3,346 research outputs found
Special values of multiple polylogarithms
Historically, the polylogarithm has attracted specialists and non-specialists alike with its lovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within the past decade, have arisen in combinatorics, knot theory and high-energy physics. More recently, we have been forced to consider multidimensional extensions encompassing the classical polylogarithm, Euler sums, and the Riemann zeta function. Here, we provide a general framework within which previously isolated results can now be properly understood. Applying the theory developed herein, we prove several previously conjectured evaluations, including an intriguing conjecture of Don Zagier
Fast methods to compute the Riemann zeta function
The Riemann zeta function on the critical line can be computed using a
straightforward application of the Riemann-Siegel formula, Sch\"onhage's
method, or Heath-Brown's method. The complexities of these methods have
exponents 1/2, 3/8 (=0.375), and 1/3 respectively. In this paper, three new
fast and potentially practical methods to compute zeta are presented. One
method is very simple. Its complexity has exponent 2/5. A second method relies
on this author's algorithm to compute quadratic exponential sums. Its
complexity has exponent 1/3. The third method employs an algorithm, developed
in this paper, to compute cubic exponential sums. Its complexity has exponent
4/13 (approximately, 0.307).Comment: Presentation simplifie
Thirty-two Goldbach Variations
We give thirty-two diverse proofs of a small mathematical gem--the
fundamental Euler sum identity zeta(2,1)=zeta(3) =8zeta(\bar 2,1). We also
discuss various generalizations for multiple harmonic (Euler) sums and some of
their many connections, thereby illustrating both the wide variety of
techniques fruitfully used to study such sums and the attraction of their
study.Comment: v1: 34 pages AMSLaTeX. v2: 41 pages AMSLaTeX. New introductory
material added and material on inequalities, Hilbert matrix and Witten zeta
functions. Errors in the second section on Complex Line Integrals are
corrected. To appear in International Journal of Number Theory. Title change
An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions
This paper sketches a technique for improving the rate of convergence of a
general oscillatory sequence, and then applies this series acceleration
algorithm to the polylogarithm and the Hurwitz zeta function. As such, it may
be taken as an extension of the techniques given by Borwein's "An efficient
algorithm for computing the Riemann zeta function", to more general series. The
algorithm provides a rapid means of evaluating Li_s(z) for general values of
complex s and the region of complex z values given by |z^2/(z-1)|<4.
Alternatively, the Hurwitz zeta can be very rapidly evaluated by means of an
Euler-Maclaurin series. The polylogarithm and the Hurwitz zeta are related, in
that two evaluations of the one can be used to obtain a value of the other;
thus, either algorithm can be used to evaluate either function. The
Euler-Maclaurin series is a clear performance winner for the Hurwitz zeta,
while the Borwein algorithm is superior for evaluating the polylogarithm in the
kidney-shaped region. Both algorithms are superior to the simple Taylor's
series or direct summation.
The primary, concrete result of this paper is an algorithm allows the
exploration of the Hurwitz zeta in the critical strip, where fast algorithms
are otherwise unavailable. A discussion of the monodromy group of the
polylogarithm is included.Comment: 37 pages, 6 graphs, 14 full-color phase plots. v3: Added discussion
of a fast Hurwitz algorithm; expanded development of the monodromy
v4:Correction and clarifiction of monodrom
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