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Zapiski Nauchnykh Seminarov LOMI, 1978, Volume 76, Pages 5–52 (Mi znsl1930)  

This article is cited in 16 scientific papers (total in 16 papers)

Theory of Eisenstein series for the group $SL(3,\mathbf R)$ and its application to a binary problem. I. Fourier expansion of the highest Eisenstein series

A. I. Vinogradov, L. A. Takhtadzhyan
Abstract: On the basis of arithmetic considerations, a Fourier expansion for the leading Eisenstein series is obtained for the principal homogeneous space of the group $SL(3,\mathbf R)$, which is automorphic with respect to the discrete group $SL(3,\mathbf Z)$. The main result is Theorem 1 in which an explicit form of the Fourier expansion is presented which generalizes the well-known formula of Selberg and Chowla. From this, in particular, there follows a proof of the analytic continuation and the functional equations for this Eisentein series which is independent of the work of Langlands. The arithmetic coefficients in the Fourier expansion which generalize the number-theoretic functions $\sigma_s(n)=\sum_{d|n,d>0}d^s$ make it possible to relate the Eisenstein series considered to the problem of finding the asymptotics as $X\to\infty$ of the sum $\sum_{n\leqslant X}\tau_3(n)\tau_3(n+k)$, where $\tau_3(n)$ is the number of solutions of the equation $d_1d_2d_3=n$ in natural numbers. Part II of the present work will be devoted to this binary problem. At the end of the paper properties of special functions used in Theorem 1 are discussed.
English version:
Journal of Soviet Mathematics, 1982, Volume 18, Issue 3, Pages 293–324
DOI: https://doi.org/10.1007/BF01084842
Bibliographic databases:
UDC: 511.3, 517.43, 519.45
Language: Russian
Citation: A. I. Vinogradov, L. A. Takhtadzhyan, “Theory of Eisenstein series for the group $SL(3,\mathbf R)$ and its application to a binary problem. I. Fourier expansion of the highest Eisenstein series”, Analytical theory of numbers and theory of functions, Zap. Nauchn. Sem. LOMI, 76, "Nauka", Leningrad. Otdel., Leningrad, 1978, 5–52; J. Soviet Math., 18:3 (1982), 293–324
Citation in format AMSBIB
\Bibitem{VinTak78}
\by A.~I.~Vinogradov, L.~A.~Takhtadzhyan
\paper Theory of Eisenstein series for the group $SL(3,\mathbf R)$ and its application to a binary problem. I.~Fourier expansion of the highest Eisenstein series
\inbook Analytical theory of numbers and theory of functions
\serial Zap. Nauchn. Sem. LOMI
\yr 1978
\vol 76
\pages 5--52
\publ "Nauka", Leningrad. Otdel.
\publaddr Leningrad
\mathnet{http://mi.mathnet.ru/znsl1930}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=527787}
\zmath{https://zbmath.org/?q=an:0476.10024|0434.10020}
\transl
\jour J. Soviet Math.
\yr 1982
\vol 18
\issue 3
\pages 293--324
\crossref{https://doi.org/10.1007/BF01084842}
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  • This publication is cited in the following 16 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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