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

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