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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.
11N37 Asymptotic results on arithmetic functions
Full Text: DOI
[1] A. E. Ingham, ”Some asymptotic formulae in theory of numbers,”J. London Math. Soc.,2, No. 3, 202–208 (1927). · JFM 53.0157.01 · doi:10.1112/jlms/s1-2.3.202
[2] T. Estermann, ”Über die Darstellung einer Zahl als Differenz von zwei Produkten,”J. Reine Angew. Math.,164, 173–182 (1931). · Zbl 0001.20302 · doi:10.1515/crll.1931.164.173
[3] D. Ismailov, ”On the asymptotics of the representation of numbers as the difference of two products,”Dokl. Akad. Nauk Tadzhik SSR,22, No. 2, 75–79 (1979).
[4] D. R. Heath-Brown, ”The fourth power moment of the Rieman zeta function,”Proc. London Math. Soc.,38, No. 3, 385–422 (1979). · Zbl 0403.10018 · doi:10.1112/plms/s3-38.3.385
[5] N. M. Timofeev, ”An analog of the Halos theorem in the case of the generalization of the additive divisor problem,”Mat. Zametki [Math. Notes],48, No. 1, 116–127 (1990). · Zbl 0708.11048
[6] N. M. Timofeev and S. T. Tulyaganov, ”Problems similar to the additive divisor problem,”Mat. Zametki [Math. Notes],64, No. 3, 443–456 (1998). · Zbl 0922.11076
[7] M. B. Khripunova, ”Sums of multiplicative functions with shifted arguments,”Mat. Zametki [Math. Notes],51, No. 4, 136–138 (1992). · Zbl 0795.11046
[8] A. G. Postnikov,Introduction to the Analytic Theory of Numbers [in Russian], Nauka, Moscow (1971). · Zbl 0231.10001
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