Lambert series and Liouville's identities

Tom 445 / 2007

A. Alaca, Ş. Alaca, E. McAfee, K. S. Williams Dissertationes Mathematicae 445 (2007), 1-72 MSC: 11R11, 11R27. DOI: 10.4064/dm445-0-1

Streszczenie

The relationship between Liouville's arithmetic identities and products of Lambert series is investigated. For example it is shown that Liouville's arithmetic formula for the sum $$ \sum_{\textstyle {(a,b,x,y) \in \mathbb{N}^{{4}}\atop ax+by=n}} (F(a-b)-F(a+b)),$$ where $n\in \mathbb{N}$ and $F:\mathbb{Z} \rightarrow \mathbb{C}$ is an even function, is equivalent to the Lambert series for $$ \bigg( \sum_{n=1}^{\infty} \frac{q^n}{1-q^n} \sin n \theta \bigg)^2 \quad\ (\theta \in \mathbb{R} ,\, |q| <1)$$ given by Ramanujan.

Autorzy

  • A. AlacaCentre for Research in Algebra and Number Theory
    School of Mathematics and Statistics
    Carleton University
    Ottawa, Ontario, Canada K1S 5B6
    e-mail
  • Ş. AlacaCentre for Research in Algebra and Number Theory
    School of Mathematics and Statistics
    Carleton University
    Ottawa, Ontario, Canada K1S 5B6
    e-mail
  • E. McAfeeCentre for Research in Algebra and Number Theory
    School of Mathematics and Statistics
    Carleton University
    Ottawa, Ontario, Canada K1S 5B6
    e-mail
  • K. S. WilliamsCentre for Research in Algebra and Number Theory
    School of Mathematics and Statistics
    Carleton University
    Ottawa, Ontario, Canada K1S 5B6
    e-mail

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