KUWAIT FOUNDATION LECTURE 92 - May 12, 2009
Multiplicative functions (and pretensions)
Andrew Granville
(Université de Montréal)
A multiplicative function f : N −→ G, a multiplicative group, satisfies f (mn) = f (m)f (n)
whenever
(m, n) = 1. We are primarily interested in understanding when mean values,
P
1
n≤N f (n), are large in absolute value – a problem which, in one form or another,
N
comes into many central themes of modern number theory.
Over the last decade, K. Soundararajan and I have investigated this theme, particularly
when N is fixed and large and we vary over f , to deduce results about character sums
and other questions. Slowly we came to realize, much in the spirit of Gowers’ “rough
classification”, that whenever such a mean value is large, f (n) seems to behave like another
multiplicative function of relatively low “complexity.” Thus, f is “pretentious”.
P
For example, if N1 n≤N f (n) is “large”, then Halasz showed that f “pretends” to be nit
for some “small” t. Halasz, of course, gave a rather precise version of this result; our main
contribution is perhaps to work and think with this looser notion of “pretentiousness.” In
our talk we show how this leads to various applications:
- Making rigorous “the” intuitive proof of the prime number theorem
- Linking zeros of different L-functions
- Improving bounds on character sums
- Showing how multiplicative functions are distributed in arithmetic progressions, in
the sprit of Gallagher’s version of the Bombieri-Vinogradov theorem
- Proving repulsion results for zeros of L-functions, by showing that “pretentiousness”
is repulsive
- Classifying when exponential sums, with multiplicative coefficients, can be large
- Showing the values of Maass forms are uniformly distributed.