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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.