Dimitris Koukoulopoulos, CRM, Montréal (Canada)

When is a multiplicative function small on average?

Let f be a multiplicative function. The main problem we will be
concerned with in this talk is understanding when f is small on
average. Halasz showed that, unless f "pretends to be" n^it for
some small t, this is true and gave quantitative estimates on the
rate of decay of the partial sums of f. The estimate provided by
Halasz's theorem is in general tight but there are functions f for
which it is far from the truth. A natural question that arises is to
classify the functions f whose partial sums are significantly smaller
than what one might predict by Halasz's theorem. More precisely,
if &Sigma_n≤x f(n) = O(x(log x)^-A) for every A, then what can we say about f?
We show that unless f pretends to be μ(n)n^it for some small t, then
this is true if and only if &Sigma_p≤x f(p) = O(x(log x)^-B) for every B. Finally,
we show how these methods can be used to give a new proof of the
prime number theorem in arithmetic progressions.