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Licensed Unlicensed Requires Authentication Published by De Gruyter March 23, 2012

Higher Kronecker “limit” formulas for real quadratic fields

  • Maria Vlasenko EMAIL logo and Don Zagier

Abstract

For every integer k ≧ 2 we introduce an analytic function of a positive real variable and give a universal formula expressing the values ζ(ℬ, k) of the zeta functions of narrow ideal classes in real quadratic fields in terms of this function and its derivatives up to order k − 1 evaluated at reduced real quadratic irrationalities associated to ℬ. We show that our functions satisfy functional equations and use these to deduce explicit formulas for the rational numbers ζ(ℬ, 1 − k). We also give an interpretation of our formula for ζ(ℬ, k) in terms of cohomology groups of SL(2, ℤ) with analytic coefficients and describe a “twisted” extension of the main formula that allows one to treat zeta values of zeta functions of ray classes rather than just ideal classes. Finally, we use our formulas to compute some zeta-values numerically and test that they are expressible as combinations of higher polylogarithm functions evaluated at algebraic arguments.

Received: 2010-03-10
Published Online: 2012-03-23
Published in Print: 2013-06

©[2013] by Walter de Gruyter Berlin Boston

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