Sommes de Gál et applications

Régis de La Bretèche (1) , Gerald Tenenbaum (2)

1 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
2 IECL - Institut Élie Cartan de Lorraine

We evaluate the asymptotic size of various sums of Gál type, in particular $$S( \M):=\sum_{m,n\in\M} \sqrt{(m,n) \over [m,n]},$$ where $\M$ is a finite set of integers. Elaborating on methods recently developed by Bondarenko and Seip, we obtain an asymptotic formula for $$\log\Big( \sup_{|\M|= N}{S( \M)/N}\Big)$$ and derive new lower bounds for localized extreme values of the Riemann zeta-function, for extremal values of some Dirichlet $L$-functions at $s=\dm$, and for large character sums.

Mots clés: Gál's theorem; GCD sums; Dirichlet polynomials; the Riemann zeta function; Dirichlet $L$-functions; character sums; resonance method

Domaines: Théorie des nombres [math.NT]

Régis de La Bretèche, Gerald Tenenbaum. Sommes de Gál et applications. Proceedings of the London Mathematical Society, 2018, 119 (1), pp.104-134. ⟨10.1112/plms.12224⟩. ⟨hal-02890627⟩