Date of Award
Doctor of Philosophy (PhD)
Hedenmalm, Lindqvist, and Seip in 1997  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 ). 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.
Chair and Committee
John E. McCarthy
Gregory Knese, Constanze Liaw, Xiang Tang, Brett Wick,
Sargent, Meredith, "Carlson's Theorem for Different Measures" (2018). Arts & Sciences Electronic Theses and Dissertations. 1575.