Abstract
We define an axiomatic class of \(L\)-functions extending the Selberg class. We show in particular that one can recast the traditional conditions of an Euler product, analytic continuation and functional equation in terms of distributional identities akin to Weil’s explicit formula. The generality of our approach enables some new applications; for instance, we show that the \(L\)-function of any cuspidal automorphic representation of \({{\mathrm{GL}}}_3(\mathbb {A}_\mathbb {Q})\) has infinitely many zeros of odd order.
Similar content being viewed by others
Notes
Selberg acknowledges in a footnote of his paper that we may take \(\lambda \) to be a half-integer in every known case. It seems likely that he did not intend for his definition to be taken as a serious attempt at generalization, but rather as a recognition of the fact that the \(\Gamma \)-factors are non-canonical because of the Legendre and Gauss multiplication formulas. We note that those identities have analogues at the finite places as well, e.g. the Legendre duplication formula is analogous to the “difference of squares” identity \(1-p^{-2s}=(1-p^{-s})(1+p^{-s})\), but this ambiguity causes no real confusion. Note also that the analogue of \(\Gamma (\lambda {s})\) is the generalized Dirichlet series \(1/(1-p^{-2\lambda {s}})\), which is not permitted under Selberg’s definition unless \(\lambda \) is a half-integer.
It is tempting to consider more general singularities as well, but we quickly find ourselves in a much larger landscape of functions that is presumably very hard to classify; for instance, the Selberg zeta-functions and their trace formulae give identities of this type with second-order singularities.
There is also an argument if favor of keeping examples like \(\zeta (2s-\frac{1}{2})\) in the definition: Shimura’s integral representation for the symmetric square \(L\)-function could be viewed as an extension of the Rankin–Selberg method to this example, and that in turn was a key ingredient in the proof of the Gelbart–Jacquet lift.
References
Avdispahić, M., Smajlović, L.: On the Selberg orthogonality for automorphic \(L\)-functions. Arch. Math. (Basel) 94(2), 147–154 (2010). doi:10.1007/s00013-009-0099-z
Booker, A.R., Hiary, G.A., Keating, J.P.: Detecting squarefree numbers. arXiv:1304.6937 (2013)
Booker, A.R., Krishnamurthy, M.: Weil’s converse theorem with poles. Int. Math. Res. Not. (2013). doi:10.1093/imrn/rnt127
Booker, A.R., Thorne, F.: Zeros of \(L\)-functions outside the critical strip. arXiv:1306.6362 (2013)
Cogdell, J.W., Piatetski-Shapiro, I.I.: Converse theorems for \({{\rm GL}_{n}}\). Inst. Hautes Études Sci. Publ. Math. 79, 157–214 (1994). http://www.numdam.org/item?id=PMIHES_1994__79__157_0
Conrey, J.B., Farmer, D.W., Keating, J.P., Rubinstein, M.O., Snaith, N.C.: Integral moments of \(L\)-functions. Proc. London Math. Soc. (3) 91(1), 33–104 (2005). doi:10.1112/S0024611504015175
Conrey, J.B., Ghosh, A.: On the Selberg class of Dirichlet series: small degrees. Duke Math. J. 72(3), 673–693 (1993). doi:10.1215/S0012-7094-93-07225-0
Iwaniec, H., Sarnak, P.: Perspectives on the analytic theory of \(L\)-functions. Geom. Funct. Anal. (Special Volume, Part II), 705–741 (2000). 2000, doi:10.1007/978-3-0346-0425-3_6. GAFA (Tel Aviv, 1999)
Kaczorowski, J., Perelli, A.: On the structure of the Selberg class. I. \(0{\le } d{\le } 1\). Acta Math. 182(2), 207–241 (1999). doi:10.1007/BF02392574
Kaczorowski, J., Perelli, A.: On the structure of the Selberg class. V. \(1<d<5/3\). Invent. Math. 150(3), 485–516 (2002). doi:10.1007/s00222-002-0236-9
Kaczorowski, J., Perelli, A.: On the structure of the Selberg class, VII: \(1<d<2\). Ann. Math. 173(3), 1397–1441 (2011). doi:10.4007/annals.2011.173.3.4
Lemke Oliver, R.J.: Multiplicative functions dictated by Artin symbols. Acta Arith. 161(1), 21–31 (2013). doi:10.4064/aa161-1-2
Luo, W., Rudnick, Z., Sarnak, P.: On the generalized Ramanujan conjecture for \({\rm GL}(n)\). In: Automorphic forms, automorphic representations, and arithmetic (Fort Worth, TX, 1996), Proc. Sympos. Pure Math., vol. 66, pp. 301–310. Am. Math. Soc., Providence, RI (1999)
Murty, M.R.: Stronger multiplicity one for Selberg’s class. In: Harmonic analysis and number theory (Montreal, PQ, 1996), CMS Conf. Proc., vol. 21, pp. 133–142. Amer. Math. Soc., Providence, RI (1997)
Raghunathan, R.: A comparison of zeros of \(L\)-functions. Math. Res. Lett. 6(2), 155–167 (1999)
Rudnick, Z., Sarnak, P.: Zeros of principal \(L\)-functions and random matrix theory. Duke Math. J. 81(2), 269–322 (1996). doi:10.1215/S0012-7094-96-08115-6. A celebration of John F. Nash, Jr
Saias, E., Weingartner, A.: Zeros of Dirichlet series with periodic coefficients. Acta Arith. 140(4), 335–344 (2009). doi:10.4064/aa140-4-4
Selberg, A.: Old and new conjectures and results about a class of Dirichlet series. In: Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989), pp. 367–385. Univ. Salerno, Salerno (1992)
Soundararajan, K.: Degree 1 elements of the Selberg class. Expo. Math. 23(1), 65–70 (2005). doi:10.1016/j.exmath.2005.01.013
Stein, E.M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43. Princeton University Press, Princeton, NJ, : With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III (1993)
Tao, T.: Lecture notes 8 for 247B. http://www.math.ucla.edu/tao/247b.1.07w/notes8.dvi
Thorne, F.: Analytic properties of Shintani zeta functions. In: Proceedings of the RIMS Symposium on automorphic representations an related topics. Kyoto (2010)
Weil, A.: Sur les “formules explicites” de la théorie des nombres premiers. Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952(Tome Supplementaire), 252–265 (1952)
Weil, A.: Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen. Math. Ann. 168, 149–156 (1967)
Wu, J., Ye, Y.: Hypothesis H and the prime number theorem for automorphic representations. Funct. Approx. Comment. Math. 37(part 2), 461–471 (2007). doi:10.7169/facm/1229619665
Acknowledgments
This work was carried out during a year-long stay at the Research Institute for Mathematical Sciences, Kyoto, Japan. It is a pleasure to thank all of the RIMS staff, in particular my host, Akio Tamagawa, for their generous hospitality. I would also like to thank Akihiko Yukie for organizing the Conference on Automorphic Forms at Kyoto University in June 2013, which provided the impetus for this work. Finally, I thank Frank Thorne for performing some computations in relation to Remarks 1.12, and Brian Conrey, David Farmer, Peter Sarnak and Akshay Venkatesh for helpful suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
The author was supported by EPSRC Fellowship EP/H005188/1.
Rights and permissions
About this article
Cite this article
Booker, A.R. \(L\)-functions as distributions. Math. Ann. 363, 423–454 (2015). https://doi.org/10.1007/s00208-015-1178-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-015-1178-z