2012/07/19 by Johan Andersson, Andersson, Johan · 1 citation
Mathematics · #11M #Analytic Number Theory Research #Analytic and geometric function theory #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1207.4624
openalex publication_date 2012/07/19 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
We prove effective results on when a function can be approximated by a Dirichlet polynomial with bounded coefficients. Assuming that Φ(n) is an increasing function we prove that the set of polynomials ∑n=2N an nit-1: N ≥ 2, |an| ≤ Φ(n), is dense in L2(0,H) if and only if ∑n=2^∞ \fraclog Φ(n) n log2 n = ∞. We also prove variants of this result for generalized Dirichlet polynomials. The main tools are theorems of Paley and Wiener related to quasianalyticity and the Pechersky rearrangement theorem. We use this result to give precise conditions on when a conjecture of Ramachandra is true and when it is false. We prove that whenever Φ(n) is a positive increasing function then limN → ∞ min|an| ≤ Φ(n) ∫0H \abs1+∑n=2N an nit-12 dt =0, if and only if the above sum is divergent. This has applications on lower bounds for moments of the Riemann zeta-functions in short intervals close to Re(s)=1, and to questions of Universality for zeta-functions on and close to their abscissa of convergence.