On the binary additive divisor problem.

*(English)*Zbl 0967.11039
Jutila, Matti (ed.) et al., Number theory. Proceedings of the Turku symposium on number theory in memory of Kustaa Inkeri, Turku, Finland, May 31-June 4, 1999. Berlin: de Gruyter. 223-246 (2001).

The binary additive divisor problem is a classical problem of analytic number theory, with a rich history and many results. It consists of the estimation of the function \(E(N,f)\), the error term in the asymptotic formula for \(\sum_{n\leq N}d(n)d(n+f)\), where \(d(n)\) is as usual the number of all positive divisors of \(n\) and the ‘shift’ parameter \(f\) is not necessarily fixed, but may vary with \(N\). In a fundamental paper [Ann. Sci. Éc. Norm. Supér. 27, 529-572 (1994; Zbl 0819.11038)] Y. Motohashi used deep methods from spectral theory to prove that, uniformly for \(1 \leq f \leq N^{2/(1+2\alpha)}\),
\[
E(N,f) \ll_\varepsilon (N(N+f))^{1/3}N^\varepsilon + (N(N+f))^{1/4}f^{1/8+\alpha/2}N^\varepsilon + f^{1/2+\alpha}N^\varepsilon.
\]
Here the constant \(\alpha\) is defined by the bound \(t_j(n) \ll_\varepsilon n^{\alpha+\varepsilon}\), where \(t_j(n)\) is generated by the Hecke series \(H_j(s)\) (the famous Ramanujan-Petersson conjecture is that one has \(\alpha = 0\)). In the present paper the author improves the above estimate for large \(f\) by proving that
\[
E(N,f) \ll_\varepsilon (N(N+f))^{1/3}N^\varepsilon + (N(N+f))^{1/4}\min\left(N^{1/4}, f^{1/8+\alpha/2}\right)N^\varepsilon.
\]
Thus one has a better result for \(f \geq N^{2/(1+4\alpha)}\), in particular one has a new result for \(N^{7/6} \leq f \leq N^{2-\varepsilon}\) with the known value \(\alpha = 5/28\). The author also indicates how the range for which the mean square estimate for \(E(N,f)\) of Y. Motohashi and the reviewer [Q. J. Math., Oxf. (2) 46, 471-483 (1995; Zbl 0847.11046)] can be extended. The proof of the author’s result is involved, but the details are clearly presented and the paper is well-written.

For the entire collection see [Zbl 0959.00053].

For the entire collection see [Zbl 0959.00053].

Reviewer: Aleksandar Ivić (Beograd)

##### MSC:

11N37 | Asymptotic results on arithmetic functions |

11F30 | Fourier coefficients of automorphic forms |

11F72 | Spectral theory; trace formulas (e.g., that of Selberg) |