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Analogues of the Gauss-Vinogradov formula on the critical line. (English) Zbl 0534.10039

Translation from Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 109, 41–82 (Russian) (1981; Zbl 0468.10023).

MSC:

11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11N35 Sieves
11E41 Class numbers of quadratic and Hermitian forms
Full Text: DOI

References:

[1] C. F. Gauss, Disquisitiones Arithmeticae, Leipzig (1801).
[2] I. M. Vinogradov, ?New method of derivation of asymptotic expressions of arithmetical functions,? Izv. Ross. Akad. Nauk, Ser. 6,11, 1347?1378 (1917) [I. M. Vinogradov, Selected Works, Moscow (1952), pp. 3?28].
[3] I. M. Vinogradov, ?On the mean value of the number of classes of properly primitive forms of negative discriminant,? Soobshch. Khark. Mat. Obshch.,16, 10?38 (1917) (Selected Works, pp. 29?53).
[4] I. M. Vinogradov, ?Number of integer points in the sphere,? Tr. Mat. Inst. Steklov. Akad. Nauk SSSR,9, 17?38 (1935).
[5] I. M. Vinogradov, ?On the number of integer points in the sphere,? Izv. Akad. Nauk SSSR, Ser. Mat.,27, 957?968 (1963).
[6] I. M. Vinogradov, ?On one asymptotic equality of quadratic forms theory,? Zh. Fiz.-Mat. Obshch. Permsk. Univ.,1, 18?28 (1918) (Selected Works, pp. 58?66).
[7] A. I. Vinogradov and L. A. Takhtadzhyan, ?On analogues of the Vinogradov-Gauss formula,? Dokl. Akad. Nauk SSSR,254, No. 6, 1298?1300 (1980).
[8] A. I. Vinogradov and L. A. Takhtadzhyan, ?Analogues of the Vinogradov-Gauss formula in the critical strip,? Tr. Mat. Inst. Steklov. Akad. Nauk SSSR,158 (1981). · Zbl 0483.12002
[9] D. Goldfeld and C. Viola, ?Mean values of L-functions associated to elliptic, Fermat and other curves at the center of the critical strip,? J. Number Theory,11, No. 3, 305?320 (1979). · Zbl 0409.10029 · doi:10.1016/0022-314X(79)90004-0
[10] A. F. Lavrik, ?Functional equation for Dirichlet’s L-functions and the problem of divisors in arithmetical progressions,? Izv. Akad. Nauk SSSR, Ser. Mat.,30, No. 2, 433?448 (1966).
[11] R. M. Kaufman, ?On A. F. Lavrik’s shortened equations,? Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,76, 433?448 (1966).
[12] H. Davenport, Multiplicative Number Theory, Chicago (1967). · Zbl 0159.06303
[13] A. I. Vinogradov, ?Kubota’s series and theta-functions,? in: Actual Problems of Analitic Number Theory [in Russian], Nauka i Tekhnika, Minsk (1973), pp. 23?48.
[14] M. Jutila, ?On mean values of Dirichlet polynomials with real characters,? Acta Arithmetica,27, 191?198 (1975). · Zbl 0261.10028
[15] H. L. Montgomery, ?Topics in multiplicative number theory,? Lect. Notes Math.,227 (1971). · Zbl 0216.03501
[16] M. Jutila, ?On mean values of L-functions and short character sums with real characters,? Preprint Univ. Turku, Finland, No. 1 (1974).
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