The Riemann zeta function ζ(s) in the real variable s was introduced by L. Eu-
ler (1737) in connection with questions about the distribution of prime numbers.
Later B. Riemann (1859) derived deeper results about the prime numbers by consid-
ering the zeta function in the complex variable. He revealed a dual correspondence
between the primes and the complex zeros of ζ(s), which started a theory to be
developed by the greatest minds in mathematics. Riemann was able to provide
proofs of his most fundamental observations, except for one, which asserts that all
the non-trivial zeros of ζ(s) are on the line Re s =
. This is the famous Riemann
Hypothesis – one of the most important unsolved problems in modern mathematics.
These lecture notes cover closely the material which I presented to graduate
students at Rutgers in the fall of 2012. The theory of the Riemann zeta function
has expanded in different directions over the past 150 years; however my goal was
limited to showing only a few classical results on the distribution of the zeros. These
results include the Riemann memoir (1859), the density theorem of F. Carlson
(1920) about the zeros off the critical line, and the estimates of G. H. Hardy - J.
E. Littlewood (1921) for the number of zeros on the critical line.
Then, in Part 2 of these lectures, I present in full detail the result of N. Levinson
(1974), which asserts that more than one third of the zeros are critical (lie on the
line Re s =
). My approach had frequent detours so that students could learn
different techniques with interesting features. For instance, I followed the stronger
construction invented by J. B. Conrey (1983), because it reveals clearly the esssence
of Levinson’s ideas.
After establishing the principal inequality of the Levinson-Conrey method, it
remains to evaluate asymptotically the second power-moment of a relevant Dirichlet
polynomial, which is built out of derivatives of the zeta function and its mollifier.
This task was carried out differently than by the traditional arguments and in
greater generality than it was needed. The main term coming from the contribution
of the diagonal terms fits with results in sieve theory and can be useful elsewhere.
I am pleased to express my deep appreciation to Pedro Henrique Pontes, who
actively participated in the course and he gave valuable mathematical comments,
which improved my presentation. He also helped significantly in editing these notes
in addition to typing them. My thanks also go to the Editors of the AMS University
Lecture Series for publishing these notes in their volumes, and in particular to Sergei
Gelfand for continuous encouragements.