A \(q\)-analogue for Euler's evaluations of the Riemann zeta function (Q2316316)

In a previous paper [Ann. Comb. 23, No. 3--4, 801--806 (2019; Zbl 1459.11051)] the author presented \(q\)-analogs of Euler's \(\zeta(4)=\pi^4/90\) and \(\zeta(6)=\pi^6/945\) which were in turn inspired by a result of Z. W. Sun concerning a \(q\)-analogue of Euler's \(\zeta(2)=\pi^2/6\). In the paper under review the author extends and generalises these results by presenting \(q\)-analogues of Euler's \(\zeta(2k)=(-1)^{k+1}2^{2k}B_{2k}\pi^{2k}/(2(2k!))\) and any positive integer \(k\). There are two main theorems. The first states that if \(k\ge 2\) is an even positive integer and \(q=e^{2\pi i \tau}\), where \(\tau\) is in the complex half-plane \(\Im(\tau)>0\), then \[ \sum_{n=0}^{\infty} \frac{2^{2k-1}q^{2n+1} P_{2k-2}^e(q^{2n+1})}{(1-q^{2n+1})^{2k}}-T_{2k}(\tau/2)=q^{k/2} d_k \prod_{n=1}^{\infty} \frac{(1-q^{2n})^{4k}}{1-q^{2n-1})^{4k}}. \] In the above, \(P_{2k-2}^e(z)\) is polynomial of degree \(2k-2\) in \(z\) with integer coefficients which are explicitly given in terms of binomial coefficients and Stirling numbers of the second kind, \(d_k\) is a certain rational multiple of the Bernoulli number \(B_{2k}\) and \(T_{2k}(\tau)\) is a weight \(2k\)-cusp form on \(\Gamma_0(4)\). In particular, \(T_{2k}(\tau/2)\) tends to \(0\) when \(q\) tends to \(1\) from inside the unit disk. There is a similar theorem for \(k\) odd. At the start \(T_{2k}(\tau)\) is not explicitly given but later it is given in terms of the \(4k\)th power of the Dirichlet \(\eta\)-function evaluated in \(q^2\) and a certain Eisenstein series \(H_{2k}(\tau)\) of weight \(2k\) whose coefficients are related to the sum of the \(2k-1\) powers of the divisors of \(n\). The proof relates the \(q\)-product on the right-hand side with the number of representations of \(n\) as a sum of \(4k\) triangular numbers which in turn is related to the above Eisenstein series \(H_{2k}(\tau)\) and the coefficients of \(T_{2k}(\tau)\) via a result from [\textit{A. Atanasov} et al., Involve 1, No. 1, 111--121 (2008; Zbl 1229.11066)]. The presence of the argument \(\tau/2\) in the case \(k\) even is to be interpreted as the fact that the difference between the first \(q\)-sum in the left-hand side and the \(q\)-product in the right-hand side is a series in \(q^2\). Based on numerical calculations for \(k=1,3,5\), the author suggests that this phenomenon likely happens for \(k\) odd as well which he leaves open.