Related papers: Approximate functional equation for the derivative…

In this article, we show that the Riemann hypothesis for an $L$-function $F$ belonging to the Selberg class implies that all the derivatives of $F$ can have at most finitely many zeros on the left of the critical line with imaginary part…

Let $f$ be a holomorphic cusp form of weight $k$ with respect to full modular group $SL_2(\mathbb{Z})$ satisfying a normalized Hecke eigenform, $L_f(s)$ the $L$-function attached to the form $f$. Good gave the approximate functional…

We obtain the approximate functional equation for the Rankin-Selberg zeta-function on the 1/2-line.

Schr\"{o}dinger equations in \emph{functional derivatives} are solved via quantized Galerkin limit of antinormal functional Feynman integrals for Schr\"{o}dinger equations in \emph{partial derivatives

We derive new approximate representations of the Lommel functions in terms of the Scorer function and approximate representations of the first derivative of the Lommel functions in terms of the derivative of the Scorer function. Using the…

For functions defined via Dirichlet/generalized Dirichlet series in some half planes of the complex plane, we give a new simple elementary approach to obtain an Approximate Functional Equation(AFE for short) for the product of functions…

Given a degree 1 function $F\in\mathcal{S}^{\sharp}$ and a real number $\alpha$, we consider the linear twist $F(s,\alpha)$, proving that it satisfies a functional equation reflecting $s$ into $1-s$, which can be seen as a Hurwitz-Lerch…

The standard twist $F(s,\alpha)$ of $L$-functions $F(s)$ in the Selberg class has several interesting properties and plays a central role in the Selberg class theory. It is therefore natural to study its finer analytic properties, for…

In this paper we prove that the Generalized Riemann Hypothesis (GRH) for functions in the class $\mathcal{S}^{\sharp\flat}$ containing the Selberg class is equivalent to a certain integral expression of the real part of the generalized Li…

Assuming the Generalized Riemann Hypothesis, we provide uniform upper and lower bounds with explicit main terms for $\log{\left|\cL(s)\right|}$ for $\sigma \in (1/2,1)$ and for functions in the Selberg class. In particular, we focus on the…

We give a closed formula for the derivative of arbitrary order of the function $\ds g(x)=\exp(f(x))$.

We present a streamlined account of a recent theorem on the classification of the $L$-functions of degree 2 and conductor 1 from the extended Selberg class. We also present a more general new result dealing with functional equations…

A simple proof is given of the known fact that an m-times continuously differentiable function on the real line can be approximated along with its derivatives by an entire function and its respective derivatives.

We obtain a new decomposition of the Riemann-Liouville operators of fractional integration as a series involving derivatives (of integer order). The new formulas are valid for functions of class $C^n$, $n \in \mathbb{N}$, and allow us to…

In this series, we investigate the calculation of mean values of derivatives of Dirichlet $L$-functions in function fields using the analogue of the approximate functional equation and the Riemann Hypothesis for curves over finite fields.…

We give a full description of the functions $F$ of degree 2 and conductor 1 in the general framework of the extended Selberg class. This is performed by means of a new numerical invariant $\chi_F$, which is easily computed from the data of…

In 1874, Mertens proved the approximate formula for partial Euler product for Riemann zeta function at $s=1$, which is called Mertens' theorem. In this paper, we generalize Mertens' theorem for Selberg class and show the prime number…

In certain classes of subharmonic functions u on C distinguished in terms of lower bounds for the Riesz measure of u, a sharp estimate is obtained for the rate of approximation by functions of the form log |f(z)|, where f is an entire…

Recently, Garunk\v{s}tis, Laurin\v{c}ikas, Matsumoto, J. & R. Steuding showed an effective universality-type theorem for the Riemann zeta-function by using an effective multi-dimensional denseness result of Voronin. We will generalize…

We study some "density function" related to the value-distribution of $L$-functions. The first example of such a density function was given by Bohr and Jessen in 1930s for the Riemann zeta-function. In this paper, we construct the density…