A simple modular proof of Farkas' arithmetic identity modulo 4 (Q1001157)

The author uses results of \textit{D. L. Vestal} jun. from [Rocky Mt. J. Math. 31, No. 2, 687--696 (2001; Zbl 0982.11020)] which use generalized Dedekind \(\eta\)-functions to represent the basis of~\(M_2(\Gamma_0(4))\) and a theorem of Jacobi on representation numbers of sums of two squares to give a simple new derivation of an identity first proved by \textit{H. Farkas} [Complex analysis and dynamical systems II, Contemp. Math. 382, 121--130 (2005; Zbl 1176.11013)], which relates the Fourier coefficients of modular Eisenstein series of weight~2 for \(\Gamma_0(4)\) and the difference between number of divisors of integers which are congruent to~1 and to~3 modulo~4.