Modularity gap for Eisenstein series (Q989433)

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.