On The Mean Value Identity Of L - Function

Let n, h, q be positive integers, q> 2 and (h, q)=1. Let χ be a Dirichlet character modulo q. The mean value of Dirichlet L-functions is defined by where L(s, χ) denotes a Dirichlet L-function.For n=1, there are a lot of results about W. P. Zhang《On the general Dedekind sums and one kind identities of Dirichlet L-functions)) gave a formula for Dirichlet L-functions, that is where n is a positive odd integer, h is a positive integer greater than 1 and S(h, n, q) denotes the general Dedekind sums. R. Ma etc.《Some notes on identities for Dirichlet L-functions)) gaved the identities for the mean value of the Dirichlet L-functions when h=1,2,3 or h|q-1. Specially, they proved that where h|q-1.In this thesis, we obtain some results on the mean value of L-functions basing on these two papers. The main results are as follows:Theorem 3.3 Let p be an odd prime, h be a positive integer with h|p+1 and X be a Dirichlet character modulo p. ThenTheorem 3.4 Let p be an odd prime, h be a positive integer with h|p-2 and X be a Dirichlet character modulo p. ThenTheorem 3.5 Let p be an odd prime, h be an integer greater than 2 with h|p+2 and χ be a Dirichlet character modulo p. ThenBesides, for n=3, we also obtain some formulas about the mean value of Dirichlet L-functions.Theorem 4.1 Let q>1 be an odd integer, h=2 and χ be a Dirichlet character modulo q. Then where Πp|q denotes the product over all different prime divisiors of q and φ(q) is the Euler function.Theorem 4.2 Let q> 2 be an integer, h=3 and χ be a Dirichlet character modulo q. When q≡1(mod 3), we have where Πp|q denotes the product over all different prime divisiors of q and φ(q) is the Euler function.Theorem 4.3 Let q> 2 be an integer, h=3 and χ be a Dirichlet character modulo q. When q≡2 (mod 3), we have where Πp|q denotes the product over all different prime divisiors of q and φ(q) is the Euler function.