In this paper, we present an optimization of Odlyzko and Schönhage algorithm that computes efficiently Zeta function at large height on the critical line, together with computation of zeros of the Riemann Zeta function thanks to an implementation of this technique. The first family of computations consists in the verification of the Riemann Hypothesis on all the first 10 non trivial zeros. The ...

Example 1.2. (a) Suppose X is a Riemann surface. Let Y ⊂ X be a (connected) open subset. Then Y is a Riemann surface, whose complex structure is given by taking all U ⊂ Y from charts of X. (b) Let P = C ∪ {∞}, homeomorphic to the real sphere. Take U1 = P\{∞} = C, U2 = P\{0} = C∗ ∪ {∞}. Define φ1(z) = z, φ2(z) = 1/z for z 6= ∞ and φ2(∞) = 0. Then φ2 ◦ φ−1 1 : C∗ → C∗ is given by z 7→ 1/z, which ...

We study that subset of the moduli space M g of stable genus g, g > 1, Riemann surfaces which consists of such stable Riemann surfaces on which a given nite group F acts. We show rst that this subset is compact. It turns out that, for general nite groups F, the above subset is not connected. We show, however, that for Z 2 actions this subset is connected. Finally, we show that even in the modul...

We consider Riemann mappings from bounded Lipschitz domains in the plane to a triangle. We show that in this case the Riemann mapping has a linear variational principle: it is the minimizer of the Dirichlet energy over an appropriate affine space. By discretizing the variational principle in a natural way we obtain discrete conformal maps which can be computed by solving a sparse linear system....

Motivated by a problem of scattering theory, the authors solve by quadratures a vector Riemann– Hilbert problem with the matrix coefficient of Chebotarev–Khrapkov type. The problem of matrix factorization reduces to a scalar Riemann–Hilbert boundary-value problem on a twosheeted Riemann surface of genus 3 that is topologically equivalent to a sphere with three handles. The conditions quenching ...

It has long been known that, under the assumption of the Riemann Hypothesis, one can give upper and lower bounds for the error ∑ p≤x log p−x in the Prime Number Theorem, such bounds being within a factor of (log x)2 of each other and this fact being equivalent to the Riemann Hypothesis. In this paper we show that, provided “Riemann Hypothesis” is replaced by “Generalized Riemann Hypothesis”, re...

The aim of this paper is to present theoretical basis for computing a representation of a compact Riemann surface as an algebraic plane curve and to compute a numerical approximation for its period matrix. We will describe a program Cars ((3]) that can be used to deene Riemann surfaces for computations. Cars allows one also to perform the Fenchel{Nielsen twist and other deformations on Riemann ...

Introduction. Since the time of Riemann, there has been a close interplay between the study of the geometry of complex algebraic curves (or, equivalently, compact Riemann surfaces) and the topology of 2-manifolds. These connections arise from the uniformization theorem, which asserts that every simply connected Riemann surface is conformally equivalent to either the Riemann sphere, the plane, o...

We study the one parameter family of genus 2 Riemann surfaces defined by the orbit of the L-shaped translation surface tiled by three squares under the Teichmüller geodesic flow. These surfaces are real algebraic curves with three real components. We are interested in describing these surfaces by their period matrices. We show that the only Riemann surface in that family admitting a non-hyperel...