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Modularity gap for Eisenstein series. (English) Zbl 1214.11098
The author considers the behavior of the modularity gap of the Eisenstein series at nonnegative real numbers and that of the Ramanujan \(q\)-series and \(q\)-zeta values near the boundary \(|q|= 1\).
The Ramanujan \(q\)-series is defined by \[ \Phi_{s-1}(q)= \sum^\infty_{n= 1} {n^{s-1} q^n\over 1- q^n}= \sum^\infty_{n=1} \sigma_{s-1}(n)q^n\quad\text{with }\sigma_{s-1}(n)= \sum_{d|n} d^{s-1}, \] the Eisenstein series is written as \[ E_s(\tau)= {\zeta(1- s)\over 2}+ \Phi_{s-1}(e^{2\pi i\tau})\quad\text{for }\tau\in\mathbb C,\;\text{Im}(\tau)> 0, \] and the modularity gap of \(E_s(\tau)\) here is defined by \(\Delta_s(\tau)= \tau^{-s}E_s(-1/\tau)- E_s(\tau)\).
First the author expresses \(\Delta_s(\tau)\) in terms of the double series \[ \Lambda_\tau(s)= \sum_{j,k\in\mathbb N}(j\tau+ k)^{-s}\quad\text{for Re}(s)> 2, \] and from that he derives the formula for \(\Delta_s(\tau)\) as \(\tau\to\mu\) with \(\mu\in\mathbb R_{>0}\). When \(\mu\) is a positive rational number, the double series \(\Lambda_\mu(s)\) has a finite expression in terms of the Hurwitz zeta function, and so is the limit.
Next the author obtains the estimate of \(\Delta_s(\tau)\) as \(\tau\to 0\) through some sector for \(\text{Re}(s)> 2\), and by using this estimate the asymptotic expressions of \(\Phi_{s-1}(q)\) and the \(q\)-analogue of the Riemann zeta function defined by \[ \zeta_q(s)- (1-q)^s \sum^\infty_{n=1} {g^{n(s-1)}\over (1- q^n)^s}\quad\text{for Re}(s)\geq 2 \] (with \(s\in\mathbb N\) in most cases) near the boundary \(|q|= 1\) are given.
Reviewer: Kaori Ota (Tokyo)

MSC:
11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
30B30 Boundary behavior of power series in one complex variable; over-convergence
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[1] B. C. Berndt, Number theory in the spirit of Ramanujan , Amer. Math. Soc., Providence, RI, 2006. · Zbl 1117.11001
[2] J. Chazy, Sur les équations différentielles du troisième ordre et d’ordre supérieur dont l’intégrale générale a ses points critiques fixes, Acta Math. 34 (1911), no. 1, 317-385. · JFM 42.0340.03 · doi:10.1007/BF02393131
[3] M. Kaneko, N. Kurokawa and M. Wakayama, A variation of Euler’s approach to values of the Riemann zeta function, Kyushu J. Math. 57 (2003), no. 1, 175-192. · Zbl 1067.11053 · doi:10.2206/kyushujm.57.175
[4] S. Koyama and N. Kurokawa, Multiple Eisenstein series and multiple cotangent functions, J. Number Theory 128 (2008), no. 6, 1769-1774. · Zbl 1168.11035 · doi:10.1016/j.jnt.2007.06.004
[5] N. Kurokawa, Limit values of Eisenstein series and multiple cotangent functions, J. Number Theory 128 (2008), no. 6, 1775-1783. · Zbl 1168.11032 · doi:10.1016/j.jnt.2007.06.003
[6] M. Noumi, Euler ni manabu , Nihonhyoronsha, Tokyo, 2007. (in Japanese).
[7] K. Ota, Derivatives of Dedekind sums and their reciprocity law, J. Number Theory 98 (2003), no. 2, 280-309. · Zbl 1038.11028 · doi:10.1016/S0022-314X(02)00046-X
[8] Yu. A. Pupyrev, Linear and algebraic in dependence of q -zeta values, Mat. Zametki 78 (2005), no. 4, 608-613; translation in Math. Notes 78 (2005), no. 3-4, 563-568. · Zbl 1160.11338 · doi:10.1007/s11006-005-0155-3
[9] V. V. Zudilin, Diophantine problems for q -zeta values, Mat. Zametki 72 (2002), no. 6, 936-940; translation in Math. Notes 72 (2002), no. 5-6, 858-862. · Zbl 1044.11066 · doi:10.1023/A:1021450231834
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