Abstract
Let K be an algebraic number field and f a complex-valued function on the ideal class group of K. Then, f extends in a natural way to the set of all non-zero ideals of the ring of integers of K and we can consider the Dirichlet series \({L(s,f) =\sum_{{\mathfrak a}} f({\mathfrak a}){\bf N}({\mathfrak a})^{-s}}\) which converges for \({{\mathfrak R}(s) >1 }\). After extending this function to \({{\mathfrak R}(s)=1}\), and in the case that f does not contain the trivial character, we study the special value L(1, f) when f is algebraic valued and K is an imaginary quadratic field. Applying Kronecker’s limit formula and Baker’s theory of linear forms in logarithms, we derive a variety of results related to the transcendence of this special value.
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Research of both authors partially supported by an NSERC grant.
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Murty, M.R., Murty, V.K. Transcendental values of class group L-functions. Math. Ann. 351, 835–855 (2011). https://doi.org/10.1007/s00208-010-0619-y
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DOI: https://doi.org/10.1007/s00208-010-0619-y