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Book Review

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Book Information:

Author: Carlos Moreno
Title: Advanced analytic number theory: L-functions
Additional book information: Mathematical Surveys and Monographs, vol. 115, American Mathematical Society, Providence, RI, 2005, xx+291 pp., ISBN 978-0-8218-3641-5, US$76.00

[Art] James Arthur, Harmonic analysis and group representations, Notices Amer. Math. Soc. 47 (2000), no. 1, 26–34. MR 1733064
[Bru] Farrell Brumley, Effective multiplicity one on 𝐺𝐿_{𝑁} and narrow zero-free regions for Rankin-Selberg 𝐿-functions, Amer. J. Math. 128 (2006), no. 6, 1455–1474. MR 2275908
[BFH] Daniel Bump, Solomon Friedberg, and Jeffrey Hoffstein, On some applications of automorphic forms to number theory, Bull. Amer. Math. Soc. (N.S.) 33 (1996), no. 2, 157–175. MR 1359575, https://doi.org/10.1090/S0273-0979-96-00654-4
[GLS] Stephen S. Gelbart, Erez M. Lapid, and Peter Sarnak, A new method for lower bounds of 𝐿-functions, C. R. Math. Acad. Sci. Paris 339 (2004), no. 2, 91–94 (English, with English and French summaries). MR 2078295, https://doi.org/10.1016/j.crma.2004.04.024
Stephen S. Gelbart and Erez M. Lapid, Lower bounds for 𝐿-functions at the edge of the critical strip, Amer. J. Math. 128 (2006), no. 3, 619–638. MR 2230919
[GeS] Stephen Gelbart and Freydoon Shahidi, Boundedness of automorphic 𝐿-functions in vertical strips, J. Amer. Math. Soc. 14 (2001), no. 1, 79–107. MR 1800349, https://doi.org/10.1090/S0894-0347-00-00351-9
[Har] M. Harris, ``Review of Introduction to the Langlands Program'', Bull. Amer. Math. Soc. (N.S.), 41 (2004), 257-266.
[IwK] Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. MR 2061214
[JSh] H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic representations. I, Amer. J. Math. 103 (1981), no. 3, 499–558. MR 618323, https://doi.org/10.2307/2374103
[Kow] E. Kowalski, ``Elementary Theory of L-functions'', An Introduction to the Langlands Program, Birkhäuser, Boston, 2003.
[Lan] R. P. Langlands, Problems in the theory of automorphic forms, Lectures in modern analysis and applications, III, Springer, Berlin, 1970, pp. 18–61. Lecture Notes in Math., Vol. 170. MR 0302614
[LWY] Jianya Liu, Yonghui Wang, and Yangbo Ye, A proof of Selberg’s orthogonality for automorphic 𝐿-functions, Manuscripta Math. 118 (2005), no. 2, 135–149. MR 2177681, https://doi.org/10.1007/s00229-005-0563-4
[Mic] P. Michel, Analytic Number Theory and Families of Automorphic L-Functions, to appear.
[Mor1] C. J. Moreno, Explicit formulas in the theory of automorphic forms, Number Theory Day (Proc. Conf., Rockefeller Univ., New York, 1976), Springer, Berlin, 1977, pp. 73–216. Lecture Notes in Math., Vol. 626. MR 0476650
[Mor2] Carlos J. Moreno, Analytic proof of the strong multiplicity one theorem, Amer. J. Math. 107 (1985), no. 1, 163–206. MR 778093, https://doi.org/10.2307/2374461
[Mur] M. Ram Murty and V. Kumar Murty, Non-vanishing of 𝐿-functions and applications, Progress in Mathematics, vol. 157, Birkhäuser Verlag, Basel, 1997. MR 1482805
[RuS] Zeév Rudnick and Peter Sarnak, Zeros of principal 𝐿-functions and random matrix theory, Duke Math. J. 81 (1996), no. 2, 269–322. A celebration of John F. Nash, Jr. MR 1395406, https://doi.org/10.1215/S0012-7094-96-08115-6
[Sar] Peter Sarnak, Nonvanishing of 𝐿-functions on ℜ(𝔰)=1, Contributions to automorphic forms, geometry, and number theory, Johns Hopkins Univ. Press, Baltimore, MD, 2004, pp. 719–732. MR 2058625
[Sel] Atle Selberg, Old and new conjectures and results about a class of Dirichlet series, Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989) Univ. Salerno, Salerno, 1992, pp. 367–385. MR 1220477
[Sha] Freydoon Shahidi, Automorphic 𝐿-functions and functoriality, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 655–666. MR 1957073

Review Information:

Reviewer: S. Gelbart
Affiliation: Weizmann Institute of Science
Email: stephen.gelbart@weizmann.ac.il

Journal: Bull. Amer. Math. Soc. 45 (2008), 169-175
MSC (2000): Primary 11Mxx, 11Rxx, 22Exx; Secondary 11Sxx
Published electronically: June 8, 2007
Review copyright: © Copyright 2007 American Mathematical Society