Ikehara-type theorem involving boundedness

TL;DR

This paper establishes a precise condition involving boundary behavior of Dirichlet series for the boundedness of normalized partial sums, linking it to convergence of associated pseudomeasures and Fourier transforms.

Contribution

It provides a necessary and sufficient condition for the boundedness of partial sums of Dirichlet series, connecting boundary limits to pseudomeasures and Fourier analysis.

Findings

Boundedness of (s_N)/N characterized by convergence to a pseudomeasure.

Optimal estimate for (s_N)/N under boundedness of (1-x)f(x).

Connection between Dirichlet series boundary behavior and distributional Fourier transforms.

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 a_1+...+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.