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2011 Frobenius difference equations and algebraic independence of zeta values in positive equal characteristic
Chieh-Yu Chang, Matthew Papanikolas, Jing Yu
Algebra Number Theory 5(1): 111-129 (2011). DOI: 10.2140/ant.2011.5.111

Abstract

By analogy with the Riemann zeta function at positive integers, for each finite field Fpr with fixed characteristic p, we consider Carlitz zeta values ζr(n) at positive integers n. Our theorem asserts that among the zeta values in the set r=1{ζr(1),ζr(2),ζr(3),}, all the algebraic relations are those relations within each individual family {ζr(1),ζr(2),ζr(3),}. These are the algebraic relations coming from the Euler–Carlitz and Frobenius relations. To prove this, a motivic method for extracting algebraic independence results from systems of Frobenius difference equations is developed.

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Chieh-Yu Chang. Matthew Papanikolas. Jing Yu. "Frobenius difference equations and algebraic independence of zeta values in positive equal characteristic." Algebra Number Theory 5 (1) 111 - 129, 2011. https://doi.org/10.2140/ant.2011.5.111

Information

Received: 27 January 2010; Revised: 18 October 2010; Accepted: 21 November 2010; Published: 2011
First available in Project Euclid: 20 December 2017

zbMATH: 1315.11066
MathSciNet: MR2833787
Digital Object Identifier: 10.2140/ant.2011.5.111

Subjects:
Primary: 11J93
Secondary: 11G09 , 11M38

Keywords: $t$-motives , algebraic independence , Frobenius difference equations , zeta values

Rights: Copyright © 2011 Mathematical Sciences Publishers

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