Lecture Room B, 4th Floor, The 3rd General Building, NTHU
(清華大學綜合三館 4樓B演講室)
Explicit Formulas for Cyclotomic Multiple Bernoulli Numbers
Ku-Yu Fan (Nagoya University)
Abstract
-adic multiple -functions, introduced by Furusho, Komori, Matsumoto, and Tsumura, are multivariable analogues of the Kubota–Leopoldt -adic -function. Furusho and Jarossay studied their special values at tuples of positive integers and expressed them as convergent infinite sums of cyclotomic multiple harmonic values. In this expansion, they introduced certain coefficients, now called cyclotomic multiple Bernoulli numbers. However, these numbers were given only through recursive relations.
In this talk, we derive a explicit formula for cyclotomic multiple Bernoulli numbers in terms of twisted multiple Bernoulli numbers. Twisted multiple Bernoulli numbers arise as special values at tuples of non-positive integers of generalized Euler–Zagier–Lerch-type complex multiple zeta functions. Our formula therefore makes the coefficients in the Furusho–Jarossay expansion explicit and connects them with special values of complex multiple zeta functions. Consequently, the positive special values of -adic multiple -functions can be expressed as convergent infinite sums of cyclotomic multiple harmonic values whose coefficients are given explicitly by non-positive special values of complex multiple zeta functions.
Organizer
Fu-Tsun Wei (NTHU)
