Hecke L-Functions and the Distribution of Totally Positive Integers

Let K be a totally real number field of degree n. We show that the number of totally positive integers (or more generally the number of totally positive elements of a given fractional ideal) of given trace is evenly distributed around its expected value, which is obtained from geometric considerations. This result depends on unfolding an integral over a compact torus. Introduction In 1921 Hecke [3] showed (among other things) that the Fourier coefficients of certain functions on a non-split torus in GL(2)/Q were given by (what we now call) the L-functions of certain Hecke characters of a real quadratic field. Interestingly, the Hecke characters which arise here are not of type A. Hecke’s key idea in that portion of his paper is to unfold the integral that computes the Fourier coefficient in amanner foreshadowing the Rankin–Selberg method. Siegel [8, Ch. II, Section 4] observed that the same method could be used to compute the hyperbolic Fourier expansion of the nonholomorphic Eisenstein series on GL(2)/Q , and to express these coefficients in terms of Hecke L-functions. Hecke used his Fourier expansion to study the distribution of the fractional parts of mα where m runs over the rational integers and α is a fixed real quadratic irrationality. Siegel used his in conjunction with Kronecker’s limit formula to derive some relationships between Dedekind’s η-function and invariants of real quadratic fields. Our main goal in this paper is to generalize Hecke’s result to an arbitrary totally real field K. To do so, we first explain the generalization of Siegel’s Fourier expansion computation to GL(n)/Q . Where Siegel worked with a real quadratic field, we let K be any number field, [K :Q] = n. We can view K as the Q-points of a non-split torus T in GL(n)/Q . Let X denote the totally geodesic subspace of the symmetric space of GL(n)/Q defined by T. We show that an automorphic form on GL(n,Q), restricted to X, has a Fourier expansion. We then consider this Fourier expansion for the standard maximal parabolic Eisenstein series E(g, s) of type (n− 1, 1). We prove that the Fourier coefficients, which are again functions of s, are in fact certain Hecke L-functions associated to K (more accurately, partial Hecke L-functions with respect Received by the editors October 25, 2004. Research supported in part by NSF grant DMS-0139287 (Ash) and by NSA grant MDA904-03-1-0012 and NSF grant DMS-0353964 (Friedberg). AMS subject classification: Primary: 11M41; secondary: 11F30, 11F55, 11H06, 11R47.