On an arithmetical function (Q1770960)

For the positive integer \(n\) one denotes by \(d(n)\) the number of its positive divisors, and by \(\sigma(n)\) their sum. \(\delta(n)\) denotes the difference between the number of those positive divisors of \(n\) which are congruent to \(1\pmod 3\) and the number of those positive divisors of \(n\) which are congruent to \(-1\pmod 3\); \(\delta\) is a multiplicative function and \(\delta(3n+2)=0\). The function \(\delta\) is interesting because of its tight connection to the functions \(\sigma\) and \(d\). Most of the results obtained in the paper are non-elementary. Some of the proofs use the function \[ f(z,y)= \exp \biggl(\frac{\pi i}{6} \biggr)y^{\frac{1}{24}}z^{\frac{1}{6}} \prod_{n=0}^{\infty}(1-y^{3n+3})(1-y^{3n+2}z)(1-y^{3n+1}/z), \] a product representation of the first order theta function [see \textit{H. M. Farkas} and \textit{I. Kra}, Theta Constants, Riemann Surfaces and the Modular Group. Graduate Studies in Mathematics. 37. Providence, RI: (American Mathematical Society) (AMS) (2001; Zbl 0982.30001)]. For \(n=3^\alpha\cdot m, \alpha\geq0\), the author denotes by \(\sigma'(n)\) the number \(\sigma(m)\) and proves the relation \(\delta(n)=3\sum_{j=1}^{n-1}\delta(j)\delta(n-j)=\sigma'(n)\). (Theorem 1). For \(a_n=\sum_{d| n}\frac{n}{d}\) with \((d,3)=1\), the paper contains the proof of several identities, among which we emphasize evaluations for \(\sum_{n=1}^{\infty}a_ny^n\) and \(3a_n-2\sigma(n)=\delta(n)+3\sum_{j=1}^{n-1}\delta(j)\delta(n-j)\), with interesting consequences such as: `\(3n+1\) is prime iff \(\sum_{j=0}^{n-1}\delta(3j+1)\delta(n-j)=n/2\). \(3n+2\) is prime iff \(\sum_{j=0}^n\delta(3j+1)\delta(3(n-j)+1)=n+1\).' The paper ends with a remark concerning the use of the function \(\delta\) to evaluate the number of integer solutions of the equation \(x^2+3y^2=N^2\).