N ov 2 00 1 Spacing of Zeros of Hecke L - Functions and the Class Number Problem

Contents: 1. A bit of history and results 2. Basic automorphic forms 3. Summation formulas 4. Convolution sums 5. Point to integral mean-values of Dirichlet's series 6. Evaluation of A(T) 7. Approximate functional equation 8. Evaluation of ℓ(s) on average 9. Estimation of x(s) on average 10. Applications The group of ideal classes Cℓ(K) of an imaginary quadratic field K = Q(√ −q) is the most fascinating finite group in arithmetic. Here we are faced with one of the most challenging problems in analytic number theory, that is to estimate the order of the group h = |Cℓ(K)|. C.F. Gauss conjectured (in a parallel setting of binary quadratic forms) that the class number h = h(−q) tends to infinity as −q runs over the negative discriminants. Hence there are only a finite number of imaginary quadratic fields with a given class number. But how many of these fields are there exactly for h = 1, or h = 2, etc. ? To answer this question one needs on effective lower bound for h in terms of q (a fast computer could be helpful as well). The problem was linked early on to the L-series (1.1) L(s, χ) = ∞ n=1 χ(n) n −s for the real character χ of conductor q (the Kronecker symbol) (1.2) χ(n) = −q n. In this connection L. Dirichlet established the formula (1.3) h = π −1 √ qL(1, χ) (we assume that −q is a fundamental discriminant, q > 4, so there are two units ±1 in the ring of integers O K ⊂ K). Rather than estimating the class number, Dirichlet inferred from (1.3) that L(1, χ) does not vanish, which property he needed to establish the equidistribution of primes in arithmetic progressions. Truly the lower bound (1.4) L(1, χ) π √ q follows from (1.3), because h 1. The Grand Riemann Hypothesis for L(s, χ) implies (1.5) (log log q) −1 ≪ L(1, χ) ≪ log log q, whence the class number varies only slightly about √ q (1.6) √ q log log q ≪ h ≪ √ q log log q. But sadly enough we may not see proofs of such estimates (which are best possible in order of magnitude) in the near future. At present we know (after J. Hadamard and C.J. de la Vallée-Poussin) that L(s, χ) = 0 for s = σ + it in the region (1.7) …