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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)

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
Full Text: DOI
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