相关论文: On sums of integrals of powers of the zeta-functio…

We investigate the Mellin transforms of \(1/\operatorname{arctanh} x\) and \(1/(\sqrt{1-x^{2}}\,\operatorname{arctanh} x)\), viewed as compactly supported functions on \((0,1)\). These transforms are closely connected with conjectures on…

We improve the universality theorem of the Riemann zeta-function in short intervals by establishing universality for significantly shorter intervals $[T,T+H]$. Assuming the Riemann Hypothesis, we prove that universality in such short…

In the context of mod-Gaussian convergence, as defined previously in our work with J. Jacod, we obtain lower bounds for local probabilities for a sequence of random vectors which are approximately Gaussian with increasing covariance. This…

We describe a numerical algorithm for evaluating the numbers of roots minus the number of poles contained in a region based on the argument principle with the function of interest being written as a Mellin transformation of a usually…

Assuming the Riemann hypothesis, we investigate the shifted moments of the zeta function \[ M_{\alpha,{\beta}}(T) = \int_T^{2T} \prod_{k = 1}^m |\zeta(\tfrac{1}{2} + i (t + \alpha_k))|^{2 \beta_k} dt \] introduced by Chandee, where…

This paper gives some results for the logarithm of the Riemann zeta-function and its iterated integrals. We obtain a certain explicit approximation formula for these functions. The formula has some applications, which are related with the…

We prove that for $s=\sigma+it$ with $\sigma\ge0$ and $0<t\le x$, we have \[\zeta(s)=\sum_{n\le x}n^{-s}+\frac{x^{1-s}}{(s-1)}+\Theta\frac{29}{14} x^{-\sigma},\qquad \frac{29}{14}=2.07142\dots\] where $\Theta$ is a complex number with…

This note introduces a new range of modified gamma and beta $k$ functions. The authors present new modified gamma and beta $k$-functions, first and second summation relations, various functionals, Mellin transforms, and integral…

Conditionally on the Riemann Hypothesis we obtain bounds of the correct order of magnitude for the 2k-th moment of the Riemann zeta-function for all positive real k < 2.181. This provides for the first time an upper bound of the correct…

We consider the sum $\sum 1/\gamma$, where $\gamma$ ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in an interval $(0,T]$, and consider the behaviour of the sum as $T \to\infty$. We show that, after subtracting a…

We intimate deeper connections between the Riemann zeta and gamma functions than often reported and further derive a new formula for expressing the value of $\zeta(2n+1)$ in terms of zeta at other fractional points. This paper also…

We study the incomplete Mellin transformation of the fractional part and the related log-sine function when composed by an affine complex map. We evaluate the corresponding integral in two different ways which yields equalities with series…

Suppose $K$ is a number field and $a_K(m)$ is the number of integral ideals of norm equal to $m$ in $K$, then for any integer $l$, we asymptotically evaluate the sum \[ \sum_{m\leqslant T} a_K^l(m) \] as $T\to\infty$. We also consider the…

Using a polylogarithmic identity, we express the values of $\zeta$ at odd integers $2n+1$ as integrals over unit $n-$dimensional hypercubes of simple functions involving products of logarithms. We also prove a useful property of those…

Assuming the Riemann Hypothesis, we provide explicit upper bounds for moduli of $S(t)$, $S_1(t)$, and $\zeta\left(1/2+\mathrm{i}t\right)$ while comparing them with recently proven unconditional ones. As a corollary we obtain a conditional…

An overview of results and problems concerning the asymptotic formula for $\int_0^T|\zeta(1/2+it)|^4dt$ is given, together with a discussion of modern methods from spectral theory used in recent work on this subject.

Already in 1734 Euler found a short explicit formula for the value of Riemann zeta function Zeta(s) when the argument s equals a positive integer 2n where n=1,2,3,. No such formula exists for odd positive integer arguments of Zeta. The…

We consider finite iterated generalized harmonic sums weighted by the binomial $\binom{2k}{k}$ in numerators and denominators. A large class of these functions emerges in the calculation of massive Feynman diagrams with local operator…

An asymptotic formula for $$ \int_{T/2}^{T}Z^2(t)Z(t+U)\,dt\qquad(0< U = U(T) \le T^{1/2-\varepsilon}) $$ is derived, where $$ Z(t) := \zeta(1/2+it){\bigl(\chi(1/2+it)\bigr)}^{-1/2}\quad(t\in\Bbb R), \quad \zeta(s) = \chi(s)\zeta(1-s) $$ is…

By applying the inverse Mellin transform to some simple closed form identities, a number of relationships are established that connect integrals containing Riemann's and Hurwitz' zeta functions ($\zeta(s)$ and $\zeta(s,a)$) and their…