Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-01-06038-5
Published electronically: April 2, 2001
MathSciNet review: 1844990
Full-text PDF

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. Benedetto, J.J. and Zimmerman, G. (1997). Sampling Multipliers and the Poisson Summation Formula. J. of Fourier Analysis and Appl. 3, 505-524. MR 99b:42011
  • 2. Dickson, L.E. (1930). Studies in the Theory of Numbers. Univ. of Chicago Press, Chicago.
  • 3. Erdelyi, A. (1981). Higher Transcendental Functions, Vol. I. Krieger Publishing Company, Melbourne, Florida. MR 84h:33001a
  • 4. Knopp, M.I. (1989). On the Fourier Coefficients of Cusp Forms Having Small Positive Weight. Proc. of Symposia in Pure Math. 49 (2), 111-127. MR 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. Lebedev, N.N. (1972). Special Functions and Their Applications. Dover, New York. MR 50:2568
  • 8. Lehner, J. (1964). Discontinuous Groups and Automorphic Functions. Amer. Math. Soc., Providence, RI. MR 29:1332
  • 9. Lipschitz, R. (1889). Untersuchung der Eigenschaften einer Gattung von unendlichen Reihen. J. Reine und Angew. Math. CV, 127-156.
  • 10. Maass, H. (1964). Modular Functions of One Complex Variable, Tata Institute, Bombay. MR 36:1392
  • 11. Pasles, P.C. (1999). Nonanalytic automorphic integrals on the Hecke groups. Acta Arith. 90 (2), 155-171. MR 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. Siegel, C.L. (1956). Die Funktionalgleichungen einiger Dirichletscher Reihen. Math. Z. 63, 363-373. MR 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: https://doi.org/10.1090/S0002-9939-01-06038-5
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

American Mathematical Society