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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. Goulds 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 $srightarrow infty$ 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 $4mathcal{C}_{n-1}p_{2n}(s)$ 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.
In this note, a criterion for a class of binomials to be permutation polynomials is proposed. As a consequence, many classes of binomial permutation polynomials and monomial complete permutation polynomials are obtained. The exponents in these monomials are of Niho type.
In 2005, Kayal suggested that Schoofs algorithm for counting points on elliptic curves over finite fields might yield an approach to factor polynomials over finite fields in deterministic polynomial time. We present an exposition of his idea and then
In this paper we give the q-extension of Euler numbers which can be viewed as interpolating of the q-analogue of Euler zeta function ay negative integers, in the same way that Riemann zeta function interpolates Bernoulli numbers at negative integers.
We compute the asymptotics of the fourth moment of the Riemann zeta function times an arbitrary Dirichlet polynomial of length $T^{{1/11} - epsilon}$
We present several formulae for the large $t$ asymptotics of the Riemann zeta function $zeta(s)$, $s=sigma+i t$, $0leq sigma leq 1$, $t>0$, which are valid to all orders. A particular case of these results coincides with the classical results of Sieg