Abstract
For a positive integer N, we derive a Kronecker type limit formula for a Dedekind type zeta function \(\zeta _{\mathcal {O}_{K}(N)}(s;[\mathfrak {a}])\) associated to a wide ideal class of a quadratic order of conductor N of a quadratic field K. As consequences, we establish a Chowla–Selberg type formula for a modular form for \(\Gamma _{0}(N)\), give a proof to a classical class number relation of a quadratic field, and find an asymptotic formula for the average value of the number of representations by a primitive positive definite quadratic form of discriminant \(N^{2}d\) with d fundamental. Furthermore, as an application, we obtain a formula for the Petersson norm of a theta function attached to a class character of the ideal group of conductor N of \(\mathcal {O}_{K}\).
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References
Blomer, V., Milićević, D.: Kloosterman sums in residue classes. JEMS 17, 51–69 (2015)
Cojocaru, A.C., Murty, M.R.: An Introduction to Sieve Methods and Their Applications. Cambridge University Press, Cambridge (2006)
Chen, I., Yui, N.: Singular values of Thompson series, groups, difference sets, and the monster (Columbus, OH, 1993). In: Arasu, K.T., Dillon, J.F., Harada, K., Sehgal, S., Solomon, R. (eds.) Ohio State Univ. Math. Res. Inst. Publ. 4, pp. 255–326. de Gruyter, Berlin (1996)
Chowla, S., Selberg, A.: On Epstein’s Zeta-function. J. Reine Angew Math. 227, 86–110 (1967)
Conrad, K.: The conductor ideal. https://kconrad.math.uconn.edu/blurbs/gradnumthy/conductor.pdf. Accessed 2 Jan 2020
Cox, D.: Primes of the form \(x^{2}+ny^{2}\). Wiley, Hoboken (1989)
Deninger, C.: On the analogue of the formula of Chowla and Selberg for real quadratic fields. J. Reine Angew Math. 351, 171–191 (1984)
Deshouillers, J.-M., Iwaniec, H.: Kloosterman sums and Fourier coefficients of cusp forms. Invent. Math. 70, 219–288 (1982)
Du, T., Yang, T.: Arithmetic Siegel–Weil formula on \(X_{0}(N)\). Adv. Math. 345, 702–755 (2019)
Gross, B., Zagier, D.: Heegner points and derivative of \(L\)-series. Invent. Math. 85, 225–320 (1986)
Hecke, E.: Über die Kroneckersche Grenzformel für reelle quadratische Körper und die Klassenzahl relativ-abelscher Körper. Verhandl. d. Naturforschenden Gesell. i. Basel 28, 363–372 (1917)
Kani, E.: The space of binary theta series. Ann. Sci. Math. Québec 36, 501–534 (2012)
Kronecker, L.: Zur Theorie der elliptischen Modulfunktionen. 4, 347–495 and 5, 1–132 (1929)
Kubota, T.: Elementary theory of Eisenstein series. Kodansha Ltd., Tokyo (1973)
Neukirch, J.: Algebraic Number Theory. Springer, Berlin (1999)
Siegel, C.L.: Lectures on Advanced Analytic Number Theory. Tata Institute, Bombay (1961)
Simard, N.: Petersson Inner Product of Theta Series, Ph.D. thesis, McGill University (2017)
Stevenhagen, P.: The Arithmetic of Number Rings. Algorithmic Number Theory, vol. 44. MSRI Publications, Cambridge (2008)
Zagier, D.: A Kronecker limit formula for real quadratic fields. Math. Ann. 213, 153–184 (1975)
Zagier, D.: Zetafunktionen und quadratische Körper: eine Einführung in die höhere Zahlentheorie, Hochschultext. Springer, Berlin 149+vi pages (1981)
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The author is supported by the Natural Science Foundation of China (Grant No. 11901586), the Natural Science Foundation of Guangdong Province (Grant No. 2019A1515011323) and the Sun Yat-sen University Research Grant for Youth Scholars (Grant No. 19lgpy244)
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Ye, D. On a Zeta Function Associated to a Quadratic Order. Results Math 75, 27 (2020). https://doi.org/10.1007/s00025-019-1153-1
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DOI: https://doi.org/10.1007/s00025-019-1153-1