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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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