Some implications of the 1907 Hurwitz formula III

When/Where:

September 23, 2014, at 1:55pm, in LIT 368

Abstract:

Let \(t(n)\) denote \(\#\) representations of integer \(n\) by \(x^2+y^2+2z^2\). Let \(n\) be odd positive integer. I use Siegel’s formula to show that

\(t(n^2) \geq 4n\)

with equality if and only if all prime divisors of \(n\) are congruent to \(1\) or \(3\) modulo \(8\).

I employ the above inequality together with a special case of the Jacobi triple product identity to establish that

\([q^{(8k+1)}]\, q^{17} \phi(q^8) \psi(q^8) \psi(q^{128}) \geq 0\)

with equality if and only if \(8k+1 = E^2\) with \(E\) being generated by \(1\) and primes congruent to \(1\) or \(3\) modulo \(8\). Here \(k\) is a non-negative integer and \(\phi(q) := \sum_{n}q^{n^2},\; \psi(q): = \sum_{n}q^{2n^2+n}\).

 

Next, I discuss the following

Corollary: The form \(4x^2 + 8y^2 + 17 z^2 +4 xz\) represents all positive odd integers not of the form \(E^2,\; (8m+3),\;(8m+5),\;(8m+7)\) where \(E\) and \(m\) are non-negative integers and \(E\) is generated by \(1\) and primes congruent to \(1\) or \(3\) modulo \(8\).