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Period(d)ness of \(L\)-values. (English) Zbl 1316.11038

Borwein, Jonathan M. (ed.) et al., Number theory and related fields. In memory of Alf van der Poorten. Based on the proceedings of the international number theory conference, Newcastle, Australia, March 12–16, 2012. New York, NY: Springer (ISBN 978-1-4614-6641-3/hbk; 978-1-4614-6642-0/ebook). Springer Proceedings in Mathematics & Statistics 43, 381-395 (2013).
Let \(q=\exp(2 \pi i \tau)\), where \(\tau\) is in the upper half-plane \(\mathrm{Im} (\tau)>0\), and let \[ \eta(\tau)=q^{1/24} \prod_{m=1}^{\infty}(1-q^m)=\sum_{n=-\infty}^{\infty} (-1)^n q^{(6n+1)^2/24} \] be the Dedekind eta function. Set \(\eta_k=\eta(k \tau)\), and \[ L(f,m)=\frac{1}{(m-1)!} \int_{0}^{1} f \log^{m-1} q \frac{dq}{q} \] for a modular form \(f(\tau)=\sum_{n=1}^{\infty} a_n q^n\). For the cusp form \(f(\tau)=\nu_4^2 \nu_8^2\) (whose \(L\)-series coincide with those of a conductor \(32\) elliptic curve) the authors prove that \[ L(f,2)=\frac{\pi}{8} \int_{0}^{1} \frac{x}{\sqrt{1-x^4}} \log \frac{1+x}{1-x} dx=0.9170506353\dots \] and \[ L(f,3)=\frac{\pi^2}{128} \int_{0}^{1} \frac{(1-\sqrt{1-x^2})^2}{(1-x^2)^{3/4}} dx \int_{0}^1 \int_{0}^1 \frac{dy dw}{1-x^2(1-(1-y^2)(1-w^2))} \] (which is \(0.9826801478\dots\)). Some further hypergeometric evaluations are also obtained.
For the entire collection see [Zbl 1266.11001].

MSC:

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11G55 Polylogarithms and relations with \(K\)-theory
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