Related papers: On Fourier and Zeta(s)
A formula connecting a moment of L-functions and a dual moment in a way that interchanges the roles of certain key parameters on both sides is known as a reciprocity relation. We establish a reciprocity relation for a first moment of GL(2)…
In this note we define L-functions of finite graphs and study the particular case of finite cycles in the spirit of a previous paper that studied spectral zeta functions of graphs. The main result is a suggestive equivalence between an…
We propose two types of extensions to Hamburger's theorems on the Dirichlet series with functional equation like the one of the Riemann zeta function, under weaker hypotheses. This builds upon the dictionary betweeen the moderate…
In this paper we set up the theory of acid zeta function and ajoint acid zeta function, based on the theory, we point out a reason to doubt the truth of the Riemann hypothesis and also as a consequence, we give out some new RH equivalences.
A well-known result, due to Dirichlet and later generalized by de la Vallee-Poussin, expresses a relationship between the sum of fractional parts and the Euler-Mascheroni constant. In this paper, we prove an asymptotic relationship between…
In this paper, we study a more general pair correlation function, $F_h(x,T)$, of the zeros of the Riemann zeta function. It provides information on the distribution of larger differences between the zeros.
In this paper we define and study a Dedekind-like zeta function for the algebra of multicomplex numbers. By using the idempotent representations for such numbers, we are able to identify this zeta function with the one associated to a…
We study relationships between spinor representations of certain Lie algebras and Lie superalgebras of differential operators on the circle and values of $\zeta$--functions at the negative integers. By using formal calculus techniques we…
This is the first of four papers that study algebraic and analytic structures associated to the Lerch zeta function. This paper studies "zeta integrals" associated to the Lerch zeta function using test functions, and obtains functional…
We study the distributions of values of the logarithmic derivatives of the Dedekind zeta functions on a fixed vertical line. The main object is determining and investigating the density functions of such value-distributions for any…
We study flat deformations of quotients of a polynomial algebra in a class of graded commutative associative algebras. Functional equations and their solutions in terms of theta functions play important role in these studies. An analog of…
As automorphic $L$-functions or Artin $L$-functions, several classes of $L$-functions have Euler products and functional equations. In this paper we study the zeros of $L$-functions which have the Euler products and functional equations. We…
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…
We present a new expansion of the zeta-function of Riemann. The current formalism -- which combines both the idea of interpolation with constraints and the concept of hypergeometric functions -- can, in a natural way, be generalised within…
Numerical investigations around a transformation of Landau's formula suggest certain statistical regularities in the distribution of zeros of the Riemann zeta function.
This paper uses cybernetic approach to study behavior of the Riemann zeta function. It is based on the elementary cybernetic concepts like feedback, transfer functions, time delays, PI (Proportional--Integral) controllers or FOPDT (First…
We review generalized zeta functions built over the Riemann zeros (in short: "superzeta" functions). They are symmetric functions of the zeros that display a wealth of explicit properties, fully matching the much more elementary Hurwitz…
We point out an interesting occurrence of the sine kernel in connection with the shifted moments of the Riemann zeta function along the critical line. We discuss rigorous results in this direction for the shifted second moment and for the…
Suppose $Y$ is a regular covering of a graph $X$ with covering transformation group $\pi = \mathbb{Z}$. This paper gives an explicit formula for the $L^2$ zeta function of $Y$ and computes examples. When $\pi = \mathbb{Z}$, the $L^2$ zeta…
Fourier sine transforms containing irrational integrands are presented. Explicit closed form expressions are shown to be related to Lommel functions and in special cases to the Fresnel integrals. Such integrals arise in the semi-classical…