Landau-Siegel zeros over a base field and effective Brauer-Siegel

Abstract

Let k be a number field. We prove that for every quadratic extension F/k, every real zero beta of L(s, chi_{F/k}) satisfies (1 - beta) log d_F >= c(k) with an effective constant c(k) > 0, and we deduce (through a reduction of Stark) that the Brauer-Siegel theorem holds effectively and uniformly over all Galois extensions K of k, in the form h_K R_K >= c(k) d_K^{1/2} / ((2 pi)^{[K:Q]} log d_K). Stark's effective cases of the Brauer-Siegel theorem were the fields without quadratic subfields, since the quadratic subfields carried the ineffectivity of Siegel's theorem; over a base field k, the first result removes it. Both rely on a uniform bound for Chebotarev sums. Let K/k be a Galois extension with group G, and suppose S is a subset of G containing the identity. For any prime p of k, let delta_p be the proportion of the primes of K above p whose Frobenius lies in S. Then the sum over Np <= X of delta_p log Np / Np is at most (|S|/|G| + epsilon) log X + O(1), uniformly in K, as soon as X >= d_K^C with C depending only on epsilon and the degree [K:Q]. The proof is Kerr's form of the OpenAI argument, simple vanishing at the points (iota theta)_{iota in S}, with theta in a box in O_K, and Masser's zero estimate. The field enters only through Masser's invariant of the lattice of points, which we compute to be [K:Q]/|S| by averaging over the Galois group of the normal closure. Over Q, the determinant method of Green, Barreto, and Kominers gives sharper Chebotarev bounds, and we explain the comparison.