439 research outputs found
Relations among the Riemann zeta and Hurwitz zeta functions as well as their products
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 behaviour of the modified Hurwitz zeta function
, , , which is also
presented here.Comment: 20 page
Multiple zeta values and Rota--Baxter algebras
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 -functions
We survey the results and the methods in the theory of universality for
various zeta and -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
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 series, to
Eisenstein series, and to general Mellin transforms.
The Hurwitz numbers occur in the Laurent expansion about the
origin of a certain Weierstrass 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 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
The \emph{Barnes -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 and and
continued meromorphically to \C. Specialized at negative integers , the
Barnes -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
-functions, Bernoulli--Barnes numbers and polynomials, which generalize
various identities of Agoh, Apostol, Dilcher, and Euler.Comment: 11 page
-Bernoulli Numbers and Polynomials Associated with Multiple -Zeta Functions and Basic -series
By using -Volkenborn integration and uniform differentiable on
, we construct -adic -zeta functions. These functions
interpolate the -Bernoulli numbers and polynomials. The value of -adic
-zeta functions at negative integers are given explicitly. We also define
new generating functions of -Bernoulli numbers and polynomials. By using
these functions, we prove analytic continuation of some basic (or -) %
-series. These generating functions also interpolate Barnes' type Changhee -Bernoulli numbers with attached to Dirichlet character as well. By applying
Mellin transformation, we obtain relations between Barnes' type % -zeta
function and new Barnes' type Changhee -Bernolli numbers. Furthermore, we
construct the Dirichlet type Changhee (or -) % -functions.Comment: 37 page
Series representations of the Riemann and Hurwitz zeta functions and series and integral representations of the first Stieltjes constant
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 values emerge as nonlinear
Euler sums in terms of generalized harmonic numbers. We additionally obtain
series and integral representations of the first Stieltjes constant
. The presentation unifies some earlier results.Comment: 28 pages, no figure
Rigorous high-precision computation of the Hurwitz zeta function and its derivatives
We study the use of the Euler-Maclaurin formula to numerically evaluate the
Hurwitz zeta function for , along with an
arbitrary number of derivatives with respect to , 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
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 - 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
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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