Farkas’ identities with quartic characters

Farkas in (On an Arithmetical Function II. Complex Analysis and Dynamical Systems II. Contemporary Mathematics, American Mathematical Society, Providence, 2005) 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 (Ramanujan J 19(1):19–27, 2009) discussed a natural generalization of the identity by introducing a quadratic character $$\chi $$ modulo a prime $$p \equiv 3 \ (\mathrm {mod}\ 4)$$ . 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 (Ramanujan J 43(1):197–213, 2017) found a finite list of identities of similar flavor using different methods. Clearly, if $$p \not \equiv 3 \ (\mathrm {mod}\ 4)$$ , 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 Guerzhoy and Raji (Ramanujan J 19(1):19–27, 2009): exact analogs of Farkas’ identity happen exactly for $$p=5$$ and 13.abstract