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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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References:
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