Lower Bounds for the Principal Genus of Definite Binary Quadratic Forms

Abstract

We apply Tatuzawa's version of Siegel's theorem to derive two lower bounds on the size of the principal genus of positive definite binary quadratic forms.

On the theorem of Conrey and Iwaniec

“…This is unconditional. However, a little arithmetic show that the interval ( 7) is nonempty only when (9) h(−D) < D 1/2 log(D) 74 .…”

“…From Lemma 2 of [9] we have that where W(x) denotes the Lambert W-function, i.e. the inverse function of f (w) = w exp(w) (see [3], [13, p. 146 and p. 348, ex 209]).…”