2016/07/04 by Ebénézer Ntienjem, Ntienjem, Ebénézer
Mathematics · #History and Theory of Mathematics #Analytic Number Theory Research #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.1607.01082
We discuss an elementary method for the evaluation of the convolution sums\n underset substack\n (l,m)\∈\ℕ02 \α ,l+\β ,m=n \n\∑\σ(l)\σ(m) for those \α,\β\∈\ℕ for which\n\gcd(\α,\β)=1 and \α\β=2\ν mho, where\n\ν\∈ 0,1,2,3 and mho is a finite product of distinct odd primes.\nModular forms are used to achieve this result. We also generalize the\nextraction of the convolution sum to all natural numbers. Formulae for the\nnumber of representations of a positive integer n by octonary quadratic forms\nusing convolution sums belonging to this class are then determined when\n\α\β\≡ 0 pmod4 or \α\β\≡ 0 pmod3. To achieve this\napplication, we first discuss a method to compute all pairs\n(a,b),(c,d)\∈\ℕ2 necessary for the determination of such formulae\nfor the number of representations of a positive integer n by octonary\nquadratic forms when \α\β has the above form and \α\β\≡\n0 pmod4 or \α\β\≡ 0 pmod3. We illustrate our approach by\nexplicitly evaluating the convolution sum for \α\β=33=3\⋅ 11, >\n\α\β=40=23\⋅ 5 and \α\β=56=23\⋅ 7, and by\nrevisiting the evaluation of the convolution sums for \α\β=10, 11,\n12, 15, 24. We then apply these convolution sums to determine formulae\nfor the number of representations of a positive integer n by octonary\nquadratic forms. In addition, we determine formulae for the number of\nrepresentations of a positive integer n when (a,b)=(1,1), (1,3), (2,3),\n(1,9).\n