Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The approximate functional equation of Hecke's Dirichlet series


Authors: T. M. Apostol and Abe Sklar
Journal: Trans. Amer. Math. Soc. 86 (1957), 446-462
MSC: Primary 10.00; Secondary 39.00
MathSciNet review: 0094319
Full-text PDF Free Access

References | Similar Articles | Additional Information

References [Enhancements On Off] (What's this?)

  • [1] T. M. Apostol, Identities involving the coefficients of certain Dirichlet series, Duke Math. J. 18 (1951), 517–525. MR 0041880 (13,15c)
  • [2] N. Čudakov, On Goldbach-Vinogradov's theorem, Ann. of Math. vol. 48 (1947) pp. 515-545.
  • [3] A. Erdélyi, et al. Higher transcendental functions, vol. 2, McGraw-Hill, 1954.
  • [4] -, Tables of integral transforms, vol. 1, McGraw-Hill. 1954.
  • [5] 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, http://dx.doi.org/10.1007/BF01211614
  • [6] -, 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.
  • [7] -, The approximate functional equations for $ \zeta (s)$ and $ {\zeta ^2}(s)$, Proc. London Math. Soc. (2) vol. 29 (1929) pp. 81-97.
  • [8] E. Hecke, Dirichlet series, Planographed lectures, Princeton, Institute for Advanced Study, 1938.
  • [9] E. Landau, Über die Anzahl der Gitterpunkte in gewissen Bereichen II, Nachr. Ges. Wiss. Gottingen (1915) pp. 209-243.
  • [10] H. S. A. Potter, Approximate equations for the Epstein zeta-function, Proc. London Math. Soc. (2) vol. 36 (1934) pp. 501-515.
  • [11] R. A. Rankin, Contributions to the theory of Ramanujan’s function 𝜏(𝑛) and similar arithmetical functions. I. The zeros of the function ∑^{∞}_{𝑛=1}𝜏(𝑛)/𝑛^{𝑠} on the line ℜ𝔰=13/2. II. The order of the Fourier coefficients of integral modular forms, Proc. Cambridge Philos. Soc. 35 (1939), 351–372. MR 0000411 (1,69d)
  • [12] R. A. Rankin, Contributions to the theory of Ramanujan’s function 𝜏(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 0001249 (1,203d)
  • [13] 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.
  • [14] Z. Suetuna, The zeros of the $ L$-functions on the critical line, Tôhoku Math. J. vol. 29 (1925) pp. 313-331.
  • [15] -, Über die approximative Funktionalgleichung die Dirichletsche $ L$-funktionen, Jap. J. Math. vol. 9 (1932) pp. 111-116.
  • [16] Tikao Tatuzawa, The approximate functional equation for Dirichlet’s 𝐿-series, Jap. J. Math. 22 (1952), 19–25 (1953). MR 0062775 (16,16c)
  • [17] E. C. Titchmarsh, The approximate functional equation for $ {\zeta ^2}(s)$, Quart. J. Math. Oxford Ser. vol. 9 (1938) pp. 109-114.
  • [18] -, Theory of functions, 2d ed., Oxford University Press, 1939.
  • [19] -, Theory of the Riemann zeta function, Oxford University Press, 1951.
  • [20] Rudolf Wiebelitz, Über approximative Funktionalgleichungen der Potenzen der Riemannschen Zetafunktion, Math. Nachr. 6 (1952), 263–270 (German). MR 0049926 (14,248f)
  • [21] 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.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 10.00, 39.00

Retrieve articles in all journals with MSC: 10.00, 39.00


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1957-0094319-3
PII: S 0002-9947(1957)0094319-3
Article copyright: © Copyright 1957 American Mathematical Society