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- W3117407098 abstract "Abstract In this paper we study two functions <m:math xmlns:m=http://www.w3.org/1998/Math/MathML><m:mrow><m:mi>F</m:mi><m:mo></m:mo><m:mrow><m:mo stretchy=false>(</m:mo><m:mi>x</m:mi><m:mo stretchy=false>)</m:mo></m:mrow></m:mrow></m:math> {F(x)} and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML><m:mrow><m:mi>J</m:mi><m:mo></m:mo><m:mrow><m:mo stretchy=false>(</m:mo><m:mi>x</m:mi><m:mo stretchy=false>)</m:mo></m:mrow></m:mrow></m:math> {J(x)} , originally found by Herglotz in 1923 and later rediscovered and used by one of the authors in connection with the Kronecker limit formula for real quadratic fields. We discuss many interesting properties of these functions, including special values at rational or quadratic irrational arguments as rational linear combinations of dilogarithms and products of logarithms, functional equations coming from Hecke operators, and connections with Stark’s conjecture. We also discuss connections with 1-cocycles for the modular group <m:math xmlns:m=http://www.w3.org/1998/Math/MathML><m:mrow><m:mi>PSL</m:mi><m:mo></m:mo><m:mrow><m:mo stretchy=false>(</m:mo><m:mn>2</m:mn><m:mo>,</m:mo><m:mi>ℤ</m:mi><m:mo stretchy=false>)</m:mo></m:mrow></m:mrow></m:math> {operatorname{PSL}(2,mathbb{Z})} ." @default.
- W3117407098 created "2021-01-05" @default.
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- W3117407098 date "2023-03-01" @default.
- W3117407098 modified "2023-10-01" @default.
- W3117407098 title "Arithmetic properties of the Herglotz function" @default.
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- W3117407098 doi "https://doi.org/10.1515/crelle-2023-0009" @default.
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