Date of Award

Spring 5-15-2018

Author's School

Graduate School of Arts and Sciences

Author's Department


Degree Name

Doctor of Philosophy (PhD)

Degree Type



Hedenmalm, Lindqvist, and Seip in 1997 [11] revitalized the modern study of Dirichlet

series by defining the space H 2 and considering it as isometrically isomorphic to the Hardy

space of the infinite polytorus H 2 (T ∞ ). This allowed a new viewpoint to be applied to

classical theorems, including Carlson’s theorem about the integral in the mean of a Dirichlet

series. Carlson’s theorem holds only for vertical lines in the right half plane, and cannot be

extended to the boundary in full generality (as shown by Saksman and Seip in [14]). However,

Carlson’s theorem can be shown to hold on the imaginary axis for a more restrictive class of

Dirichlet series, and we shall do so.

The main result contained in this dissertation is a generalized version of Carlson’s the-

orem: given a Borel probability measure on the polytorus, a measure is constructed on the

imaginary axis so that the integral in the mean is equal to the integral on the polytorus.

Chapter 1 contains background material on Dirichlet series, including questions of con-

vergence, the Bohr lift, and spaces of Dirichlet series. Chapter 2 is the statement and proof

of the main result generalizing Carlson’s theorem.

Finally, Chapter 3 is a small result about weighted spaces of Dirichlet series from work

done jointly with Houry Melkonian.


English (en)

Chair and Committee

John E. McCarthy

Committee Members

Gregory Knese, Constanze Liaw, Xiang Tang, Brett Wick,


Permanent URL:

Included in

Mathematics Commons