Q-series identities and values of certain L-functions

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

Research output: Contribution to journalArticle

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Abstract

As usual, define Dedekind's eta-function η(z) by the infinite product η(z}:=q1/24n=1 (1-qn) (q:=e2πix throughout). In a recent paper, D. Zagier proved that (note: empty products equal 1 throughout) ∑n=0(η(24x)-q(1-q24)(1-q 48)⋯(1-q24n))=η(24x))D(q)+E(q), where the series D(q) and E(q) are defined by D(q) = -1/2+∑n=1 q24n/1-q24n = -1/2+∑n=1d(n)q24n = -1/2+q24+2q48+2q72+3q96+... E(q) = 1/2∑n=1(12/n)nqn2=1/2q-5/2q 25-7/2q49+11/2q121+... 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(√2), these identities can be used to show that -2e-t/8n=0 (1-e-2t) (1-e-4t)⋯(1-e-2nt)/1+e-t(1+e -3t)⋯(1+e-(2n+1)t) = ∑n=0(-1/8)n ·L(χ2,-2n-1)·tn/n

Original languageEnglish (US)
Pages (from-to)395-419
Number of pages25
JournalDuke Mathematical Journal
Volume108
Issue number3
DOIs
StatePublished - Jan 1 2001

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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