Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A generalization of the Lipschitz summation formula and some applications


Authors: Paul C. Pasles and Wladimir de Azevedo Pribitkin
Journal: Proc. Amer. Math. Soc. 129 (2001), 3177-3184
MSC (2000): Primary 11F30, 11F37, 42A99
Published electronically: April 2, 2001
MathSciNet review: 1844990
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract:

The Lipschitz formula is extended to a two-variable form. While the theorem itself is of independent interest, we justify its existence further by indicating several applications in the theory of modular forms.


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

  • 1. John J. Benedetto and Georg Zimmermann, Sampling multipliers and the Poisson summation formula, J. Fourier Anal. Appl. 3 (1997), no. 5, 505–523. Dedicated to the memory of Richard J. Duffin. MR 1491931 (99b:42011), http://dx.doi.org/10.1007/BF02648881
  • 2. Dickson, L.E. (1930). Studies in the Theory of Numbers. Univ. of Chicago Press, Chicago.
  • 3. Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vol. I, Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981. Based on notes left by Harry Bateman; With a preface by Mina Rees; With a foreword by E. C. Watson; Reprint of the 1953 original. MR 698779 (84h:33001a)
  • 4. Marvin I. Knopp, On the Fourier coefficients of cusp forms having small positive weight, Theta functions—Bowdoin 1987, Part 2 (Brunswick, ME, 1987) Proc. Sympos. Pure Math., vol. 49, Amer. Math. Soc., Providence, RI, 1989, pp. 111–127. MR 1013169 (90j:11035)
  • 5. Knopp, M.I. (1993). Modular Functions in Analytic Number Theory, 2nd edition. Chelsea, New York.
  • 6. Knopp, M.I. and Robins, S., Easy Proofs of Riemann's Functional Equation for $\zeta(s)$ and of Lipschitz Summation. Proc. Amer Math. Soc., posted on February 2, 2001, PII S 0002-9939(01)06033-6 (to appear in print).
  • 7. N. N. Lebedev, Special functions and their applications, Dover Publications, Inc., New York, 1972. Revised edition, translated from the Russian and edited by Richard A. Silverman; Unabridged and corrected republication. MR 0350075 (50 #2568)
  • 8. Joseph Lehner, Discontinuous groups and automorphic functions, Mathematical Surveys, No. VIII, American Mathematical Society, Providence, R.I., 1964. MR 0164033 (29 #1332)
  • 9. Lipschitz, R. (1889). Untersuchung der Eigenschaften einer Gattung von unendlichen Reihen. J. Reine und Angew. Math. CV, 127-156.
  • 10. Hans Maass, Lectures on modular functions of one complex variable, Notes by Sunder Lal. Tata Institute of Fundamental Research Lectures on Mathematics, No. 29, Tata Institute of Fundamental Research, Bombay, 1964. MR 0218305 (36 #1392)
  • 11. Paul C. Pasles, Nonanalytic automorphic integrals on the Hecke groups, Acta Arith. 90 (1999), no. 2, 155–171. MR 1709052 (2000i:11074)
  • 12. Pasles, P.C. (2000). Convergence of Poincaré series with two complex coweights. Contemp. Math., 251, AMS, Providence, 453-461.
  • 13. Pasles, P.C. (2000). A Hecke correspondence theorem for nonanalytic automorphic integrals. J. Number Theory 83 (2), 256-281. CMP 2000:16
  • 14. Pribitkin, W. (1995). The Fourier coefficients of modular forms and modular integrals having small positive weight. Doctoral Dissertation, Temple University, Philadelphia.
  • 15. Pribitkin, W. (1999). The Fourier coefficients of modular forms and Niebur modular integrals having small positive weight, I. Acta Arith. 91 (4), 291-309. CMP 2000:07
  • 16. Pribitkin, W. (2000). The Fourier coefficients of modular forms and Niebur modular integrals having small positive weight, II. Acta Arith. 93 (4), 343-358. CMP 2000:13
  • 17. Carl Ludwig Siegel, Die Funktionalgleichungen einiger Dirichletscher Reihen, Math. Z. 63 (1956), 363–373 (German). MR 0074533 (17,602c)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11F30, 11F37, 42A99

Retrieve articles in all journals with MSC (2000): 11F30, 11F37, 42A99


Additional Information

Paul C. Pasles
Affiliation: Department of Mathematical Sciences, Villanova University, Villanova, Pennsylvania 19085
Email: pasles@member.ams.org

Wladimir de Azevedo Pribitkin
Affiliation: Department of Mathematics, Princeton University, 607 Fine Hall, Princeton, New Jersey 08544
Email: w_pribitkin@msn.com, wladimir@princeton.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06038-5
PII: S 0002-9939(01)06038-5
Keywords: Poisson summation formula, Lipschitz summation formula, modular forms, automorphic forms, Eisenstein series, Hecke correspondence.
Received by editor(s): March 20, 2000
Published electronically: April 2, 2001
Communicated by: Dennis A. Hejhal
Article copyright: © Copyright 2001 American Mathematical Society