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Zapiski Nauchnykh Seminarov LOMI, 1984, Volume 134, Pages 84–116
(Mi znsl4743)
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This article is cited in 3 scientific papers (total in 3 papers)
Zeta function of the additive divisor problem and the spectral expansion of the automorphic Laplacian
A. I. Vinogradov, L. A. Takhtadzhyan
Abstract:
The representation for zeta function of the additive divisor problem $\zeta_k(s)=\sum_{n=1}^\infty\frac{\tau(n)\tau(n+k)}{n^s}$, $\operatorname{Re}s>1$, in terms of spectral data of the automorphic Laplacian is presented. With its help the meromorphic continuation of $\zeta_k(s)$ into the whole
complex plane is proved and an estimate of the order of $\zeta_k(s)$ in the critical strip $0<\operatorname{Re}s\leqslant1$ is obtained. Using the method of complex integration the asymptotic formula
$$
\sum_{n\leqslant x}\tau(n)\tau(n+k)=xP_k(\log x)+O(x^{\frac23+\varepsilon}),\quad\varepsilon>0,
$$
is derived where $P_k(x)$ is a quadratic polynomial.
Citation:
A. I. Vinogradov, L. A. Takhtadzhyan, “Zeta function of the additive divisor problem and the spectral expansion of the automorphic Laplacian”, Automorphic functions and number theory. Part II, Zap. Nauchn. Sem. LOMI, 134, "Nauka", Leningrad. Otdel., Leningrad, 1984, 84–116
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https://www.mathnet.ru/eng/znsl4743 https://www.mathnet.ru/eng/znsl/v134/p84
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