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.

Keywords:

Eisenstein series, toroidal integral, Fourier series, Hecke $L$-function, totally positive integer, trace Eisenstein series, toroidal integral, Fourier series, Hecke $L$-function, totally positive integer, trace

MSC Classifications:

11M41, 11F30, 11F55, 11H06, 11R47 show english descriptions Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}
Fourier coefficients of automorphic forms
Other groups and their modular and automorphic forms (several variables)
Lattices and convex bodies [See also 11P21, 52C05, 52C07]
Other analytic theory [See also 11Nxx] 11M41 - Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}
11F30 - Fourier coefficients of automorphic forms
11F55 - Other groups and their modular and automorphic forms (several variables)
11H06 - Lattices and convex bodies [See also 11P21, 52C05, 52C07]
11R47 - Other analytic theory [See also 11Nxx]

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