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

Farkas在（关于算术函数II。复杂分析和动力学系统II。当代数学，美国数学学会，普罗维登斯，2005年）中引入了算术函数\（\ delta \）并发现了一个包含\（\ delta \）和一个除数函数\（\ sigma'\）的和。第一命名的作者和Raji在（拉马努金J，19（1）：19-27，2009）通过引入一个二次字符讨论的身份的一个自然推广\（\智\）模素\（P \当量3 \ （\ mathrm {mod} \ 4）\）。特别是，事实证明，除了Farkas考虑的原始情况\（p = 3 \）之外，Farkas身份的精确模拟（在某种意义上）仅发生在\（p = 7 \）。最近，对于小的复合模量的二次特征，Williams在（Ramanujan J 43（1）：197–213，2017）中发现了使用不同方法的相似风味身份的有限列表。显然，如果\（p \ not \ equiv 3 \（\ mathrm {mod} \ 4）\），则字符\（\ chi \）不是二次的，甚至不是偶数。在本文中，我们证明在某些条件下，偶数字符都不存在Farkas身份的类似物。假设\（\ chi \）是奇四次的，我们得出的结果与Guerzhoy和Raji（Ramanujan J 19（1）：19–27，2009）的结果令人惊讶地相似：Farkas身份的确切相似恰好发生于\（ p = 5 \）和13.摘要