The approximate functional equation of Hecke’s Dirichlet series
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- by T. M. Apostol and Abe Sklar PDF
- Trans. Amer. Math. Soc. 86 (1957), 446-462 Request permission
References
- T. M. Apostol, Identities involving the coefficients of certain Dirichlet series, Duke Math. J. 18 (1951), 517–525. MR 41880, DOI 10.1215/S0012-7094-51-01842-X N. Čudakov, On Goldbach-Vinogradov’s theorem, Ann. of Math. vol. 48 (1947) pp. 515-545. A. Erdélyi, et al. Higher transcendental functions, vol. 2, McGraw-Hill, 1954. —, Tables of integral transforms, vol. 1, McGraw-Hill. 1954.
- G. H. Hardy and J. E. Littlewood, The zeros of Riemann’s zeta-function on the critical line, Math. Z. 10 (1921), no. 3-4, 283–317. MR 1544477, DOI 10.1007/BF01211614 —, The approximate functional equation in the theory of the zeta-function, with applications to the divisor problems of Dirichlet and Piltz, Proc. London Math. Soc. (2) vol. 21 (1922) pp. 39-74. —, The approximate functional equations for $\zeta (s)$ and ${\zeta ^2}(s)$, Proc. London Math. Soc. (2) vol. 29 (1929) pp. 81-97. E. Hecke, Dirichlet series, Planographed lectures, Princeton, Institute for Advanced Study, 1938. E. Landau, Über die Anzahl der Gitterpunkte in gewissen Bereichen II, Nachr. Ges. Wiss. Gottingen (1915) pp. 209-243. H. S. A. Potter, Approximate equations for the Epstein zeta-function, Proc. London Math. Soc. (2) vol. 36 (1934) pp. 501-515.
- R. A. Rankin, Contributions to the theory of Ramanujan’s function $\tau (n)$ and similar arithmetical functions. I. The zeros of the function $\sum ^\infty _{n=1}\tau (n)/n^s$ on the line ${\mathfrak {R}}s=13/2$. II. The order of the Fourier coefficients of integral modular forms, Proc. Cambridge Philos. Soc. 35 (1939), 351–372. MR 411, DOI 10.1017/S0305004100021095
- R. A. Rankin, Contributions to the theory of Ramanujan’s function $\tau (n)$ and similar arithmetical functions. III. A note on the sum function of the Fourier coefficients of integral modular forms, Proc. Cambridge Philos. Soc. 36 (1940), 150–151. MR 1249, DOI 10.1017/s0305004100017114 C. L. Siegel, Über Riemanns Nachlass zur analytische Zahlentheorie, Quellen und Studien zur Geschichte der Math., Astron. und Physik, Abt. B: Studien, vol. 2 (1932) pp. 45-80. Z. Suetuna, The zeros of the $L$-functions on the critical line, Tôhoku Math. J. vol. 29 (1925) pp. 313-331. —, Über die approximative Funktionalgleichung die Dirichletsche $L$-funktionen, Jap. J. Math. vol. 9 (1932) pp. 111-116.
- Tikao Tatuzawa, The approximate functional equation for Dirichlet’s $L$-series, Jpn. J. Math. 22 (1952), 19–25 (1953). MR 62775, DOI 10.4099/jjm1924.22.0_{1}9 E. C. Titchmarsh, The approximate functional equation for ${\zeta ^2}(s)$, Quart. J. Math. Oxford Ser. vol. 9 (1938) pp. 109-114. —, Theory of functions, 2d ed., Oxford University Press, 1939. —, Theory of the Riemann zeta function, Oxford University Press, 1951.
- Rudolf Wiebelitz, Über approximative Funktionalgleichungen der Potenzen der Riemannschen Zetafunktion, Math. Nachr. 6 (1952), 263–270 (German). MR 49926, DOI 10.1002/mana.19520060503 J. R. Wilton, An approximate functional equation for the product of two $\zeta$-functions, Proc. London Math. Soc. (2) vol. 31 (1930) pp. 11-17.
Additional Information
- © Copyright 1957 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 86 (1957), 446-462
- MSC: Primary 10.00; Secondary 39.00
- DOI: https://doi.org/10.1090/S0002-9947-1957-0094319-3
- MathSciNet review: 0094319