439 research outputs found

    Relations among the Riemann zeta and Hurwitz zeta functions as well as their products

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    Several relations are obtained among the Riemann zeta and Hurwitz zeta functions, as well as their products. A particular case of these relations give rise to a simple re-derivation if the important results of [11]. Also, a relation derived here provides the starting point of a novel approach which in a series of companion papers yields a formal proof of the Lindel\"{o}f hypothesis. Some of the above relations motivate the need for analysing the large α\alpha behaviour of the modified Hurwitz zeta function ζ1(s,α)\zeta_1(s,\alpha), s∈Cs\in \mathbf{C}, α∈R\alpha\in \mathbf{R}, which is also presented here.Comment: 20 page

    Multiple zeta values and Rota--Baxter algebras

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    We study multiple zeta values and their generalizations from the point of view of Rota--Baxter algebras. We obtain a general framework for this purpose and derive relations on multiple zeta values from relations in Rota--Baxter algebras

    A survey on the theory of universality for zeta and LL-functions

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    We survey the results and the methods in the theory of universality for various zeta and LL-functions, obtained in these forty years after the first discovery of the universality for the Riemann zeta-function by Voronin.Comment: 49page

    Bernoulli identities, zeta relations, determinant expressions, Mellin transforms, and representation of the Hurwitz numbers

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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 Lettington's identity is provided, together with its generalizations to the Hurwitz and Lerch zeta functions, hence to Dirichlet LL series, to Eisenstein series, and to general Mellin transforms. The Hurwitz numbers H~n\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.Comment: 40 pages, no figure

    Relations for Bernoulli--Barnes Numbers and Barnes Zeta Functions

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    The \emph{Barnes ζ\zeta-function} is \zeta_n (z, x; \a) := \sum_{\m \in \Z_{\ge 0}^n} \frac{1}{\left(x + m_1 a_1 + \dots + m_n a_n \right)^z} defined for ℜ(x)>0\Re(x) > 0 and ℜ(z)>n\Re(z) > n and continued meromorphically to \C. Specialized at negative integers −k-k, the Barnes ζ\zeta-function gives \zeta_n (-k, x; \a) = \frac{(-1)^n k!}{(k+n)!} \, B_{k+n} (x; \a) where B_k(x; \a) is a \emph{Bernoulli--Barnes polynomial}, which can be also defined through a generating function that has a slightly more general form than that for Bernoulli polynomials. Specializing B_k(0; \a) gives the \emph{Bernoulli--Barnes numbers}. We exhibit relations among Barnes ζ\zeta-functions, Bernoulli--Barnes numbers and polynomials, which generalize various identities of Agoh, Apostol, Dilcher, and Euler.Comment: 11 page

    qq-Bernoulli Numbers and Polynomials Associated with Multiple qq-Zeta Functions and Basic LL-series

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    By using qq-Volkenborn integration and uniform differentiable on Z\mathbb{Z}%_{p}, we construct pp-adic qq-zeta functions. These functions interpolate the qq-Bernoulli numbers and polynomials. The value of pp-adic qq-zeta functions at negative integers are given explicitly. We also define new generating functions of qq-Bernoulli numbers and polynomials. By using these functions, we prove analytic continuation of some basic (or qq-) LL% -series. These generating functions also interpolate Barnes' type Changhee % q -Bernoulli numbers with attached to Dirichlet character as well. By applying Mellin transformation, we obtain relations between Barnes' type qq% -zeta function and new Barnes' type Changhee qq-Bernolli numbers. Furthermore, we construct the Dirichlet type Changhee (or qq-) LL% -functions.Comment: 37 page

    Series representations of the Riemann and Hurwitz zeta functions and series and integral representations of the first Stieltjes constant

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    We develop series representations for the Hurwitz and Riemann zeta functions in terms of generalized Bernoulli numbers (N\"{o}rlund polynomials), that give the analytic continuation of these functions to the entire complex plane. Special cases yield series representations of a wide variety of special functions and numbers, including log Gamma, the digamma, and polygamma functions. A further byproduct is that ζ(n)\zeta(n) values emerge as nonlinear Euler sums in terms of generalized harmonic numbers. We additionally obtain series and integral representations of the first Stieltjes constant γ1(a)\gamma_1(a). The presentation unifies some earlier results.Comment: 28 pages, no figure

    Rigorous high-precision computation of the Hurwitz zeta function and its derivatives

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    We study the use of the Euler-Maclaurin formula to numerically evaluate the Hurwitz zeta function ζ(s,a)\zeta(s,a) for s,a∈Cs, a \in \mathbb{C}, along with an arbitrary number of derivatives with respect to ss, to arbitrary precision with rigorous error bounds. Techniques that lead to a fast implementation are discussed. We present new record computations of Stieltjes constants, Keiper-Li coefficients and the first nontrivial zero of the Riemann zeta function, obtained using an open source implementation of the algorithms described in this paper.Comment: 15 pages, 2 figure

    Integral representations of functions and Addison-type series for mathematical constants

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    We generalize techniques of Addison to a vastly larger context. We obtain integral representations in terms of the first periodic Bernoulli polynomial for a number of important special functions including the Lerch zeta, polylogarithm, Dirichlet LL- and Clausen functions. These results then enable a variety of Addison-type series representations of functions. Moreover, we obtain integral and Addison-type series for a variety of mathematical constants.Comment: 36 pages, no figure

    A Dynamical Key to the Riemann Hypothesis

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    We investigate a dynamical basis for the Riemann hypothesis (RH) that the non-trivial zeros of the Riemann zeta function lie on the critical line x = 1/2. In the process we graphically explore, in as rich a way as possible, the diversity of zeta and L-functions, to look for examples at the boundary between those with zeros on the critical line and otherwise. The approach provides a dynamical basis for why the various forms of zeta and L-function have their non-trivial zeros on the critical line. It suggests RH is an additional unprovable postulate of the number system, similar to the axiom of choice, arising from the asymptotic behavior of the primes as tends to infinity.Comment: 44 pages, 49 figure
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