Ramanujan’s Formula for ζ(2n + 1)

  • Bruce C. Berndt
  • Armin Straub


Ramanujan made many beautiful and elegant discoveries in his short life of 32 years, and one of them that has attracted the attention of several mathematicians over the years is his intriguing formula for ζ(2n + 1). To be sure, Ramanujan’s formula does not possess the elegance of Euler’s formula for ζ(2n), nor does it provide direct arithmetical information. But, one of the goals of this survey is to convince readers that it is indeed a remarkable formula. In particular, we discuss the history of Ramanujan’s formula, its connection to modular forms, as well as the remarkable properties of the associated polynomials. We also indicate analogues, generalizations and opportunities for further research.


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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Illinois at Urbana–ChampaignUrbanaUSA
  2. 2.Department of Mathematics and StatisticsUniversity of South AlabamaMobileUSA

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