We present a new, simple proof, based upon Poisson summation, of the Lipschitz summation formula. A conceptually easy corollary is the functional relation for the Hurwitz zeta function. As a direct consequence we obtain a short, motivated proof of Riemann’s functional equation for ζ(s). Introduction We present a short and motivated proof of Riemann’s functional equation for Riemann’s zeta funct...

Let Q2m be the generalized quaternion group of order 2 m and DN the dihedral group of order 2N . We classify the orbits in Q2m and D n pm (p prime) under the Hurwitz action. 1 The Hurwitz Action Let G be a group. For a, b ∈ G, let a = bab and a = bab. The Hurwitz action on G (n ≥ 2) is an action of the n-string braid group Bn on G . Recall that Bn is given by the presentation Bn = 〈σ1, . . . , ...

It is proved that the commutator subgroup of the fundamental group of the complement of any plane affine irreducible Hurwitz curve (respectively, any plane affine irreducible pseudoholomorphic curve) is finitely presented. It is shown that there exists a pseudo-holomorphic curve (a Hurwitz curve) in CP whose fundamental group of the complement is not Hopfian and, respectively, this group is not...

We use bounds of mixed character sums modulo a square-free integer q of a special structure to estimate the density of integer points on the hypersurface f1(x1) + . . . + fn(xn) = ax k1 1 . . . x kn n for some polynomials fi ∈ Z[X] and nonzero integers a and ki, i = 1, . . . , n. In the case of f1(X) = . . . = fn(X) = X 2 and k1 = . . . = kn = 1 the above hypersurface is known as the Markoff-Hu...

Hurwitz loci have played a basic role in the study of themoduli space of curves at least since 1872 when Clebsch proved that Mg is irreducible by showing that a certain Hurwitz space is connected. Hurwitz cycles on Mg are essential in the work of Harris and Mumford [HM] on the Kodaira dimension of Mg and are expected to govern the length of minimal affine stratifications of Mg. Faber and Pandha...

Contents 1. Introduction 1 2. Structure algebra 4 2.1. Definition of a structure algebra 4 2.2. Semisimple structure algebras 7 2.3. Structure algebra of a finite group 11 3. Cuts of stratified surfaces 15 3.1. Stratified surfaces 15 3.2. Cut systems 17 3.3. Basic and simple surfaces 21 3.4. Neighboring complete cut systems 25 4. Two-dimensional topological field theory 29 4.1. Definition of Kl...

Define the Hecke group by «."((i r ".')> We call G (oo) the G -rationals, and M G (oo) the G -irrationals. The problem we treat here is the approximation of G -irrationals by Gq -rationals. Let M (a) be the upper bound of numbers c for which \a k/m\ < l/cm for all G -irrationals and infinitely many k/m e G (oo). Set h' = infQ M (a). We call h' the Hurwitz constant for by using A -continued frac...

In this study, we obtain new classes of linear codes over Hurwitz integers equipped with a new metric. We refer to the metric as Hurwitz metric. The codes with respect to Hurwitz metric use in coded modulation schemes based on quadrature amplitude modulation (QAM)-type constellations, for which neither Hamming metric nor Lee metric. Also, we define decoding algorithms for these codes when up to...

A number of authors have considered mean values of powers of the modulus of the Hurwitz zeta function ζ(s, a), see [3, 4, 5, 6, 7]. In this paper, the mean of the function itself is considered. First a functional equation relating the Riemann zeta function to sums of the values of the Hurwitz zeta function at rational values of a is derived. This functional equation underlies the vanishing of t...

We discuss the equivalence between the categories of certain ribbon graphs and subgroups of the modular group Γ and use this equivalence to construct exponentially large families of not Hurwitz equivalent simple braid monodromy factorizations of the same element. As an application, we also obtain exponentially large families of topologically distinct algebraic objects such as extremal elliptic ...