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Problems similar to the additive divisor problem

Authors:

Institution:

(1) Vladimir Pedagogical State University, USSR;(2) Mathematics Institute, Academy of Sciences of Uzbekistan, Uzbekistan

Abstract:

For multiplicative functions ƒ(n), let the following conditions be satisfied: ƒ(n)≥0 ƒ(p ^r)≤A ^r,A>0, and for anyε>0 there exist constants $A_\varepsilon$ ,α>0 such that $f(n) \leqslant A_\varepsilon n^\varepsilon$ and Σ_p≤x ƒ(p) lnp≥αx. For such functions, the following relation is proved:

$\sum\limits_{n \leqslant x} {f(n)} \tau (n - 1) = C(f)\sum\limits_{n \leqslant x} {f(n)lnx(1 + 0(1))}$

. Hereτ(n) is the number of divisors ofn andC(ƒ) is a constant. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 443–456, September, 1998. The work of the first author was supported by the Russian Foundation for Basic Research.

Keywords:

multiplicative function additive divisor problem Riemann zeta function Euler function primitive characters

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