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

摘要

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 qp in the expansion of q ∏_(j=1)~∞(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)/(p~9-1)-p~4((-n')/p)(p~9[λ_p/2]-1)/(p~9-1)], wheren = 2~λ ∏_P p~(λp) 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 (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 (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.