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Mathematics of Computation

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Explicit smoothed prime ideals theorems under GRH

Authors: Loïc Grenié and Giuseppe Molteni
Journal: Math. Comp. 85 (2016), 1875-1899
MSC (2010): Primary 11R42; Secondary 11Y40
Published electronically: October 6, 2015
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Abstract: Let $ \psi _{\mathbb{K}}$ be the Chebyshev function of a number field $ \mathbb{K}$. Let $ \psi ^{(1)}_{\mathbb{K}}(x):=\int _{0}^{x}\psi _{\mathbb{K}}(t)\,\mathrm {d} t$ and $ \psi ^{(2)}_{\mathbb{K}}(x):=2\int _{0}^{x}\psi ^{(1)}_{\mathbb{K}}(t)\,\mathrm {d} t$. We prove under GRH (Generalized Riemann Hypothesis) explicit inequalities for the differences
$ \vert\psi ^{(1)}_{\mathbb{K}}(x) - \tfrac {x^2}{2}\vert$ and $ \vert\psi ^{(2)}_{\mathbb{K}}(x) - \tfrac {x^3}{3}\vert$. We deduce an efficient algorithm for the computation of the residue of the Dedekind zeta function and a bound on small-norm prime ideals.

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Additional Information

Loïc Grenié
Affiliation: Dipartimento di Ingegneria gestionale, dell’informazione e della produzione, Università di Bergamo, viale Marconi 5, 24044 Dalmine, Italy

Giuseppe Molteni
Affiliation: Dipartimento di Matematica, Università di Milano, via Saldini 50, 20133 Milano, Italy

Received by editor(s): October 18, 2013
Received by editor(s) in revised form: January 15, 2014, June 6, 2014, and January 14, 2015
Published electronically: October 6, 2015
Article copyright: © Copyright 2015 American Mathematical Society