On the Lindelöf Hypothesis for the Riemann Zeta function and Piltz divisor problem

2024/06/01 by Lahoucine Elaissaoui, Elaissaoui, Lahoucine
Mathematics · #11B65 #11M06 #11N56 #30B10 #30B30 #30B40 #30B50 #Analytic Number Theory Research #Approximation Theory and Sequence Spaces #FOS: Mathematics #Mathematical Approximation and Integration #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2406.00331

openalex publication_date 2024/06/01 · openalex created_date 2024/06/06 · openalex updated_date 2026/07/28

Abstract

In order to well understand the behaviour of the Riemann zeta function inside the critical strip, we show; among other things, the Fourier expansion of the ζ^k(s) (k ∈ ℕ) in the half-plane \Re s > 1/2 and we deduce a necessary and sufficient condition for the truth of the Lindelöf Hypothesis. Moreover, if Δ_kdenotes the error term in the Piltz divisor problem then for almost all x≥ 1 and any given k ∈ ℕ we have Δ_k(x) = lim_{ρ→ 1^-}∑_n=0^+∞(-1)ⁿℓ_n,kL_n(log(x))ρⁿ where (ℓ_n,k)_n and L_n denote, respectively, the Fourier coefficients of ζ^k(s) and Laguerre polynomials.

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