On The Siegel-Tatuzawa Theorem

In this paper, we apply the upper bound estimate of L(1, x) of Louboutin [22],[23] and the residue at s = 1 of the Dedekind zeta function of an algebraic number field given by Ji and Lu [21] to study the lower bound of real primitive Dirichlet L-function at s = 1.In Chapter one, we present some basic definitions, notions and facts that will be used later in this paper, and we give an estimate ofζ(5/2).In Chapter two, we use the estimate ofζ(5/2) to obtain an estimate of L(1,x) if L(s,x) has no zero in some interval, then we give some lemmas having been proved before, which will be used later in the proof of Theorem 3.2.In Chapter three, we apply the lemmas in Chapter two and some arithmetic theory of biquadratic number field to get a new lower bound of real primitive L-function ats = 1.In Chapter four, we consider the exceptional quadratic number field in Theorem 3.2 be imaginary quadratic number field with class number h₀ and give another lower bound of real primitive L-function at s = 1 with respect to h₀.