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Register a new userA trio of Bernoulli relations, their implications for the Ramanujan polynomials and the zeta constants
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Authors
Matthew C. Lettington
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We study the interplay between recurrences for zeta related functions at integer values, `Minor Corner Lattice Toeplitz determinants and integer composition based sums. Our investigations touch on functional identities due to Ramanujan and Grosswald, the transcendence of the zeta function at odd integer values, the Li Criterion for the Riemann Hypothesis and pseudo-characteristic polynomials for zeta related functions. We begin with a recent result for zeta(2s) and some seemingly new Bernoulli relations, which we use to obtain a generalised Ramanujan polynomial and properties thereof.
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The Riemann zeta identity at even integers of Lettington, along with his other Bernoulli and zeta relations, are generalized. Other corresponding recurrences and determinant relations are illustrated. Another consequence is the application to sums of double zeta values. A set of identities for the Ramanujan and generalized Ramanujan polynomials is presented. An alternative proof of Lettingtons identity is provided, together with its generalizations to the Hurwitz and Lerch zeta functions, hence to Dirichlet $L$ series, to Eisenstein series, and to general Mellin transforms. The Hurwitz numbers $tilde{H}_n$ occur in the Laurent expansion about the origin of a certain Weierstrass $wp$ function for a square lattice, and are highly analogous to the Bernoulli numbers. An integral representation of the Laurent coefficients about the origin for general $wp$ functions, and for these numbers in particular, is presented. As a Corollary, the asymptotic form of the Hurwitz numbers is determined. In addition, a series representation of the Hurwitz numbers is given, as well as a new recurrence.
In this paper, we consider Barnes multiple Bernoulli and poly-Bernoulli mixed-type polynomials. From the properties of Sheffer sequences of these polynomials arising from umbrral calculus, we derive new and interesting identities.
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 work we introduce a new polynomial representation of the Bernoulli numbers in terms of polynomial sums allowing on the one hand a more detailed understanding of their mathematical structure and on the other hand provides a computation of $B_{2n}$ as a function of B$_{2n-2}$ only. Furthermore, we show that a direct computation of the Riemann zeta-function and their derivatives at k $in mathbb Z$ is possible in terms of these polynomial representation. As an explicit example, our polynomial Bernoulli number representation is applied to fast approximate computations of $zeta$(3), $zeta$(5) and $zeta$(7).
The Stieltjes constants $gamma_k(a)$ appear in the regular part of the Laurent expansion for the Hurwitz zeta function $zeta(s,a)$. We present summatory results for these constants $gamma_k(a)$ in terms of fundamental mathematical constants such as the Catalan constant, and further relate them to products of rational functions of prime numbers. We provide examples of infinite series of differences of Stieltjes constants evaluating as volumes in hyperbolic $3$-space. We present a new series representation for the difference of the first Stieltjes constant at rational arguments. We obtain expressions for $zeta(1/2)L_{-p}(1/2)$, where for primes $p>7$, $L_{-p}(s)$ are certain $L$-series, and remarkably tight bounds for the value $zeta(1/2)$, $zeta(s)=zeta(s,1)$ being the Riemann zeta function.
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