On the Balasubramanian-Ramachandra method close to Re(s)=1

Abstract

We study the problem on how to get good lower estimates for the integral $${\int }_T^{T+H}|\zeta (\sigma +it)|dt,$$ when $H\ll 1$ is small and $\sigma$ is close to $1$, as well as related integrals for other Dirichlet series, by using ideas related to the Balasubramanian-Ramachandra method. We use kernel-functions constructed by the Paley-Wiener theorem as well as the kernel function of Ramachandra. We also notice that the Fourier transform of Ramachandra's Kernel-function is in fact a $K$-Bessel function. This simplifies some aspects of Balasubramanian-Ramachandra method since it allows use of the theory of Bessel-functions.

15 pages

References in corpus (4)

• On the multiplicity of zeros of the zeta-function
• On the zeta function on the line Re(s) = 1
• On generalized Hardy classes of Dirichlet series
• On a problem of Ramachandra and approximation of functions by Dirichlet polynomials with bounded coefficients