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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].

11N37 Asymptotic results on arithmetic functions
11F30 Fourier coefficients of automorphic forms
11F72 Spectral theory; trace formulas (e.g., that of Selberg)