相关论文: On the analytic continuation of the Riemann zeta f…

In this paper, we will give a new proof for a known result of the mean square of Riemann zeta-function.

Several results are obtained concerning multiplicities of zeros of the Riemann zeta-function $\zeta(s)$. They include upper bounds for multiplicities, showing that zeros with large multiplicities have to lie to the left of the line $\sigma…

Associated to classical semi-simple groups and their maximal parabolics are genuine zeta functions. Naturally related to Riemann's zeta and governed by symmetries, including that of Weyl, these zetas are expected to satisfy the Riemann…

Assuming the Riemann Hypothesis we show that there exist infinitely many consecutive zeros of the Riemann zeta-function whose gaps are greater than 2.9 times the average spacing.

This article describes a sequence of rational functions which converges locally uniformly to the zeta function. The numerators (and denominators) of these rational functions can be expressed as characteristic polynomials of matrices that…

We define a generalisation of the completed Riemann zeta function in several complex variables. It satisfies a functional equation, shuffle product identities, and has simple poles along finitely many hyperplanes, with a recursive structure…

This brief note explicates some mathematical details of Phys. Rev. Lett. 118, 130201 (2017), by showing how a version of the operator of that paper can be rigorously constructed on a well-defined linear space of functions.

Assuming the Riemann Hypothesis we obtain an upper bound for the moments of the Riemann zeta-function on the critical line. Our bound is nearly as sharp as the conjectured asymptotic formulae for these moments. The method extends to moments…

We consider the known functional identity on the Weierstrass sigma function. A complete classification of odd entire functions which satisfy the same identity is obtained.

In this paper, for any positive integer $\ell\geq2,$ we define $\ell$-generalized Fibonacci zeta function. We then study its analytic continuation to the whole complex plane $\mathbb{C}.$ Further, we compute a possible list of singularities…

The GUE Hypothesis, which concerns the distribution of zeros of the Riemann zeta-function, is used to evaluate some integrals involving the logarithmic derivative of the zeta-function. Some connections are shown between the GUE Hypothesis…

We develop approximations for the Riemann zeta function that enable high-precision computation within the critical strip and other vertical strips. These approximations combine the main sum of the Riemann-Siegel formula with a simple…

By some hypergeometric summation theorems, the authors establish a series of new infinite summation formulas involving generalized harmonic numbers related to Riemann-Zeta function, with three different patterns.

Making use of inverse Mellin transform techniques for analytical continuation, an elegant proof and an extension of the zeta function regularization theorem is obtained. No series commutations are involved in the procedure; nevertheless the…

An analog of the Riemann hypothesis is proved in this paper. Some new integral equations for the functions $\pi(x)$ and $R(x)$ follows. A new effect that is shown is that these function - with essentially different behavior - are the…

In this paper we give some interesting identities between Euler numbers and zeta functions. Finally we will give the new values of Euler zeta function at positive even integers.

We build on a recent paper on Fourier expansions for the Riemann zeta function. We establish Fourier expansions for certain $L$-functions, and offer series representations involving the Whittaker function $W_{\gamma,\mu}(z)$ for the…

We introduce a new generalization of Stirling numbers of the second kind and analyze their properties, including generating functions, integral representations, and recurrence relations. These numbers are used to approximate Riemann zeta…

We express the Riemann zeta function $\zeta\left(s\right)$ of argument $s=\sigma+i\tau$ with imaginary part $\tau$ in terms of three absolutely convergent series. The resulting simple algorithm allows to compute, to arbitrary precision,…

From eigensolutions of the harmonic oscillator or Kepler-Coulomb Hamiltonian we extend the functional equation for the Riemann zeta function and develop integral representations for the Riemann xi function that is the completed classical…