Sums of reciprocals of additive functions running over short intervals

J.-M. De Koninck¹ ; I. Kátai²

¹ Département de mathématiques Université Laval Québec G1K 7P4, Canada
² Computer Algebra Department Eötvös Loránd University Pázmány Péter Sétány I//C 1117 Budapest, Hungary

Colloquium Mathematicum, Tome 107 (2007) no. 2, pp. 317-326.

Résumé

Letting $f(n)=A\log n+t(n)$, where $t(n)$ is a small additive function and $A$ a positive constant, we obtain estimates for the quantities $\sum _{x \le n \le x+H} 1/f(Q(n))$ and $\sum _{x \le p \le x+H} 1/f(Q(p))$, where $H=H(x)$ satisfies certain growth conditions, $p$ runs over prime numbers and $Q$ is a polynomial with integer coefficients, whose leading coefficient is positive, and with all its roots simple.

Détail

DOI : 10.4064/cm107-2-11

Keywords: letting log where small additive function positive constant obtain estimates quantities sum sum where satisfies certain growth conditions runs prime numbers polynomial integer coefficients whose leading coefficient positive its roots simple