Explicit averages of non-negative multiplicative functions: going beyond the main term

Olivier Ramaré (1) , P. Akhilesh (2)

1 I2M - Institut de Mathématiques de Marseille
2 Harish-Chandra Research Institute

We produce an explicit formula for averages of the type Sigma(d <= D) (g star 1)(d) / d, where star is the Dirichlet convolution and g a function that vanishes at infinity (more precise conditions are needed, a typical example of an acceptable function is g(m) - mu(m) / m). This formula enables one to exploit the changes of sign of g(m). We use this formula for the classical family of sieve-related functions G(q)(D) = Sigma(d <= D, (d,q) = 1) mu(2) (d) / (sic))(d) for an integer parameter q, improving noticeably on earlier results. The remainder of the paper deals with the special case q = 1 to show how to practically exploit the changes of sign of the Mobius function. It is proven in particular that broken vertical bar G(1) (D) - log D - c(0) broken vertical bar <= 4 / root D and broken vertical bar G(1) (D) - log D - c(0) broken vertical bar <= 18.4 / root D log D) when D > 1, for a suitable constant c(0)

Mots clés: multiplicative functions; explicit evaluations

Domaines: Théorie des nombres [math.NT]

Olivier Ramaré, P. Akhilesh. Explicit averages of non-negative multiplicative functions: going beyond the main term. Colloquium Mathematicum, 2017, 147 (2), pp.275 - 313. ⟨10.4064/cm6080-4-2016⟩. ⟨hal-01550555⟩