  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
$13$.
