Abstract
One obtains a nontrivial estimate for the mean square variant of Yu. V. Linnik's conjecture on Kloosterman's sum of sums.
Similar content being viewed by others
Literature cited
H. Bateman and A. Erdélyi, Higher Transcendental Functions, Vol. II, McGraw-Hill, New York (1953).
N. V. Kuznetsov, “Petersson's conjecture for forms of weight zero and Linnik's conjecture,” Preprint No. 02, Khabarovsk Complex Sci. Res. Inst., Far Eastern Scientific Center, Khabarovsk (1977).
K. Prachar, Primzahlverteilung, Springer-Verlag, Berlin (1957).
N. V. Proskurin, “Summation formulas for general Kloosterman sums,” Zap. Nauch. Sem. Leningr. Otd. Mat. Inst.,82, 103–135 (1979).
N. V. Proskurin, “Estimates for the eigenvalues of Hecke operators in the space of parabolic forms of weight O,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,82, 136–143 (1979).
N. V. Proskurin, “Convolutions of Dirichlet series with Fourier coefficients of parabolic forms of weight O,” Zap. Nauchn. Sem Ins. Leningr. Otd. Mat. Inst. (1979) (in press).
E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, Clarendon Press, Oxford (1951).
L. D. Faddeev, “Expansion in eigenfunctions of the Laplace operator on the fundamental domain of a discrete group on the Lobachevskii plane,” Tr. Mosk. Mat. Obshch.,17, 323–349 (1967).
R. W. Bruggeman, “Fourier coefficients of cusp forms,” Invent. Math.,45, No. 1, 1–18 (1978).
P. Deligne, “La conjecture de Weil. I,” Publ. Math. I.H.E.S.,43, 273–307 (1974).
Th. Estermann, “Vereinfachter Beweis eines Satzes von Kloosterman,” Abh. Math. Sem. Univ. Hamburg,7, 82–98 (1930).
J. Lehner, Discontinuous Groups and Automorphic Functions, Amer. Math. Soc., Providence (1964).
Ju. V. Linnik, “Additive problems and eigenvalues of the modular operators,” in: Proc. Int. Congr. Math. August 1962, Djursholm, Uppsala (1963), pp. 270–284.
H. Maass, “Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen,” Math. Ann.,121, No. 2, 141–183 (1949).
H. Petersson, “Über die Entwicklungskoeffizienten der automorphen Formen,” Acta Math.,58, 169–215 (1932).
H. Petersson, “Über Modulfunktionen und Partionenprobleme,” Abg. Deutsch. Akad. Wiss. Berlin. Kl. Math. Phys. Techn., H.2 (1954).
A. Weil, “On some exponential sums,” Proc. Nat. Acad. Sci. USA,34, 204–207 (1948).
Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 91, pp. 94–118, 1979.
Rights and permissions
About this article
Cite this article
Proskurin, N.V. Yu. V. Linnik's conjecture. J Math Sci 17, 2147–2162 (1981). https://doi.org/10.1007/BF01567593
Issue Date:
DOI: https://doi.org/10.1007/BF01567593