q-series identities and values of certain L-functions

Abstract

As usual, define Dedekind's eta-function η(z) by the infinite product

$$ \eta(z):=q^{1/24}\prod_{n=1}^{\infty} \big(1-q^n\big) \quad \big(q:=e^{2\pi i z} \text{ throughout}\big). $$

In a recent paper, D. Zagier proved that (note: empty products equal 1 throughout)

$$ \sum_{n=0}^{\infty} \Big(\eta(24z)-q\big(1-q^{24}\big) \big(1-q^{48}\big)\cdots \big(1-q^{24n}\big)\Big) =\eta(24z)D(q)+E(q), $$

where the series D(q) and E(q) are defined by

$\begin{align*} D(q)&=-\frac{1}{2}+\sum_{n=1}^{\infty}\frac{q^{24n}}{1-q^{24n}}\\ &=-\frac{1}{2}+\sum_{n=1}^{\infty}d(n)q^{24n}\\ &=-\frac{1}{2}+q^{24}+2q^{48}+2q^{72}+3q^{96}+\cdots\quad,\\ E(q)&=\frac{1}{2}\sum_{n=1}^{\infty} \bigg(\frac{12}{n}\bigg)nq^{n^2} =\frac{1}{2}q-\frac{5}{2}q^{25}- \frac{7}{2}q^{49}+\frac{11}{2}q^{121}+\cdots\quad. \end{align*}$

Here d(n) denotes the number of positive divisors of n. We obtain two infinite families of such identities and describe some consequences for L-functions and partitions. For example, if χ₂ is the Kronecker character for ℚ($\sqrt{2}$), these identities can be used to show that

$\begin{align*} &-2e^{-t/8}\sum_{n=0}^{\infty} \frac{\big(1-e^{-2t}\big)\big(1-e^{-4t}\big) \cdots\big(1-e^{-2nt}\big)} {\big(1+e^{-t}\big)\big(1+e^{-3t}\big) \cdots\big(1+e^{-(2n+1)t}\big)} \\ &\hspace{110pt}=\sum_{n=0}^{\infty} \bigg(\frac{-1}{8}\bigg)^n\cdot L(\chi_{2},-2n-1)\cdot \frac{t^{n}}{n!}. \end{align*}$