On the Mean Value of L(1/2, χ ) FW Real Characters

Abstract

Asymptotic formulae are derived for the sums ZL''(i,x), where k=l or 2 and x runs over all Dirichlet characters defined by Kronecker's symbol ^ with d being restricted to fundamental discriminants in the interval [-A'.O) or (0,jr]. Also, asymptotic formulae are proved for the sums where Zp('')=(") is defined by the Legendre symbol and p is restricted to primes p^X with/) = i; (mod 4),

Mean values of biquadratic zeta functions

“…Jutila [11], verifying a conjecture of Goldfeld and Viola [10], showed that (1.1) 0<±d<X L(1/2, χ d ) ∼ c 1 X log X + c 2 x + O(x 3/4+ ) for certain constants c 1 , c 2 . Takhtadzjan and Vinogradov [15] establish a similar result.…”

Random Matrix Theory and L-Functions at s = 1/2

“…Having the generating function, Msp(N, 8), it is a short step to find the value distributions of both log Z(U, 0) and…”

“…(47) is the success of a very similar conjecture relating moments of the Riemann zeta function to averages over U(N) [7]. The only difference is that in the case of (8) the average was along the critical line rather than over a family of functions. This is not a significant difference, however, and Conrey and Farmer in fact suggest that we think of the Riemann zeta function as a unitary family (with zeros showing the statistics of the eigenvalues of matrices from U(N» in its own right, where we are averaging over special values of the family {(1/2 +it)} as t ranges over the real numbers.…”

“…Conrey and Farmer [6] have extended this idea that the low-lying zeros of families of Lfunctions show particular statistics to the study of the mean values of the L-functions L , (8) within families at the central point s =1/2. They have found evidence that the symmetry type to which the low-lying zeros subscribe also determines the behaviour of these mean values.…”

Mean value of the class number in function fields revisited

“…But we can also consider the problem of averaging over fundamental discriminants. In 1979, Goldfeld and Viola [10] put forward a few conjectures about averages over fundamental discriminants and Jutila [13] proved such conjecture in 1981 for s = 1 2 . Then, in 1985, Goldfeld and Hoffstein [9], using Eisentein series of 1 2 -integral weight, generalized Jutila's result and proved a result valid for all s with Re(s) ≥ 1 2 .…”