Alexander Berkovich

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 x2+y2+2z2. Let n be odd positive integer. I use Siegel’s formula to show that

t(n2)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)]q17ϕ(q8)ψ(q8)ψ(q128)0

with equality if and only if 8k+1=E2 with E being generated by 1 and primes congruent to 1 or 3 modulo 8. Here k is a non-negative integer and ϕ(q):=nqn2,ψ(q):=nq2n2+n.

 

Next, I discuss the following

Corollary: The form 4x2+8y2+17z2+4xz represents all positive odd integers not of the form E2,(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.