A problem of Erdös on sums of two squarefull numbers
A relation between the Riemann zeta-function and the hyperbolic laplacian
A remark on product of Dirichlet L-functions
While trying to understand the methods and the results of [3], especially in Section 2, we stumbled on an identity (*) below, which looked worth recording since we could not locate it in the literature. We would like to thank Dinesh Thakur and Dipendra Prasad for their comments.
A scattering theory for automorphic functions
A simple proof of the mean fourth power estimate for and
A singular series average and Goldbach numbers in short intervals
A statistical density theorem for L-functions with applications
A study of the mean value of the error term in the mean square formula of the Riemann zeta-function in the critical strip
Let be the error term in the mean square formula of the Riemann zeta-function in the critical strip . It is an analogue of the classical error term . The research of has a long history but the investigation of is quite new. In particular there is only a few information known about for . As an exploration, we study its mean value . In this paper, we give it an Atkinson-type series expansion and explore many of its properties as a function of .
A variant of Selberg's asymptotic formula.
A zero density result for the Riemann zeta function
We prove an explicit bound for N(σ,T), the number of zeros of the Riemann zeta function satisfying ℜ𝔢 s ≥ σ and 0 ≤ ℑ𝔪 s ≤ T. This result provides a significant improvement to Rosser's bound for N(T) when used for estimating prime counting functions.
A χ-analogue of a formula of Ramanujan for ζ(1/2)
Addition of : application to arithmetic.
An alternative approach to a theorem of Tom Meurman
An Alternative Form of the Functional Equation for Riemann’s Zeta Function, II
This paper treats about one of the most remarkable achievements by Riemann, that is the symmetric form of the functional equation for . We present here, after showing the first proof of Riemann, a new, simple and direct proof of the symmetric form of the functional equation for both the Eulerian Zeta function and the alternating Zeta function, connected with odd numbers. A proof that Euler himself could have arranged with a little step at the end of his paper “Remarques sur un beau rapport entre...
An application of Mellin-Barnes' type integrals to the mean square of Lerch zeta-functions.
We shall establish full asymptotic expansions for the mean squares of Lerch zeta-functions, based on F. V. Atkinson's device. Mellin-Barnes' type integral expression for an infinite double sum will play a central role in the derivation of our main formulae.
An application of Mellin-Barnes type integrals to the mean square of Lerch zeta-functions (II).
For the Lerch zeta-function Φ(s,x,λ) defined below, the multiple mean square of the form (1.1), together with its discrete and Irbid analogues, (1.2) and (1.3) are investigated by means of Atkinson's [2] dissection method applied to the product Φ(u,x,λ)Φ(υ,x,-λ), where u and υ are independent complex variables (see (4.2)). A complete asymptotic expansion of (1.1) as Im s → ±∞ is deduced from Theorem 1, while those of (1.2) and (1.3) as q → ∞ and (at the same time) as Im s → ±∞ are deduced from Theorems...
An application of the Hooley-Huxley contour
An arithmetical mapping and applications to Ω-results for the Riemann zeta function
An asymptotic series for the mean value of Dirichlet L-functions.
An effective order of Hecke-Landau zeta functions near the line σ = 1. I