On some estimates involving the binary additive divisor problem. (English) Zbl 0847.11046

Ivić, Aleksandar; Motohashi, Yoichi

Q. J. Math., Oxf. II. Ser. 46, No. 184, 471-483 (1995).

Let \(E(x; f)\) be the error term in the asymptotic formula for \(\sum_{n\leq x} d(n) d(n+ f)\). Then it is shown that for every \(\varepsilon> 0\) one has \[ \sum_{f\leq F} E^2(X; f)\ll_\varepsilon F^{1/3} X^{4/3+ \varepsilon}\text{ and } \sum_{f\leq F} \Biggl( \int^{2X}_X E(x; f) dx\Biggr)^2\ll_\varepsilon FX^{3+ \varepsilon}, \] uniformly for \(F\leq X^{1/2- \varepsilon}\). The proofs use spectral large sieve inequalities, together with an explicit formula for \(E(x; f)\) due to the second author [Ann. Sci. Éc. Norm. Supér., IV. Sér. 27, 529-572 (1994; Zbl 0819.11038)].

Reviewer: D.R.Heath-Brown (Oxford)

MSC:

11N37	Asymptotic results on arithmetic functions
11F72	Spectral theory; trace formulas (e.g., that of Selberg)

Keywords:

binary additive divisor problem; spectral theory; error term; asymptotic formula; spectral large sieve inequalities