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A higher order Levin-Fainleib theorem

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Alisa Sedunova
Université de Montréal/MPIM
Mit, 22/02/2023 - 14:30 - 15:30
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

Hybrid. Contact: Pieter Moree (moree @

Landau was the first to obtain an asymptotic formula for the number of integers up to a given number that are sum of two coprime squares, where he used analytic method. Later, this procedure was further developed by Delange and Selberg allowing them to obtain asymptotic for partial sums of arithmetic functions whose Dirichlet series can be written in terms of complex powers of the Riemann zeta-function. In 1967 Levin and Fainleib using differential equation established the logarithmic density of the same set by an elementary argument under more general conditions. We generalise the Levin and Fainleib approach and allow more general hypotheses as well. When restricted to some non-negative multiplicative function, say f, bounded on primes and that vanishes on non square-free integers, our result provides us with an asymptotic for the mean value of f(n)/n when summed over n <x provided that we have the mean value over primes, namely, f(p)(logp)/p summed over p < Q.

This is a joint work with Olivier Ramare and Rita Sharma.

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