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

Abstract

Let r k ( n) denote the number of representations of an integer n as a sum of k squares. We prove that for odd primes p, r 11( p 2)= 330 31 ( p 9+1)−22(−1) ( p−1)/2 p 4+ 352 31 H( p), where H( p) is the coefficient of q p in the expansion of q ∏ j=1 ∞ (1−(−q) j ) 16(1−q 2j) 4+32q 2 ∏ j=1 ∞ (1−q 2j) 28 (1−(−q) j ) 8 . This result, together with the theory of modular forms of half integer weight is used to prove that r 11(n)=r 11(n′) 2 9⌊λ/2⌋+9−1 2 9−1 ∏ p p 9⌊λ p/2⌋+9 −1 p 9−1 −p 4 −n′ p p 9⌊λ p/2⌋ −1 p 9−1 , where n=2 λ ∏ p p λ p is the prime factorisation of n, n′ is the square-free part of n, and n′ is of the form 8 k+7. The products here are taken over the odd primes p, and ( n p ) is the Legendre symbol. We also prove that for odd primes p, r 13( p 2)= 4030 691 ( p 11+1)−26p 5+ 13936 691 τ( p), where τ( n) is Ramanujan's τ function, defined by q ∏ j=1 ∞ (1−q j ) 24= ∑ n=1 ∞ τ(n)q n . A conjectured formula for r 2k+1( p 2) is given, for general k and general odd primes p.