Carleson's convergence theorem for Dirichlet series
2003 (English)In: Pacific Journal of Mathematics, ISSN 0030-8730, Vol. 208, no 1, 85-109 p.Article in journal (Refereed) Published
A Hilbert space of Dirichlet series is obtained by considering the Dirichlet series f(s) = Sigma(n=1)(infinity) a(n)n(-s) that satisfy Sigma(n=0)(infinity) \a(n)\(2) < +&INFIN;. These series converge in the half plane Re s > 1/2 and define a functions that are locally L-2 on the boundary Re s > 1/2. An analog of Carleson's celebrated convergence theorem is obtained: Each such Dirichlet series converges almost everywhere on the critical line Re s = 1/2. To each Dirichlet series of the above type corresponds a trigonometric series Sigma(n=1)(infinity) a(n)chi(n), where chi 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-infinity, 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 chi.
Place, publisher, year, edition, pages
2003. Vol. 208, no 1, 85-109 p.
IdentifiersURN: urn:nbn:se:kth:diva-22153ISI: 000180227100007OAI: oai:DiVA.org:kth-22153DiVA: diva2:340851
QC 201005252010-08-102010-08-10Bibliographically approved