1,294 research outputs found
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
Geometric non-vanishing
We consider -functions attached to representations of the Galois group of
the function field of a curve over a finite field. Under mild tameness
hypotheses, we prove non-vanishing results for twists of these -functions by
characters of order prime to the characteristic of the ground field and by
certain representations with solvable image. We also allow local restrictions
on the twisting representation at finitely many places. Our methods are
geometric, and include the Riemann-Roch theorem, the cohomological
interpretation of -functions, and some monodromy calculations of Katz. As an
application, we prove a result which allows one to deduce the conjecture of
Birch and Swinnerton-Dyer for non-isotrivial elliptic curves over function
fields whose -function vanishes to order at most 1 from a suitable
Gross-Zagier formula.Comment: 46 pages. New version corrects minor errors. To appear in Inventiones
Mat
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