On the 2m-th power mean of Dirichlet L-functions with the weight of trigonometric sums

Let p be a prime, chi denote the Dirichlet character modulo p, f (x) = a(0) + a(1)x + ... + a(k)x(k) is a k-degree polynomial with integral coefficients such that (p, a(0), a(1),...,a(k)) = 1, for any integer m, we study the asymptotic property of Sigma(chi not equal chi 0)vertical bar Sigma(p-1)(a=1)chi(a)e (f(a)/p)vertical bar(2) vertical bar L(1,chi)vertical bar(2m), where e(y) = e(2 pi iy). The main purpose is to use the analytic method to study the 2m-th power mean of Dirichlet L-functions with the weight of the general trigonometric sums and give an interesting asymptotic formula. This result is an extension of the previous results.