# Arithmetic properties of the Herglotz function

@inproceedings{Radchenko2020ArithmeticPO, title={Arithmetic properties of the Herglotz function}, author={Danylo V. Radchenko and Don Zagier}, year={2020} }

In this paper we study two functions F (x) and 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…

## 4 Citations

### Extended higher Herglotz functions I. Functional equations

- Mathematics
- 2021

Abstract. In 1975, Don Zagier obtained a new version of the Kronecker limit formula for a real quadratic field which involved an interesting function F (x) which is now known as the Herglotz…

### A Kronecker limit formula for indefinite zeta functions

- Mathematics
- 2020

Indefinite zeta function are Mellin transforms of indefinite theta function (a la Zwegers) and specialise to differenced ray class zeta functions of real quadratic fields. We prove a Kronecker limit…

### Explicit transformations of certain Lambert series

- MathematicsResearch in the Mathematical Sciences
- 2022

An exact transformation, which we call the master identity , is obtained for the first time for the series $$\sum _{n=1}^{\infty }\sigma _{a}(n)e^{-ny}$$ ∑ n = 1 ∞ σ a ( n ) e - n y for $$a\in…

### Extended Higher Herglotz function \textup{II}

- Mathematics
- 2022

. Very recently, Radchenko and Zagier revived the theory of Herglotz functions. The main goal of the article is to show that one of the formulas on page 220 of Ramanujan’s Lost Notebook actually…

## References

SHOWING 1-10 OF 28 REFERENCES

### Higher Kronecker “limit” formulas for real quadratic fields

- Mathematics
- 2012

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…

### A generalization of the Chowla-Selberg formula and the zeta functions of quadratic orders

- Mathematics
- 1990

The purpose of this note is to present an identity which generalizes a formula of Chowla and Selberg on the periods o CM elliptic curves in connection with the zeta functions of imaginary quadratic…

### Hecke operators on weighted Dedekind symbols

- Mathematics
- 2004

Abstract Dedekind symbols generalize the classical Dedekind sums (symbols). The symbols are determined uniquely by their reciprocity laws up to an additive constant. There is a natural isomorphism…

### Curious and Exotic Identities for Bernoulli Numbers

- Mathematics
- 2014

Bernoulli numbers, which are ubiquitous in mathematics, typically appear either as the Taylor coefficients of x=tanx or else, very closely related to this, as special values of the Riemann zeta…

### A Kronecker limit formula for real quadratic fields

- Mathematics
- 1975

Then ((s, A) is (after analytic continuation) a meromorphic function of s with a simple pole at s = 1 as its only singularity. Moreover, the residue of ((s, A) at s = 1 is independent of the ideal…

### On Artin's conjecture for odd 2-dimensional representations

- Mathematics
- 1994

On the experimental verification of the artin conjecture for 2-dimensional odd galois representations over Q liftings of 2-dimensional projective galois representations over Q.- A table of…

### PERIODS OF PARABOLIC FORMS AND p-ADIC HECKE SERIES

- Mathematics
- 1973

The author proves an algebraicity theorem for the periods of parabolic forms of any weight for the full modular group, gives explicit formulas for the coefficients of the forms, and constructs -adic…

### Genus character L ‐functions of quadratic orders and class numbers

- MathematicsJournal of the London Mathematical Society
- 2020

An explicit form of genus character L ‐functions of quadratic orders is presented in full generality. As an application, we generalize a formula due to Hirzebruch and Zagier on the class number of…

### A Tribute to Emil Grosswald: Number Theory and Related Analysis

- Mathematics
- 1993

Emil Grosswald was a mathematician of great accomplishment and remarkable breadth of vision. This volume pays tribute to the span of his mathematical interests, which is reflected in the wide range…