19,925 research outputs found
A Computer-Algebra-Based Formal Proof of the Irrationality of ζ(3)
International audienceThis paper describes the formal verification of an irrationality proof of ζ(3), the evaluation of the Riemann zeta function, using the Coq proof assistant. This result was first proved by Apéry in 1978, and the proof we have formalized follows the path of his original presentation. The crux of this proof is to establish that some sequences satisfy a common recurrence. We formally prove this result by an a posteriori verification of calculations performed by computer algebra algorithms in a Maple session. The rest of the proof combines arithmetical ingredients and some asymptotic analysis that we conduct by extending the Mathematical Components libraries. The formalization of this proof is complete up to a weak corollary of the Prime Number Theorem
Irrationality of some p-adic L-values
We give a proof of the irrationality of the -adic zeta-values
for and . Such results were recently obtained by F.Calegari as
an application of overconvergent -adic modular forms. In this paper we
present an approach using classical continued fractions discovered by
Stieltjes. In addition we show irrationality of some other -adic -series
values, and values of the -adic Hurwitz zeta-function
Rational approximations to algebraic Laurent series with coefficients in a finite field
In this paper we give a general upper bound for the irrationality exponent of
algebraic Laurent series with coefficients in a finite field. Our proof is
based on a method introduced in a different framework by Adamczewski and
Cassaigne. It makes use of automata theory and, in our context, of a classical
theorem due to Christol. We then introduce a new approach which allows us to
strongly improve this general bound in many cases. As an illustration, we give
few examples of algebraic Laurent series for which we are able to compute the
exact value of the irrationality exponent
Measures of irrationality for hypersurfaces of large degree
We study various measures of irrationality for hypersurfaces of large degree
in projective space and other varieties. These include the least degree of a
rational covering of projective space, and the minimal gonality of a covering
family of curves. The theme is that positivity properties of canonical bundles
lead to lower bounds on these invariants. In particular, we prove that if X is
a very general smooth hypersurface of dimension n and degree d \ge 2n+1, then
any dominant rational mapping from X to projective n-space must have degree at
least d-1. We also propose a number of open problems, and we show how our
methods lead to simple new proofs of results of Ran and Beheshti-Eisenbud.Comment: Major revision of first version, combining it with previously
separate appendix of Bastianelli and De Poi. Extended section of open
problems added, as well as new proofs of results of Ran and
Beheshti-Eisenbud. Dedicated to J\'anos Koll\'ar on the occasion of his
sixtieth birthda
- …