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The binary additive divisor problem. (English) Zbl 0819.11038
The author gives a comprehensive account of the sums \[ D(N; f)= \sum_{n=1}^ N d(n) d(n+f) \qquad \text{and} \qquad D(N)= \sum_{n=1}^{N-1} d(n) d(N-n), \] where \(d(n)\) is the number of divisors of \(n\), and \(f\) \((\geq 1)\) is an integer which is not necessarily fixed. By using deep methods from the spectral theory of the non-Euclidean Laplacian [see his fundamental paper on \(\int_ 0^ T |\zeta (1/2+ it) |^ 4 dt\) in Acta Math. 170, 181-220 (1993; Zbl 0784.11042)] the author obtains first a rigorous proof of a formula of N. V. Kuznetsov [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 129, 43-84 (1983; Zbl 0507.10016)] for a more general sum than \(D(N; f)\). From this formula he deduces, uniformly for \(1\leq f\leq N^{10/7}\), \[ D(N; f)= 6\pi^{-2} \int_ 0^{N/f} m(x; f)dx+ O\{N^{1/3} (N+f)^{1/3+ \varepsilon}+ f^{9/40} (N^ 2+ fN)^{1/4+ \varepsilon}+ f^{7/10} N^ \varepsilon\}. \] In this remarkable result, where the range for \(f\) is the largest one known, \(m(x; f)\) is an explicit function. As a corollary it follows that \(E(N; f)\ll N^{2/3+ \varepsilon}\) uniformly for \(1\leq f\leq N^{20/27}\), where \(E(N; f)\) is the classical error term in the asymptotic formula for \(D(N; f)\). He also proves that \[ E(N)= D(N)- NP_ 2 (\log N)\ll N^{7/10+ \varepsilon}, \] where \(P_ 2 (x)\) is a suitable quadratic function in \(x\). This improves the exponent \(3/4\) of H. Halberstam [Trans. Am. Math. Soc. 84, 338- 351 (1957; Zbl 0087.043)]. The author’s bounds for \(E(N; f)\) and \(E(N)\) are in fact given also in general form as to depend on the bounds for Fourier coefficients of holomorphic and non-holomorphic cusp forms. In his last result, the author proves the conjecture of the reviewer that \(E(N; f)= \Omega (N^{1/2})\) for fixed \(f\geq 1\).
One cannot describe, in a short review such as this one is, the wealth and the depth of the methods used in the proofs of the author’s results.
Reviewer: A.Ivić (Beograd)

MSC:
11N37 Asymptotic results on arithmetic functions
11N75 Applications of automorphic functions and forms to multiplicative problems
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
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