Number Theory Seminar

Thursday, January 26, 2017 1:30 pm - 1:30 pm EST (GMT -05:00)

Sacha Mangerel, University of Toronto

"Some Applications of Pretentiousness in the Theory of Dirichlet Characters"

"Pretentious" methods in analytic number theory, as introduced by Granville and Soundararajan, are powerful tools in the analysis of mean values and correlations of multiplicative functions. In this talk, we will give two applications of these tools.

i) For a non-principal Dirichlet character $\chi$ modulo $q$, define $M(\chi) = \max_{t} |\sum_{n \leq t} \chi(n)|$. The classical Polya--Vinogradov inequality asserts that $M(\chi) \ll \sqrt{q} \log q$ unconditionally, and on the Generalized Riemann Hypothesis (GRH), Montgomery and Vaughan showed that $M(\chi) \ll \sqrt{q} \log \log q$. We discuss a recent improvement, in joint work with Y. Lamzouri, to both of these results in the case that $\chi$ has odd order. This improvement on GRH is best possible up to a factor of $\log \log \log \log q$.

One of the key ingredients in the proof of the upper bounds is a new Halasz-type inequality for logarithmic mean values of completely multiplicative functions.

ii) Time permitting, we will also discuss the following rigidity theorem related to binary correlations of Dirichlet characters.

If $f : \mathbb{N} \rightarrow \mathbb{C}$ is a 1-bounded multiplicative function for which there is a primitive Dirichlet character $\chi$ of conductor $q$ such that 
$$
 \sum_{n \leq x} f(n) \overline{f(n + h)} = (1 + o(1)) \sum_{n \leq x} \chi(n) \overline{\chi(n + h)}
$$ 
for all shifts $|h| \leq H$, then if $H \rightarrow \infty$ with 
$x$ then $f$ ``pretends'' to be $\chi$ in a precise sense. This is 
joint work with O. Klurman.

MC 5403