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A trio of Bernoulli relations, their implications for the Ramanujan polynomials and the special values of the Riemann zeta function

M. C. Lettington (2013)

Acta Arithmetica

We study the interplay between recurrences for zeta related functions at integer values, 'Minor Corner Lattice' Toeplitz determinants and integer composition based sums. Our investigations touch on functional identities due to Ramanujan and Grosswald, the transcendence of the zeta function at odd integer values, the Li Criterion for the Riemann Hypothesis and pseudo-characteristic polynomials for zeta related functions. We begin with a recent result for ζ(2s) and some seemingly new Bernoulli relations,...

A tutorial on conformal groups

Ian Porteous (1996)

Banach Center Publications

Our concern is with the group of conformal transformations of a finite-dimensional real quadratic space of signature (p,q), that is one that is isomorphic to ℝ p , q , the real vector space ℝ p + q , furnished with the quadratic form x ( 2 ) = x · x = - x 1 2 - x 2 2 - . . . - x p 2 + x p + 1 2 + . . . + x p + q 2 , and especially with a description of this group that involves Clifford algebras.

A two-dimensional continued fraction algorithm with Lagrange and Dirichlet properties

Christian Drouin (2014)

Journal de Théorie des Nombres de Bordeaux

A Lagrange Theorem in dimension 2 is proved in this paper, for a particular two dimensional continued fraction algorithm, with a very natural geometrical definition. Dirichlet type properties for the convergence of this algorithm are also proved. These properties proceed from a geometrical quality of the algorithm. The links between all these properties are studied. In relation with this algorithm, some references are given to the works of various authors, in the domain of multidimensional continued...

A two-dimensional univoque set

Martijn de Vrie, Vilmos Komornik (2011)

Fundamenta Mathematicae

Let J ⊂ ℝ² be the set of couples (x,q) with q > 1 such that x has at least one representation of the form x = ∑ i = 1 ∞ c i q - i with integer coefficients c i satisfying 0 ≤ c i < q , i ≥ 1. In this case we say that ( c i ) = c ₁ c ₂ . . . is an expansion of x in base q. Let U be the set of couples (x,q) ∈ J such that x has exactly one expansion in base q. In this paper we deduce some topological and combinatorial properties of the set U. We characterize the closure of U, and we determine its Hausdorff dimension. For (x,q) ∈ J, we also prove new properties...

A valuation criterion for normal basis generators of Hopf-Galois extensions in characteristic p

Nigel P. Byott (2011)

Journal de Théorie des Nombres de Bordeaux

Let S / R be a finite extension of discrete valuation rings of characteristic p &gt; 0 , and suppose that the corresponding extension L / K of fields of fractions is separable and is H -Galois for some K -Hopf algebra H . Let 𝔻 S / R be the different of S / R . We show that if S / R is totally ramified and its degree n is a power of p , then any element ρ of L with v L ( ρ ) ≡ - v L ( 𝔻 S / R ) - 1 ( mod n ) generates L as an H -module. This criterion is best possible. These results generalise to the Hopf-Galois situation recent work of G. G. Elder for Galois extensions.

Currently displaying 1001 – 1020 of 16555