Related papers: On Dirichlet series and functional equations
We consider the multiple Dirichlet series associated to the $k$th moment of real Dirichlet $L$-functions, and prove that it has a meromorphic continuation to a specific region in $\mathbb{C}^{k+1}$, which is conditional under the…
In this paper I give an evaluation of a functional integral by means of a series in functional derivatives, first of all we propose a differential equation of first order and solve it by iterative methods, to obtain a series for the…
We propose a refinement of the random matrix model for a certain family of $L$-functions over $\mathbb F_q[u]$, using techniques that we hope will eventually apply to an arbitrary family of $L$-functions. This consists of a probability…
The Lambert W function has utility for solving various exponential and logarithmic equations arranged in the form of $g(x)e^{g(x)}$. Using the Lambert W function and tetration, a variety of categorized inversion formulas are presented.…
Report II is concerned with the extended results of distance function wavelets (DFW). The fractional DFW transforms are first addressed relating to the fractal geometry and fractional derivative, and then, the discrete Helmholtz-Fourier…
We establish an integral representation for the Dirichlet generating function of the coefficients of Euler's pentagonal number theorem. The Bromwich-type integral enables analytic continuation to the entire complex plane, filling a gap in…
The characteristic functions of multivariate Feller processes with generator of affine type, and with smooth symbol functions have an explicit representation in terms of power series with rational number coefficients and with monmoms…
One of the main objectives of the current paper is to revisit the well known Laurent series expansions of the Riemann zeta function $\zeta(s)$, Hurwitz zeta function $\zeta(s,a)$ and Dirichlet $L$-function $L(s,\chi)$ at $s=1$. Moreover, we…
We consider a Dirichlet series $D(F,G;s)$ attached to two automorphic forms $F$ and $G$ of an orthogonal group of real signature $(2,4)$, involving their Fourier--Jacobi coefficients. When $F$ is a Hecke eigenform and $G$ a lift of a…
Solutions of nonlinear functional equations are generally not expressed as a finite number of combinations and compositions of elementary and known special functions. One of the approaches to study them is, firstly, to find formal solutions…
In reaction rate theory, in production-destruction type models and in reaction-diffusion problems when the total derivatives are replaced by fractional derivatives the solutions are obtained in terms of Mittag-Leffler functions and their…
We investigate a Dirichlet series involving the Fourier-Jacobi coefficients of two cusp forms $F,G$ for orthogonal groups of signature $(2,n+2)$. In the case when $F$ is a Hecke eigenform and $G$ is a Maass lift of a Poincar\'e series, we…
Using the method of multiple Dirichlet series, we develop L-functions ratios conjecture with one shift in both the numerator and denominator in certain ranges for quadratic families of Dirichlet and Hecke L-functions of primerelated moduli…
In this work at first the relation the Mittag-Lefler function to the exponential is given. The results are applied to the construction of the solution of Cauchy problem for ordinary linear operator differential equations with constant…
In this paper we study the mean values and zeroes of Dirichlet series of a view $\sum_{n}a_n n^{-s}$ with complex coefficients. There was introduced some class of Dirichlet series including such widely used series as the Riemann…
This paper is concerned with the study of the fractional finite sums theory. We present the classes of functions for which it is possible to characterize the constant related to the derivative of fractional sums (denominated by essence of a…
Given $k\in\mathbb N$, we study the vanishing of the Dirichlet series $$D_k(s,f):=\sum_{n\geq1} d_k(n)f(n)n^{-s}$$ at the point $s=1$, where $f$ is a periodic function modulo a prime $p$. We show that if $(k,p-1)=1$ or $(k,p-1)=2$ and…
Using Cauchy's Integral Theorem as a basis, what may be a new series representation for Dirichlet's function $\eta(s)$, and hence Riemann's function $\zeta(s)$, is obtained in terms of the Exponential Integral function $E_{s}(i\kappa)$ of…
This paper deals with the solution of unified fractional reaction-diffusion systems. The results are obtained in compact and elegant forms in terms of Mittag-Leffler functions and generalized Mittag-Leffler functions, which are suitable for…
We prove that the Dirichlet $L$-functions associated with Dirichlet characters in $\mathbb{F}_{q}[x]$ are universal. That is, given a modulus of high enough degree, $L$-functions with characters to this modulus can be found that approximate…