A Higher Order Levin-Fainleib Theorem

Un théorème de Levin-Fainleib d'ordre supérieur

Olivier Ramaré (1, 2, 3) , Alisa Sedunova (4) , Ritika Sharma

1 CNRS - Centre National de la Recherche Scientifique
2 I2M - Institut de Mathématiques de Marseille
3 AMU - Aix Marseille Université
4 SPBU - Saint Petersburg University

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 $\sum_{n\le X}f(n)/n$ with error term $\mathcal{O}((\log X)^{\kappa-h-1+\varepsilon})$ (for any positive $\varepsilon>0$) as soon as we have $\sum_{p\le Q}f(p)(\log p)/p=\kappa\log Q+\eta+\mathcal{O}(1/(\log 2Q)^h)$ for a non-negative $\kappa$ and some non-negative integer h. The method generalizes the 1967-approach of Levin and Fainleib and uses a differential equation.

Domaines: Théorie des nombres [math.NT]

Olivier Ramaré, Alisa Sedunova, Ritika Sharma. A Higher Order Levin-Fainleib Theorem. Proceedings Mathematical Sciences, 2023, 133 (1), pp.1-15. ⟨hal-03538130⟩