# Binomial Polynomials mimicking Riemann's Zeta Function

## TL;DR

This paper constructs and analyzes polynomial functions related to the Riemann zeta function, demonstrating their zeros lie on the critical line and exploring their properties through hypergeometric functions, combinatorics, and functional equations.

## Contribution

It introduces new polynomial families mimicking the Riemann zeta function's critical line zeros, extending classical results with hypergeometric and combinatorial techniques.

## Key findings

- Zeros of the polynomials lie on the critical line or real axis.
- Polynomials satisfy a functional equation similar to the Riemann xi function.
- Certain polynomial combinations yield integers with only odd prime factors.

## Abstract

The (generalised) Mellin transforms of certain Chebyshev and Gegenbauer functions based upon the Chebyshev and Gegenbauer polynomials, have polynomial factors $p_n(s)$, whose zeros lie all on the `critical line' $\Re\,s=1/2$ 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 $_3F_2(1)$ 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 $p_n(s;\beta)=\pm p_n(1-s;\beta)$, 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 $q_n(s).$ The denominator of the rational form has singularities on the negative real axis, and so $q_n(s)$ has the same `critical zeros' as the `critical polynomial' $p_n(s)$. Moreover as $s\rightarrow \infty$ along the positive real axis, $q_n(s)\rightarrow 1$ from below, mimicking $1/\zeta(s)$ on the positive real line. In the case of the Chebyshev parameters we deduce the simpler S:2/1 binomial form, and with $\mathcal{C}_n$ the $n$th Catalan number, $s$ an integer, we show that polynomials $4\mathcal{C}_{n-1}p_{2n}(s)$ and $\mathcal{C}_{n}p_{2n+1}(s)$ yield integers with only odd prime factors. The results touch on analytic number theory, special function theory, and combinatorics.