The Riemann zeta-function is employed to generate universally overconvergent power series.

In this paper, we apply the saddle-point method in conjunction with the theory of the Nörlund-Rice integrals to derive precise asymptotic formula for the generalized Li coefficients established by Omar and Mazhouda. Actually, for any function $F$ in the Selberg class $\mathcal{S}$ and under the Generalized Riemann Hypothesis, we have $$ \lambda_{F}(n)=\frac{d_{F}}{2}n\log n+c_{F}n+O(\sqrt{n}\log n), $$ with $$ c_{F}=\frac{d_{F}}{2}(\gamma-1)+\frac{1}{2}\log(\lambda Q_{F}^{2}),\ \lambda=\prod_{j=1}^{r}\lambda_{j}^{2\lambda_{j}}, $$ where $\gamma$ is the Euler's constant and the notation is as below.

In this paper we derive formulas for summation of series involving J.~Bourget's generalization of Bessel functions of integer order, as well as the analogous generalizations by H.~M.~Srivastava. These series are expressed in terms of the Riemann $\z$ function and Dirichlet functions $\eta$, $\la$, $\b$, and can be brought into closed form in certain cases, which means that the infinite series are represented by finite sums.

We investigate the problem of determining elements in the Selberg class by means of their Dirichlet series coefficients at primes.

In this paper, we study the modularity of certain elliptic surfaces by determining their $L$-series through their monodromy groups.

We define generalized Mellin transforms of mixed cusp forms, show their convergence, and prove that the function obtained by such a Mellin transform of a mixed cusp form satisfies a certain functional equation. We also prove that a mixed cusp form can be identified with a holomorphic form of the highest degree on an elliptic variety.