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On the Ingham divisor problem. (English. Russian original) Zbl 0989.11045
Math. Notes 68, No. 3, 370-377 (2000); translation from Mat. Zametki 68, No. 3, 429-438 (2000).
Let $$d(n)$$ be the divisor function, and let $$F(n)$$ be a multiplicative function of the form $$F(n)=\sum_{d\mid n}f(d)$$, with $$f(d)\ll d^{-\alpha}$$. Then it is shown that $\sum_{n\leq x}F(n) d(n)d(n+1)= xQ(\log x) +O(x^{5/6+\varepsilon}) +O(x^{1-\alpha/6+\varepsilon})$ for any $$\varepsilon >0$$. Here $$Q(\dots)$$ is a quadratic polynomial, and the implied constant depends on $$F$$ and $$\varepsilon$$. The leading term of $$Q$$ is found explicitly in the case in which $$F$$ is strongly multiplicative.
When $$F$$ is identically 1 one recovers a result of the reviewer [D. R. Heath-Brown, Proc. Lond. Math. Soc. (3) 38, 385-422 (1979; Zbl 0403.10018)]. The proof uses an estimate for $\sum_{n\leq x, n \equiv 1\pmod q}d(n), \quad (q\leq x^{2/3}).$ This estimate, which depends on Weil’s bound for the Kloosterman sum, is taken from the reviewer’s work cited above.
##### MSC:
 11N37 Asymptotic results on arithmetic functions
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##### References:
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