Related papers: On the $a$-points of the derivatives of the Rieman…
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.…
We prove an equivalent of the Riemann hypothesis in terms of the functional equation (in its asymmetrical form) and the $a$-points of the zeta-function, i.e., the roots of the equation $\zeta(s)=a$, where $a$ is an arbitrary fixed complex…
In this paper, we present some results on the $a$-points of the symmetric sum of the Euler-Zagier multiple zeta function. Our first three results are for the $a$-points free region of the function. The fourth result is the Riemann-von…
For any $a\in\mathbb{C}$, the zeros of $\zeta(s)-a$, denoted by $\rho_a=\beta_a+i\gamma_a$, are called $a$-points of the Riemann zeta function $\zeta(s)$. In this paper, we reformulate some basic results about the $a$-points of $\zeta(s)$…
Fujii investigated the uniform distribution of various sequences associated with the non-trivial zeros of the Riemann zeta function by evaluating certain exponential sums over these zeros. In this paper, we present analogous results for a…
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…
We study Riemann-type functional equations with respect to value-distribution theory and derive implications for their solutions. In particular, for a fixed complex number $a\neq0$ and a function from the Selberg class $\mathcal{L}$, we…
A simple and elementary derivation of values at integer points for the Riemann's zeta and related functions is reported.
The derivative of the Riemann zeta function was computed numerically on several large sets of zeros at large heights. Comparisons to known and conjectured asymptotics are presented.
Four propositions are considered concerning the relationship between the zeros of two combinations of the Riemann zeta function and the function itself. The first is the Riemann hypothesis, while the second relates to the zeros of a…
The Riemann hypothesis, stating that the real part of all non-trivial zero points fo the zeta function must be $\frac{1}{2}$, is one of the most important unproven hypothesises in number theory. In this paper we will proof the Riemann…
In this paper we give criteria about estimation of derivatives of the Riemann Zeta Function on the line $\sigma=1$.
We show that the higher derivatives of the Riemann zeta function may be expressed in terms of integrals involving the digamma function. Related integrals for the Stieltjes constants are also shown. We also present a formula for the…
In one of his posthumous papers, conserved in G\"ottingen, Riemann considers the derivatives of $\log\zeta(s)$ at the point $1/2$, giving explicit values for them. Around 2010 we shared Riemann's value of the second derivative with some…
We show that the $n$th derivative of the Riemann zeta function, when summed over the non-trivial zeros of zeta, is real and positive/negative in the mean for $n$ odd/even, respectively. We show this by giving a full asymptotic expansion of…
We prove a formula for the Mangoldt function which relates it to a sum over all the non-trivial zeros of the Riemann zeta function, in addition we analize a truncated version of it.
Assuming the Riemann hypothesis, we obtain upper and lower bounds for moments of the Riemann zeta-function averaged over the extreme values between its zeros on the critical line. Our bounds are very nearly the same order of magnitude. The…
Assuming the Riemann hypothesis, we obtain a formula for the mean value of the $k$-derivative of $\zeta'/\zeta$, depending on the pair correlation of zeros of the Riemann zeta-function. This formula allows us to obtain new equivalences to…
The Riemann hypothesis, conjectured by Bernhard Riemann in 1859, claims that the non-trivial zeros of $\zeta(s)$ lie on the line $\Re(s) =1/2$. The density hypothesis is a conjectured estimate $N(\lambda, T) =O\bigl(T\sp{2(1-\lambda)…
We compute the one-level density of the non-trivial zeros of the Riemann zeta-function weighted by $|\zeta(\frac12+it)|^{2k}$ for $k=1$ and, for test functions with Fourier support in $(-\frac12,\frac12)$, for $k=2$. As a consequence, for…