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Binomial polynomials mimicking Riemann's Zeta Function

Coffey, Mark W. and Lettington, Matthew C. 2017. Binomial polynomials mimicking Riemann's Zeta Function. arXiv , arXiv:1703.09251.

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The (generalised) Mellin transforms of certain Chebyshev and Gegenbauer functions based upon the Chebyshev and Gegenbauer polynomials, have polynomial factors pn(s), whose zeros lie all on the `critical line' Rs=1/2 or on the real axis (called critical polynomials). The transforms are identified in terms of combinatorial sums related to H. W. Gould's S:4/3, S:4/2 and S:3/1 binomial coefficient forms. Their `critical polynomial' factors are then identified as variants of the S:4/1 form, and more compactly in terms of certain 3F2(1) hypergeometric functions. Furthermore, we extend these results to a 1-parameter family of polynomials with zeros only on the critical line. These polynomials possess the functional equation pn(s;β)=±pn(1−s;β), similar to that for the Riemann xi function. It is shown that via manipulation of the binomial factors, these `critical polynomials' can be simplified to an S:3/2 form, which after normalisation yields the rational function qn(s). The denominator of the rational form has singularities on the negative real axis, and so qn(s) has the same `critical zeros' as the `critical polynomial' pn(s). Moreover as s→∞ along the positive real axis, qn(s)→1 from below, mimicking 1/ζ(s) on the positive real line. In the case of the Chebyshev parameters we deduce the simpler S:2/1 binomial form, and with Cn the nth Catalan number, s an integer, we show that polynomials 4Cn−1p2n(s) and Cnp2n+1(s) yield integers with only odd prime factors. The results touch on analytic number theory, special function theory, and combinatorics.

Item Type: Article
Date Type: Submission
Status: Published
Schools: Mathematics
Subjects: Q Science > QA Mathematics
Publisher: Cornell University
ISSN: 2331-8422
Last Modified: 13 Mar 2020 04:06

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