Ikehara-type theorem involving boundedness

2008/07/03 by Jacob Korevaar, Korevaar, Jacob
Mathematics · #40E05 #Advanced Harmonic Analysis Research #Analytic Number Theory Research #FOS: Mathematics #Holomorphic and Operator Theory #Number Theory (math.NT) #math.NT #msc:40E05

paper · pdf · doi:10.48550/arxiv.0807.0537

6 pages

arxiv created 2008/07/03 · openalex publication_date 2008/07/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Consider any Dirichlet series sum a_n/n^z with nonnegative coefficients a_n and finite sum function f(z)=f(x+iy) when x is greater than 1. Denoting the partial sum a₁+...+a_N by s_N, the paper gives the following necessary and sufficient condition in order that (s_N)/N remain bounded as N goes to infinity. For x tending to 1 from above, the quotient q(x+iy)=f(x+iy)/(x+iy) must converge to a pseudomeasure q(1+iy), the distributional Fourier transform of a bounded function. The paper also gives an optimal estimate for (s_N)/N under the "real condition" that (1-x)f(x) remain bounded as x tends to 1 from above.

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