Carleson’s convergence theorem for Dirichlet series

Håkan Hedenmalm and Eero Saksman

Vol. 208 (2003), No. 1, 85–109

DOI: 10.2140/pjm.2003.208.85

Abstract

A Hilbert space of Dirichlet series is obtained by considering the Dirichlet series f(s) = ∑ _n=1^∞a_nn^−s that satisfy ∑ _n=0^∞|a_n|² < +∞. These series converge in the half plane Res > and define a functions that are locally L² on the boundary Res = . An analog of Carleson’s celebrated convergence theorem is obtained: Each such Dirichlet series converges almost everywhere on the critical line Res = . To each Dirichlet series of the above type corresponds a “trigonometric” series ∑ _n=1^∞a_nχ(n), where χ is a multiplicative character from the positive integers to the unit circle. The space of characters is naturally identified with the infinite-dimensional torus 𝕋^∞, where each dimension comes from a a prime number. The second analog of Carleson’s theorem reads: The above “trigonometric” series converges for almost all characters χ.

Milestones

Received: 25 May 2001

Published: 1 January 2003