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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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