Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSummatory relations and prime products for the Stieltjes constants, and other related results
66
0
0.0
(
0
)
Authors
Mark W. Coffey
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
We consider the summatory function of the number of prime factors for integers $leq x$ over arithmetic progressions. Numerical experiments suggest that some arithmetic progressions consist more number of prime factors than others. Greg Martin conjectured that the difference of the summatory functions should attain a constant sign for all sufficiently large $x$. In this paper, we provide strong evidence for Greg Martins conjecture. Moreover, we derive a general theorem for arithmetic functions from the Selberg class.
In this paper we study products of quadratic residues modulo odd primes and prove some identities involving quadratic residues. For instance, let $p$ be an odd prime. We prove that if $pequiv5pmod8$, then $$prod_{0<x<p/2,(frac{x}{p})=1}xequiv(-1)^{1+r}pmod p,$$ where $(frac{cdot}{p})$ is the Legendre symbol and $r$ is the number of $4$-th power residues modulo $p$ in the interval $(0,p/2)$. Our work involves class number formula, quartic Gauss sums, Stickelbergers congruence and values of Dirichlet L-series at negative integers.
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.
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 present some new identities for multiple polylogarithms (abbr. MPLs) and multiple harmonic star sums (abbr. MHSSs) by using the methods of iterated integral computations of logarithm functions. Then, by applying these formulas obtained, we establish some explicit relations between Kaneko-Yamamoto type multiple zeta values (abbr. K-Y MZVs), multiple zeta values (abbr. MZVs) and MPLs. Further, we find some explicit relations between MZVs and multiple zeta star values (abbr. MZSVs). Furthermore, we define an Ap{e}ry-type variant of MZSVs $zeta^star_B({bf k})$ (called multiple zeta $B$-star values, abbr. MZBSVs) which involve MHSSs and central binomial coefficients, and establish some explicit connections among MZVs, alternating MZVs and MZBSVs by using the method of iterated integrals. Finally, some interesting consequences and illustrative examples are presented.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
