Related papers: Mean square of zeta function, circle problem and d…
We show that it is possible to approximate the zeta-function of a curve over a finite field by meromorphic functions which satisfy the same functional equation and moreover satisfy (respectively do not satisfy) the analogue of the Riemann…
In this paper we introduce new generalizations of the zeta function, the Tricomi functions; their main properties are studied. This opens the way to a deeper, better application of these functions both in the theory of special functions,…
Assume the Riemann hypothesis. On the right-hand side of the critical strip, we obtain an asymptotic formula for the discrete mean square of the Riemann zeta-function over imaginary parts of its zeros.
We have dealt with the Euler's alternating series of the Riemann zeta function to define a regularized ratio appeared in the functional equation even in the critical strip and showed some evidence to indicate the hypothesis. We briefly…
In this paper, we expand the theory of Weierstrassian elliptic functions by introducing auxiliary zeta functions $\zeta_\lambda$, zeta differences of first kind $\Delta_\lambda$ and second kind $\Delta_{\lambda,\mu}$ where…
This work is dedicated to the promotion of the results Hadamard, Landau E., Walvis A., Estarmann T and Paul R. Chernoff for pseudo zeta functions. The properties of zeta functions are studied, these properties can lead to new regularities…
It is proved that, for $T^\epsilon\le G = G(T) \le {1\over2}\sqrt{T}$, $$ \int_T^{2T}\Bigl(I_1(t+G)-I_1(t)\Bigr)^2 dt = TG\sum_{j=0}^3a_j\log^j \Bigl({\sqrt{T}\over G}\Bigr) + O_\epsilon(T^{1+\epsilon}+ T^{1/2+\epsilon}G^2) $$ with some…
We study the function $\Delta_k(x):=\sum_{n\leq x} d_k(n) - \mbox{Res}_{s=1} ( \zeta^k(s) x^s/s )$, where $k\geq 3$ is an integer, $d_k(n)$ is the $k$-fold divisor function, and $\zeta(s)$ is the Riemann zeta-function. For a large parameter…
We show that an almost trivial inequality for the first and second mean of a random variable can be used to give non-trivial improvements on deep results. As applications we improve on results on lower bounds for the Riemann zeta-function…
The aim of this paper is to show how zeta functions and excision in cyclic cohomology may be combined to obtain index theorems. In the first part, we obtain a local index formula for "abstract elliptic pseudodifferential operators"…
We examine an elliptic curve constructed in an earlier paper from a certain representation of $\SL(2,\Z)$ on the space of convergent Dirichlet series. The curve is observed to be a modular curve for $\Gamma^1(15)$ and a certain orbit of…
We study the Hurwitz-type analogue of Schur multiple zeta-functions involving shifting parameters. We extend various formulas, known for ordinary Schur multiple zeta-functions, to the case of Hurwitz type. We also mention unpublished…
In a recent paper, Saxena et al. [1] developed the solutions of three generalized fractional kinetic equations in terms of Mittag-Leffler functions. The object of the present paper is to further derive the solution of further generalized…
We define a generalized class of modified zeta series transformations generating the partial sums of the Hurwitz zeta function and series expansions of the Lerch transcendent function. The new transformation coefficients we define within…
We study the interplay between recurrences for zeta related functions at integer values, `Minor Corner Lattice' Toeplitz determinants and integer composition based sums. Our investigations touch on functional identities due to Ramanujan and…
Explicit bounds on the tails of the zeta function $\zeta$ are needed for applications, notably for integrals involving $\zeta$ on vertical lines or other paths going to infinity. Here we bound weighted $L^2$ norms of tails of $\zeta$. Two…
In this paper, we give a formula that connects two variants of multiple zeta values; multitangent functions and symmetric multiple zeta values. As an application of this formula, we give two results. First, we prove Bouillot's conjecture on…
We determine the classical approximation constants $w_{3}(\zeta),w_{3}^{\ast}(\zeta),\lambda_{3}(\zeta)$ such as the uniform constants $\widehat{w}_{3}(\zeta),\widehat{w}_{3}^{\ast}(\zeta),\widehat{\lambda}_{3}(\zeta)$ associated to real…
We obtain a new proof of Hurwitz's formula for the Hurwitz zeta function $\zeta(s, a)$ beginning with Hermite's formula. The aim is to reveal a nice connection between $\zeta(s, a)$ and a special case of the Lommel function $S_{\mu,…
We find a representation for the Maclaurin coefficients of the Hurwitz zeta-function in terms of semi-convergent series involving the Bernoulli polynomials and the Stirling numbers of the first kind. In particular, this gives a…