Mizuno, Y. The Rankin-Selberg convolution for Cohen’s Eisenstein series of half integral weight. (English) Zbl 1082.11025 Abh. Math. Semin. Univ. Hamb. 75, 1-20 (2005). In the theory of modular forms of half-integral weight one can define both holomorphic and non-holomorphic Eisenstein series. The Fourier expansion of these series can also be formulated arithmetically; the coefficients are non-zero at \(d\) when \(d\) is the discriminant of a quadratic field. In this case the Fourier coefficent at \(d\) is in essence a special value of the zeta-function of the field. This was observed by H. Cohen. The object of this paper is to determine the analytic properties of the “convolution product” of two such series. This is done by using an appropriate variant of the Rankin-Selberg method. The interest in these products is that they appear in the explicit formulae for Koecher-Maass series of Siegel-Eisenstein series. Reviewer: Samuel James Patterson (Göttingen) Cited in 6 Documents MSC: 11F37 Forms of half-integer weight; nonholomorphic modular forms 11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations 11F12 Automorphic forms, one variable Keywords:half-integral weight; Rankin-Selberg method; Eisenstein series; Koecher-Maas series; Siegel-Eisenstein series; quadratic fields PDFBibTeX XMLCite \textit{Y. Mizuno}, Abh. Math. Semin. Univ. Hamb. 75, 1--20 (2005; Zbl 1082.11025) Full Text: DOI References: [1] Cohen, H., Sums involving the values at negative integers ofL-functions of quadratic characters, Math. Ann., 217, 3, 271-285 (1975) · Zbl 0311.10030 [2] Gupta, S. D., The Rankin-Selberg method on congruence subgroups, Illinois Journal of Math, 44-1, 95-103 (2000) · Zbl 0951.11019 [3] T. Ibukiyama, A survey on the new proof of Saito-Kurokawa lifting after Duke and Imamoglu (in Japanese). In:Report of the fifth summer school of number theory “Introduction to Siegel modular forms”, 1997, pp. 134-176. [4] Ibukiyama, T.; Katsurada, H., An explicit form of Koecher Maass Dirichlet series associated with Siegel Eisenstein series, RIMS Kokyuroku, 956, 41-51 (1996) · Zbl 1043.11521 [5] T. Ibukiyama andH. Saito, On zeta functions associated to symmetric matrices (II). MPI preprint 97-37. · Zbl 1275.11085 [6] Kohnen, W., Modular forms of half-integral weight of Г_0(4), Math. Ann., 28, 249-266 (1980) · Zbl 0416.10023 [7] Kubota, T., Elementary theory of Eisenstein series (1973), New York-London-Sydney: Kodansha Ltd., Tokyo; Halsted Press, New York-London-Sydney · Zbl 0268.10012 [8] M. Miyawaki, Master thesis (in Japanese), 1999, Osaka University. http://www.math.wani.osaka-u.ac. jp/group/numberth/graduate/⇝ papers/paper.html [9] Shimura, G., On the holomorphy of certain Dirichlet series, Proc. London Math. Soc., 31, 1, 79-98 (1975) · Zbl 0311.10029 [10] Zagier, D., The Rankin-Selberg method for automorphic functions which are not of rapid decay, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 28, 415-437 (1981) · Zbl 0505.10011 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.