Logarithmic derivatives of theta functions
The authors give two new proofs of the identity \[ \sum_{n=0}^\infty \delta(3n+1)x^n= \prod_{n=1}^\infty \frac{(1-x^{3n})^3} {(1-x^n)} \] where \(\delta(n)= d_1(n)- d_2(n)\) is the number of divisors of \(n\) congruent to \(i\bmod 3\). The main tool of the proofs is the theory of theta functions with characteristics: \[ \theta\biggl[ {{\varepsilon}\atop{\varepsilon'}}\biggr](\zeta,\tau)= \sum_{N\in\mathbb Z^g}\exp(2\pi i) \Biggl(\frac12 \biggl(N+\frac\varepsilon2\biggr)^t \tau\biggl(N+ \frac\varepsilon2\biggr)+ \biggl(N+\frac\varepsilon2\biggr) \biggl(\zeta+ \frac{\varepsilon'}{2} \biggr)\Biggr), \] where \(\zeta\in\mathbb C^g\), \(\tau\) a symmetric \(g\times g\) matrix with positive definite imaginary part and \(\varepsilon,\varepsilon'\) vectors in \(\mathbb R^g\). As an application, they express the number of solutions of the Diophantine equation \(x^2+3y^2=N\) in terms of \(\delta(N)\).
scientific article; zbMATH DE number 5003411