an:00867856
Zbl 0847.11046
Ivi??, Aleksandar; Motohashi, Yoichi
On some estimates involving the binary additive divisor problem
EN
Q. J. Math., Oxf. II. Ser. 46, No. 184, 471-483 (1995).
00030522
1995
j
11N37 11F72
binary additive divisor problem; spectral theory; error term; asymptotic formula; spectral large sieve inequalities
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. ??c. Norm. Sup??r., IV. S??r. 27, 529-572 (1994; Zbl 0819.11038)].
D.R.Heath-Brown (Oxford)
Zbl 0819.11038