Arithmetic properties of the Herglotz-Zagier-Novikov function
Abstract
In this article, we undertake the study of the function $\mathscr{F}(x;u,v)$, which we refer to as the Herglotz-Zagier-Novikov function. This function appears in Novikov's work on the Kronecker limit formula, which was motivated by Zagier's paper where he obtained the Kronecker limit formula in terms of the Herglotz function $F(x)$. Two, three, and six-term functional equations satisfied by $\mathscr{F}(x;u,v)$ are exhibited. These are cohomological relations coming from the action of an involution and SL$_2(\mathbb{Z})$ on $\mathbb{C}\times {\mathbb{D}_1}^2$ (the unit circle ${\mathbb{D}_1})$. We also provide the special values of $\mathscr{F}(x;u,v)$ at rational arguments of $x$. Importantly, $\mathscr{F}(x;u,v)$ serves as a unified generalization of three other interesting functions, namely $F(x)$, $J(x)$, and $T(x)$, which also appear in various Kronecker limit formulas and are previously studied by Cohen, Herglotz, Muzaffar and Williams, and Radchenko and Zagier. Consequently, our study not only reveals the numerous elegant properties of $\mathscr{F}(x;u,v)$ but also helps us to further develop the theories of functions related to its special cases such as $J(x)$ and $T(x)$.
- Publication:
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arXiv e-prints
- Pub Date:
- September 2023
- DOI:
- 10.48550/arXiv.2309.10634
- arXiv:
- arXiv:2309.10634
- Bibcode:
- 2023arXiv230910634C
- Keywords:
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- Mathematics - Number Theory;
- Mathematics - Classical Analysis and ODEs
- E-Print:
- To appear in Advances in Mathematics