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On Goldfeld's Proof of Siegel's Theorem

In this paper, we give a detailed account of Goldfeld's proof of Siegel's theorem. Particularly, we present complete proofs of the nontrivial assumptions made in his paper.


I. Introduction
As stated in Theorem 1, Siegel's theorem is a result in multiplicative number theory concerning the lower bound of Dirichlet L-functions associated with quadratic primitive characters.
Theorem 1 (Siegel). For all ε > 0 there exists C(ε) > 0 such that for any primitive quadratic character χ modulo q > 1, By applying the mean value theorem for differentiable real-valued functions * , Theorem 1 allows us to establish a zero-free region † for L(s, χ): Theorem 2 (Siegel). For all ε > 0 there exists C(ε) > 0 such that for any primitive quadratic character χ modulo q > 1, its associated L(s, χ) is free of zeros in Siegel's theorem in the form of Theorem 2 is significant because it provides the prime number theorem in arithmetic progressions with an error term independent of the choice of modulus: . Let A be any fixed positive number and π(x; q, a) denote the number of primes that are ≡ a (mod q) and ≤ x. Then for all (q, a) = 1 we have π(x; q, a) = 1 ϕ(q) From a historical perspective, Theorem 1 is an outcome of Siegel's [5] investigation on the algebraic properties of the Dedekind zeta function associated with the quartic number field K = Q( √ q1, √ q2): where χ1, χ2 denote primitive quadratic characters modulo q1, q2 respectively. Siegel's original proof of Theorem 1 uses algebraic number theory, but purely analytic proofs have been developed by Estermann [2] and Goldfeld [3].
Although Goldfeld's method can lead to a proof of Theorem 1, he did not justify all the steps he took in the derivation. Notably, Goldfeld states without justification that x s s(s + 1)(s + 2)(s + 3)(s + 4) ds (5) holds for all 0 < β < 1. The right hand side of (5) is undenably positive, but if the ≪ constant relies on the choice of q2 then the subsequent steps in his paper will not lead to a valid proof of Theorem 1, so in this paper, we present a more complete version of Goldfeld's proof that addresses these unproven assumptions.

II. Plan for the Proof
Following the convention, we let λ denote the residue of f (s) at s = 1.
Basically, we attack the problem by giving lower estimate for λ. When its lower bound is combined with the upper bounds of various L(1, χ), Theorem 1 comes out.
To obtain a positive underestimate of λ, Goldfeld applied a variant of Perron's formula to f (s), so the lower bound emerges from an application of residue theorem.
Since Perron's formula connects partial sums with Dirichlet seris, it would be necessary to investigate both of them before proving Theorem 1. In section III, we perform an extensive study of the partial sum associated with f (s) to deduce Theorem 4. Let an be the Dirichlet series coefficient of f (s) and A0(x, w) denote the partial sum of f (s): A0(x, w) = n≤x an n w and Then for every nonnegative integer k, there exists an absolute constant c k and x k such Then, in section IV, we study the analytic properties of f (s) to derive a sharp upper bound for f (s) on the right half plane: Finally, in section V, we combine Theorem 4 and Theorem 5 via Perron's formula to deduce Theorem 1.

III. Arithmetical properties of f (s)
Before investigating the partial sum, we first focus on the properties of an. Lemma 1. a1 = 1 and an ≥ 0 for all n ∈ Z + .
Proof. It follows from the properties of Dirichlet series and Dirichlet convolution that an is multiplicative, meaning a1 = 1. Taking logarithms on both sides, we see This indicates that the Dirichlet series coefficients for log f (s) is nonnegative, so are the those of f (s). Q.E.D.
Performing an induction on Corollary 1, we can prove Theorem 4.
Proof of Theorem 4. Suppose Theorem 4 is true for k = m − 1 ≥ 0, then by definition The rightmost quantity is ≫m x m , so we conclude there exists admissible cm > 0 and Remark. Since constants appearing on the right hand side of (11) only depends on m, Theorem 4 virtually provides positive lower bounds for A k (x, w) that are independent of χ1 and χ2.
Remark. We can develop an argument analogous to the proof of Lemma 3 to deduce for all σ ≥ 0 and ε > 0, ζ(s) ≪ε |t| 1/2+ε (14) Plugging Lemma 3 and (14) into (4), we obtain Theorem 4. § We use the convention s = σ + it (where σ, t ∈ R) throughout the entire paper ¶ See §5.65 of [6] V. Proof of Theorem 1 Applying Perron's formula [7], we see that for all x ≥ 1, Integrating on both side of (15) for three times and applying Theorem 4, we have the following result: Lemma 4 (Justification of (5)). There exists absolute constants X, M > 0 such that the following inequality holds uniformly for x ≥ X.
Remark. Lemma 5 explains why the implied constant in Theorem 1 is not effectively computable because χ1 and β cannot be determined within finitely many steps.