ON THE THEOREM OF CONREY AND IWANIEC

ABSTRACT. An exposition on ‘Spacing of zeros of Hecke L-functions and the class number problem ’ by Conrey and Iwaniec; any errors are my own. HISTORY In number theory, binary quadratic forms ax2+ bxy+ cy2 = Q(x, y) have a long and glorious history. Motivated by Fermat’s and Euler’s questions about the range of such functions (Which odd primes p can be written as p = x2 + y2? Or generally p = x2 + ny2?), one defines equivalence relation which identifies forms with the same range: Q ′ ∼ Q if Q′(x, y) = Q((x, y)γ) for some γ in SL(2,Z). The form Q is positive definite if the dis-criminant −D = b2 − 4ac < 0; a calculation shows the discriminant is an invariant of the equivalence class. Gauss showed the classes of fixed discriminant −D form an abelian group C(−D) of finite order h(−D), and speculated [6, Art. 303] about the growth of this class number as a function of the discriminant:... the series of [discriminants] corresponding to the same given classification (i.e. the given number of both genera and classes) always seems to terminate with a finite number... However rigorous proofs of these ob-servations seem to be very difficult. Dirichlet’s analytic class number formula tells us (1) L(1,χ−D) = pih(−D) D1/2 where L(s,χ−D) is defined, for Re(s)> 1 by L(s,χ−D) = n=1 χ−D(n)n−s