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An analog of the prime number theorem for finite fields via truncated polylogarithm expansions


Authors: Niko Rebenich, T. Aaron Gulliver, Stephen Neville and Ulrich Speidel
Journal: Math. Comp.
MSC (2010): Primary 05A16, 11T06; Secondary 11G55, 11M35, 11Y35, 40A05
DOI: https://doi.org/10.1090/mcom/3247
Published electronically: May 5, 2017
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Abstract: An exponentially accurate asymptotic expansion of the truncated polylogarithm function is derived that leads to an asymptotic formula for enumerating monic irreducible polynomials over finite fields. This formula is analogous to the asymptotic expansion formula of the classical prime counting function. Results are presented which show that it is more accurate than previous results in the literature while requiring very little computational effort. Asymptotic expansions of the Lerch transcendent, Eulerian polynomials, and polylogarithms of negative integer order are also given. The accuracy of the proposed approach is verified via numerical results.


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Additional Information

Niko Rebenich
Affiliation: Department of Electrical and Computer Engineering, University of Victoria, P.O. Box 1700, STN CSC Victoria, British Columbia V8W 2Y2, Canada
Email: niko@ece.uvic.ca

T. Aaron Gulliver
Affiliation: Department of Electrical and Computer Engineering, University of Victoria, P.O. Box 1700, STN CSC Victoria, British Columbia V8W 2Y2, Canada
Email: agullive@ece.uvic.ca

Stephen Neville
Affiliation: Department of Electrical and Computer Engineering, University of Victoria, P.O. Box 1700, STN CSC Victoria, British Columbia V8W 2Y2, Canada
Email: sneville@ece.uvic.ca

Ulrich Speidel
Affiliation: Department of Computer Science, The University of Auckland, Private Bag 92019, Auckland, New Zealand
Email: ulrich@cs.auckland.ac.nz

DOI: https://doi.org/10.1090/mcom/3247
Keywords: Lerch transcendent, polylogarithm, Eulerian polynomials, divergent series, superasymptotic expansion, prime polynomial counting function, irreducible polynomials, aperiodic necklaces.
Received by editor(s): March 31, 2016
Received by editor(s) in revised form: June 3, 2016, and September 7, 2016
Published electronically: May 5, 2017
Article copyright: © Copyright 2017 American Mathematical Society