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Mathematics of Computation

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Explicit versions of the prime ideal theorem for Dedekind zeta functions under GRH

Authors: Loïc Grenié and Giuseppe Molteni
Journal: Math. Comp. 85 (2016), 889-906
MSC (2010): Primary 11R42; Secondary 11Y40
Published electronically: October 7, 2015
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Abstract: Let $ \psi _{\mathbb{K}}$ be the Chebyshev function of a number field $ \mathbb{K}$. Under the Generalized Riemann Hypothesis we prove an explicit upper bound for $ \vert\psi _{\mathbb{K}}(x)-x\vert$ in terms of the degree and the discriminant of $ \mathbb{K}$. The new bound improves significantly on previous known results.

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

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

Received by editor(s): December 16, 2013
Received by editor(s) in revised form: April 28, 2014
Published electronically: October 7, 2015
Article copyright: © Copyright 2015 American Mathematical Society