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

2020/08/07 by Johan Andersson, Andersson, Johan
Computer Science · Mathematics · Physics and Astronomy · #11M #Advanced Algebra and Geometry #Advanced Mathematical Theories and Applications #Complex Variables (math.CV) #FOS: Mathematics #Matrix Theory and Algorithms #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2008.03041

openalex publication_date 2020/08/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the problem on how to get good lower estimates for the integral ∫_T^T+H |ζ(σ+it)| dt, when H ≪ 1 is small and σ 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.

Citations

• Non universality on the critical line
• On a problem of Ramachandra and approximation of functions by Dirichlet polynomials with bounded coefficients
• On generalized Hardy classes of Dirichlet series

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