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On the ingham divisor problem

Authors:

A I Pavlov

Institution:

(1) V. A. Steklov Mathematics Institute, Russian Academy of Sciences, USSR

Abstract:

The main result of this paper is the following theorem. Suppose thatτ(n) = ∑_d|n l and the arithmetical functionF satisfies the following conditions:

1)	the functionF is multiplicative;
2)	ifF(n) = ∑_d\|n f(d), then there exists an α>0 such that the relationf(n)=O(n ^−α) holds asn→∞.

Then there exist constantsA ₁,A ₂, andA ₃ such that for any fixed \g3\s>0 the following relation holds:

$\sum\limits_{n \leqslant x} {r(n)} r(n + 1)F(n) = A_1 xln^{\text{2}} x + A_{\text{2}} xlnx + A_{\text{3}} x + O(x^{5/6 + \varepsilon } + x^{1 - \alpha /6 + \varepsilon } ), x \to \infty .$

. Moreover, if for any primep the inequality \vbf(p)\vb\s<1 holds and the functionF is strongly multiplicative, thenA ₁\s>0. Translated fromMatematicheskie Zametki, Vol. 68, No. 3, pp. 429–438, September, 2000.

Keywords:

Ingham divisor Ingham’ s equality arithmetical function Heath-Brown theorem

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