Some implications of the 1907 Hurwitz formula

When/Where:

September 9, 2014, at 1:55pm, in LIT 368

Abstract:

I start with a brief review of Siegel’s formula for the number of representations of a positive integer by a genera of a quadratic form. I show how this formula implies the 1907 observation by Hurwitz, who proposed that \(\left|\{ (x,y,z) \in \mathbb{Z}^3 \mid n^2 = x^2+y^2+z^2 \}\right|\) may be expressed as a simple finite function of the divisors of \(n \in \mathbb{N}\). I use this formula together with the Jacobi triple product identity to prove (among other things) that \(9x^2+ 16y^2 +36z^2 + 16yz+ 4xz + 8xy\) represents exclusively all positive integers not of the form

• \(4^a(8m+7)\);
• \(4^a(8m+3)\), \(a=0,1,2\);
• \(4^a(4m+2)\), \(a=0,1,2\);
• \(4^a(8m+5)\), \(a=0,1\);
• \(M^2\);
• or \(4M^2\);

where \(a,m,M\) are non-negative integers and \(M\) is generated by 1 and primes congruent to \(1\pmod{4}\).