A q-analog of Jacobi's two squares formula and its applications

2018/01/09 by Caballero, José Manuel Rodríguez
#11C08 #11E25 #11N13 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1801.03134

Abstract

We consider a q-analog r₂(n, q) of the number of representations of an integer as a sum of two squares r₂(n). This q-analog is generated by the expansion of a product that was studied by Kronecker and Jordan. We generalize Jacobi's two squares formula from r₂(n) to r₂(n, q). We characterize the signs in the coefficients of r₂(n, q) using the prime factors of n. We use r₂(n, q) to characterize the integers which are the length of the hypotenuse of a primitive Pythagorean triangle.

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