On the number of representations of certain integers as sums of 11 or 13 squares

Let rk(n) denote the number of representations of an integer n as a sum of k squares. We prove that for odd primes p, where H( p) is the coefficient of qp in the expansion of This result, together with the theory of modular forms of half integer weight is used to prove that where is the prime factorisation of n, n′ is the square-free part of n, and n′ is of the form 8k+7. The products here are taken over the odd primes p, and is the Legendre symbol. We also prove that for odd primes p, where τ(n) is Ramanujanʹs τ function, defined by . A conjectured formula for r2k+1( p2) is given, for general k and general odd primes p.