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Events happening on (June 22, 2026)

• 22
  Jun

  ON THE ARITHMETIC OF SPECIAL VALUES OF THE RIEMANN ZETA FUNCTION

  Date: June 22, 2026
  Time: 04:00 PM
  Location: https://lums-edu-pk.zoom.us/j/98597450368?pwd=PZvGvgELNfpPHIs0UxW44GpIj0SVkk.1

  Speaker: Dr. Waqar Ali Shah
  

  The Riemann zeta function occupies a central place in number theory. Its values at positive even integers are rational multiples of powers of π, while comparatively little is known about its values at positive odd integers. On the other hand, its values at negative integers are described in terms of Bernoulli numbers, which satisfy a rich system of congruences dating back to Kummer’s nineteenth-century work on Fermat’s Last Theorem. In this talk, I will give a brief overview of classical results concerning values of the zeta function at integers and explain how the congruences satisfied by its values at negative integers can be reinterpreted from the analytic viewpoint of p-adic numbers, where p is a prime. If time permits, I will discuss an algebraic counterpart of this analytic perspective arising from questions concerning the failure of unique factorization in cyclotomic fields, and how a deep connection between these viewpoints developed into the subject now known as Iwasawa theory.

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