# Farkas' identities with quartic characters

## TL;DR

This paper investigates generalizations of Farkas' identities involving divisor functions and characters, proving that such identities only occur for specific primes when using odd quartic characters.

## Contribution

It establishes that Farkas' identities have analogs only for primes 5 and 13 when using odd quartic characters, extending previous results to new character types.

## Key findings

- Exact analogs occur only for p=5 and 13 with odd quartic characters.
- No analogs exist for even characters under certain conditions.
- Farkas' identities are highly restrictive and depend on specific prime moduli.

## Abstract

Farkas in \cite{Farkas} introduced an arithmetic function $\delta$ and found an identity involving $\delta$ and a sum of divisor function $\sigma'$. The first-named author and Raji in \cite{Guerzhoy} discussed a natural generalization of the identity by introducing a quadratic character $\chi$ modulo a prime $p \equiv 3 \pmod 4$. In particular, it turns out that, besides the original case $p=3$ considered by Farkas, an exact analog (in a certain precise sense) of Farkas' identity happens only for $p=7$. Recently, for quadratic characters of small composite moduli, Williams in \cite{Williams} found a finite list of identities of similar flavor using different methods. Clearly, if $p \not \equiv 3 \pmod 4$, the character $\chi$ is either not quadratic or even. In this paper, we prove that, under certain conditions, no analogs of Farkas' identity exist for even characters. Assuming $\chi$ to be odd quartic, we produce something surprisingly similar to the results from \cite{Guerzhoy}: exact analogs of Farkas' identity happen exactly for $p=5$ and $13$.