Abstract
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
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Guerzhoy, P., Wong, K.L. Farkas’ identities with quartic characters. Ramanujan J 54, 679–697 (2021). https://doi.org/10.1007/s11139-020-00296-5
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DOI: https://doi.org/10.1007/s11139-020-00296-5