Skip to main content Accessibility help
Hostname: page-component-544b6db54f-rcd7l Total loading time: 0.277 Render date: 2021-10-24T04:20:34.895Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }


Published online by Cambridge University Press:  21 May 2018

Graduate School of Technology, Industrial and Social Sciences, Tokushima University, 2-1, Minami-josanjima-cho, Tokushima, 770-8506, Japan email


The author gives the analytic properties of the Rankin–Selberg convolutions of two half-integral weight Maass forms in the plus space. Applications to the Koecher–Maass series associated with nonholomorphic Siegel–Eisenstein series are given.

© 2018 Foundation Nagoya Mathematical Journal  

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


Arakawa, T., Dirichlet series related to the Eisenstein series on the Siegel upper half-plane , Comment. Math. Univ. St. Pauli 27(1) (1978), 2942.Google Scholar
Arakawa, T., Ibukiyama, T. and Kaneko, M., Bernoulli Numbers and Zeta Functions, Springer Monographs in Mathematics, Springer, Tokyo, 2014, xii+274 pp. With an appendix by Don Zagier.Google Scholar
Böcherer, S., Bemerkungen über die Dirichletreihen von Koecher und Maass , Mathematica Göttingensis, Schriftenreihe des SFB Geometrie und Analysis, Heft 68 (1986), 36.Google Scholar
Biro, A., A relation between triple products of weight 0 and weight 1/2 cusp forms , Israel J. Math. 182 (2011), 61101.CrossRefGoogle Scholar
Chinta, G., Friedberg, S. and Hoffstein, J., “ Multiple Dirichlet series and automorphic forms ”, in Multiple Dirichlet Series, Automorphic Forms, and Analytic Number Theory, Proc. Sympos. Pure Math. 75 , American Mathematical Society, Providence, RI, 2006, 341.CrossRefGoogle Scholar
Chinta, G. and Gunnells, P., Weyl group multiple Dirichlet series constructed from quadratic characters , Invent. Math. 167(2) (2007), 327353.CrossRefGoogle Scholar
Cohen, H., Sums involving the values at negative integers of L-functions of quadratic characters , Math. Ann. 217 (1975), 271285.CrossRefGoogle Scholar
Diamantis, N. and Goldfeld, D., A converse theorem for double Dirichlet series and Shintani zeta functions , J. Math. Soc. Japan 66(2) (2014), 449477.CrossRefGoogle Scholar
Duke, W. and Imamoḡlu, Ö., A converse theorem and the Saito–Kurokawa lift , Int. Math. Res. Not. IMRN (7) (1996), 347355.CrossRefGoogle Scholar
Goldfeld, D. and Hoffstein, J., Eisenstein series of 1/2-integral weight and the mean value of real Dirichlet L-series , Invent. Math. 80(2) (1985), 185208.CrossRefGoogle Scholar
Hashim, A. and Ram Murty, M., On Zagier’s cusp form and the Ramanujan 𝜏 function , Proc. Indian Acad. Sci. Math. Sci. 104(1) (1994), 9398.CrossRefGoogle Scholar
Ibukiyama, T. and Katsurada, H., “ Koecher–Maass series for real analytic Siegel Eisenstein series ”, in Automorphic Forms and Zeta Functions, Proceedings of the conference in memory of Tsuneo Arakawa, World Scientific, Hackensack, NJ, USA, 2006, 170197.CrossRefGoogle Scholar
Ibukiyama, T. and Saito, H., On zeta functions associated to symmetric matrices, II: functional equations and special values , Nagoya Math. J. 208 (2012), 265316.CrossRefGoogle Scholar
Katok, S. and Sarnak, P., Heegner points, cycles and Maass forms , Israel J. Math. 84(1–2) (1993), 193227.CrossRefGoogle Scholar
Kaufhold, G., Dirichletsche Reihe mit Funktionalgleichung in der Theorie der Modulfunktion 2. Grades , Math. Ann. 137 (1959), 454476.CrossRefGoogle Scholar
Lebedev, N., Special Functions and their Applications, Revised edition (ed. Silverman, R. A.) Dover Publications, Inc., New York, 1972, xii+308 pp. translated from the Russian. Unabridged and corrected republication.Google Scholar
Kohama, H. and Mizuno, Y., Kernel functions of the twisted symmetric square of elliptic modular forms , Mathematika 64 (2018), 184210.CrossRefGoogle Scholar
Luo, W., Rudnick, Z. and Sarnak, P., The variance of arithmetic measures associated to closed geodesics on the modular surface , J. Mod. Dyn. 3(2) (2009), 271309.CrossRefGoogle Scholar
Maass, H., Konstruktion ganzer Modulformen halbzahliger Dimension mit V-Multiplikatoren in einer und zwei Variablen , Abhandlungen aus dem Mathematischen Seminar der Hansischen Universitat 12 (1937), 133162.CrossRefGoogle Scholar
Maass, H., Siegel’s Modular Forms and Dirichlet Series, Lecture Notes in Mathematics. 216 , Springer, Berlin–New York, 1971, v+328 pp.CrossRefGoogle Scholar
Matthes, R., Rankin–Selberg method for real analytic cusp forms of arbitrary real weight , Math. Z. 211(1) (1992), 155172.CrossRefGoogle Scholar
Miyake, T., Modular Forms, x+335 pp. Springer, Berlin, 1989.CrossRefGoogle Scholar
Mizuno, Y., The Rankin–Selberg convolution for real analytic Cohen’s Eisenstein series of half integral weight , J. Lond. Math. Soc. (2) 78 (2008), 183197.CrossRefGoogle Scholar
Mizuno, Y., Koecher–Maass series for positive definite Fourier coefficients of real analytic Siegel–Eisenstein series of degree 2 , Bull. Lond. Math. Soc. 41 (2009), 10171028.CrossRefGoogle Scholar
Mizuno, Y., Dirichlet series associated with square of class numbers of binary quadratic forms , Math. Z. 272(3–4) (2012), 11151135.CrossRefGoogle Scholar
Mizuno, Y., On characterization of Siegel cusp forms of degree 2 by the Hecke bound , Mathematika 61(1) (2015), 89100.CrossRefGoogle Scholar
Müller, W., The Rankin–Selberg method for non-holomorphic automorphic forms , J. Number Theory 51(1) (1995), 4886.CrossRefGoogle Scholar
Narkiewicz, W., Number Theory, xii, 371 p World Scientific, Singapore, 1983, Transl. from the Polish by S. Kanemitsu.Google Scholar
Pitale, A., Jacobi Maass forms , Abh. Math. Semin. Univ. Hambg. 79(1) (2009), 87111.CrossRefGoogle Scholar
Rademacher, H., On the Phragmén–Lindelöf theorem and some applications , Math. Z. 72 (1959/1960), 192204.CrossRefGoogle Scholar
Sato, F., Zeta functions of (SL2 × SL2 × GL2, M 2M 2) associated with a pair of Maass cusp forms , Comment. Math. Univ. St. Pauli 55(1) (2006), 7795.Google Scholar
Shimura, G., On modular forms of half integral weight , Ann. of Math. (2) 97 (1973), 440481.CrossRefGoogle Scholar
Shimura, G., On the holomorphy of certain Dirichlet series , Proc. Lond. Math. Soc. (3) 31(1) (1975), 7998.CrossRefGoogle Scholar
Shimura, G., Elementary Dirichlet Series and Modular Forms, Springer Monographs in Mathematics, Springer, New York, 2007, viii+147 pp. ISBN: 978-0-387-72473-7.CrossRefGoogle Scholar
Siegel, C., Die Funktionalgleichungen einiger Dirichletscher Reihen , Math. Z. 63 (1956), 363373.CrossRefGoogle Scholar
Siegel, C., Advanced Analytic Number Theory, Second edition, Tata Institute of Fundamental Research Studies in Mathematics. 9 , v+268 pp. Tata Institute of Fundamental Research, Bombay, 1980.Google Scholar
Suzuki, T., Distributions with automorphy and Dirichlet series , Nagoya Math. J. 73 (1979), 157169.CrossRefGoogle Scholar
Sturm, J., Special values of zeta functions, and Eisenstein series of half integral weight , Amer. J. Math. 102(2) (1980), 219240.CrossRefGoogle Scholar
Wen, J., Shintani zeta functions and Weyl group multiple Dirichlet series. Thesis (Ph.D.) State University of New York at Stony Brook. 2014. 64 pp. ISBN: 978-1321-11773-8 (See also arXiv:1311.2132, Bhargava Integer Cubes and Weyl Group Multiple Dirichlet Series).Google Scholar
Zagier, D., Nombres de classes et formes modulaires de poids 3/2 , C. R. Acad. Sci. Paris Ser. A-B 281(21, Ai) (1975), A883A886.Google Scholar
Zagier, D., “ Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields ”, in Modular Functions of One Variable VI, Lecture Notes in Mathematics. 627 , Springer, Berlin, 1977, 105169.CrossRefGoogle Scholar
Zagier, D., The Rankin–Selberg method for automorphic functions which are not of rapid decay , J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), 415437.Google Scholar
Zagier, D., “ Eisenstein series and the Riemann zeta function ”, in Automorphic Forms, Representation Theory and Arithmetic (Bombay, 1979), Tata Inst. Fund. Res. Studies in Math. 10 , Tata Inst. Fundamental Res., Bombay, 1981, 275301.CrossRefGoogle Scholar

Send article to Kindle

To send this article to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the or variations. ‘’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Available formats

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Available formats

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *