Duke Mathematical Journal

q-series identities and values of certain L-functions

George E. Andrews, Jorge Jiménez-Urroz, and Ken Ono

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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\hs,\\ 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\hs. \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*}

Article information

Duke Math. J. Volume 108, Number 3 (2001), 395-419.

First available in Project Euclid: 5 August 2004

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Digital Object Identifier

Zentralblatt MATH identifier

Primary: 11F20: Dedekind eta function, Dedekind sums
Secondary: 11B65: Binomial coefficients; factorials; $q$-identities [See also 05A10, 05A30]


Andrews, George E.; Jiménez-Urroz, Jorge; Ono, Ken. q -series identities and values of certain L -functions. Duke Math. J. 108 (2001), no. 3, 395--419. doi:10.1215/S0012-7094-01-10831-4. http://projecteuclid.org/euclid.dmj/1091737179.

Export citation


  • G. E. Andrews, Ramanujan's ``lost'' notebook, V: Euler's partition identity, Adv. Math. 61 (1986), 156--164. MR 87i:11137
  • --. --. --. --., ``Mock theta functions'' in Theta Functions (Brunswick, Maine, 1987), Part 2, Proc. Sympos. Pure Math. 49, Amer. Math. Soc., Providence, 1989, 283--298. MR 90h:33005
  • --------, The Theory of Partitions, Cambridge Math. Lib., Cambridge Univ. Press, Cambridge, 1998. MR 99c:11126
  • G. E. Andrews, F. Dyson, and D. Hickerson, Partitions and indefinite quadratic forms, Invent. Math. 91 (1988), 391--407. MR 89f:11071
  • H. Cohen, $q$-identities for Maass waveforms, Invent. Math. 91 (1988), 409--422. MR 89f:11072
  • P. Erdös, On an elementary proof of some asymptotic formulas in the theory of partitions, Ann. of Math. (2) 43 (1942), 437--450. MR 4:36a
  • N. J. Fine, Basic Hypergeometric Series and Applications, Math. Surveys Monogr. 27, Amer. Math. Soc., Providence, 1988. MR 91j:33011
  • G. Gasper and M. Rahman, Basic Hypergeometric Series, Encyclopedia Math. Appl. 35, Cambridge Univ. Press, Cambridge, 1990. MR 91d:33034
  • W. J. Leveque, Topics in Number Theory, Vol. 1, Addison-Wesley, Reading, Mass., 1956. MR 18:283d
  • P. A. MacMahon, Combinatory Analysis, Chelsea, New York, 1960. MR 25:5003
  • D. Zagier, Vassiliev invariants and a strange identity related to the Dedekind eta-function, to appear in Topology.