathcal{O} ( T^{\frac83(1-\sigma)} (\log T)^5 )$. Ramar\'{e} recently proved an explicit version of this estimate. We discuss a generalization of the m...

This paper is divided into two independent parts. The first part presents new integral and series representations of the Riemaan zeta function. An equivalent formulation of the Riemann hypothesis is given and few results on this formulation are briefly outlined. The second part exposes a totally different approach. Using the new series representation of the zeta function of the first part, exact information on its zeros is provided.

The main aim of this paper is twofold. First we generalize, in a novel way, most of the known non-vanishing results for the derivatives of the Riemann zeta function by establishing the existence of an infinite sequence of regions in the right half-plane where these derivatives cannot have any zeros; and then, in the rare regions of the complex plane that do contain zeros of the k-th derivative of the zeta function, we describe a unexpected phenomenon, which implies great regu...

The paper reviews existing results about the statistical distribution of zeros for the three main types of zeta functions: number-theoretical, geometrical, and dynamical. It provides necessary background and some details about the proofs of main results.

We give results on zeros of a polynomial of $\zeta(s),\zeta'(s),\ldots,\zeta^{(k)}(s)$. First, we give a zero free region and prove that there exist zeros corresponding to the trivial zeros of the Riemann zeta function. Next, we estimate the number of zeros whose imaginary part is in $(1,T)$. Finally, we study the distribution of the real part and the imaginary part of zeros, respectively.

Let $\pi S(t)$ denote the argument of the Riemann zeta-function at the point $\frac12+it$. Assuming the Riemann Hypothesis, we sharpen the constant in the best currently known bounds for $S(t)$ and for the change of $S(t)$ in intervals. We then deduce estimates for the largest multiplicity of a zero of the zeta-function and for the largest gap between the zeros.

In a letter to Weierstrass Riemann asserted that the number $N_0(T)$ of zeros of $\zeta(s)$ on the critical line to height $T$ is approximately equal to the total number of zeros to this height $N(T)$. Siegel studied some posthumous papers of Riemann trying to find a proof of this. He found a function $\mathop{\mathcal R }(s)$ whose zeros are related to the zeros of the function $\zeta(s)$. Siegel concluded that Riemann's papers contained no ideas for a proof of his assertion...