MEAN SQUARE VALUE OF L-FUNCTIONS AT s = 1 FOR NON-PRIMITIVE CHARACTERS, DEDEKIND SUMS AND BOUNDS ON RELATIVE CLASS NUMBERS

Let d(0) be a given square-free integer. We give an explicit formula Md-0 (p) = A(d(0))(1+ Bd(0) (p)/p) for the quadratic mean value at s = 1 of the Dirichlet L-functions associated with the non-primitive odd Dirichlet characters modulo d(0)p induced by the odd Dirichlet characters modulo an odd prime p. Here d(0) (sic) A(d(0)) is an explicit multiplicative arithmetic function and B-d0 (f) is a twisted sum over the divisors d of d(0) of Dedekind sums s(h, df). To prove that f -> Bd(0) (f) is d(0)-periodic, we find a new and closed formula for the Dedekind sums f (sic) s(a+bf, c+df), for fixed integers a, b, c and d. We deduce explicit upper bounds for relative class numbers of cyclotomic number fields.