Hurwitz groups are the nontrivial finite quotients 2 3 7 of the ( 2 , 3 , 7 ) triangle group (x, y\x — y = (xy) = 1) . This paper gives a brief survey of such groups, their significance, and some of their properties, together with a description of all examples known to the author.

We provide an explicit formula for the Tornheim double series T (a, 0, c) in terms of an integral involving the Hurwitz zeta function. For integer values of the parameters, a = m, c = n, we show that in the most interesting case of even weight N := m + n the Tornheim sum T (m, 0, n) can be expressed in terms of zeta values and the family of integrals

Using functional properties of the Hurwitz zeta function and symbolic derivatives of the trigonometric functions, the function ζ(2n + 1, p/q) is expressed in several ways in terms of other mathematical functions and numbers, including in particular the Glaisher numbers. 2000 Mathematics Subject Classification. Primary 11M35, 33B99. Secondary 11B75, 33E20.

We use the following convention: A, B, C, A1, B1, C1, A2, B2, C2 are points of E2 T, l1, m1, n1 are real numbers, and X, Y , Z are subsets of E2 T. Let us consider X, Y . We introduce X is parallel to Y as a synonym of X misses Y . Let us consider X, Y , Z. We say that X, Y , Z are concurrent if and only if: (Def. 1) X is parallel to Y and Y is parallel to Z and Z is parallel to X or there exis...

The Hurwitz space is a compactification of the space of rational functions of a given degree. We study the intersection of various strata of this space with its boundary. A study of the cohomology ring of the Hurwitz space then allows us to obtain recurrence relations for certain numbers of ramified coverings of a sphere by a sphere with prescribed ramifications. Generating functions for these ...

In this article we describe the cell decompositions of the moduli space of Riemann surfaces due to Harer, Mumford and Penner and its relationship to a Hurwitz problem. The cells possess natural linear structures and with respect to this they can be described as rational convex polytopes. We show how to effectively calculate the number of lattice points and the volumes over all cells. These calc...

In this paper, the neutrix limit is used to extend the definition of the Hurwitz zeta function ζ(α, x) and its partial derivatives to the whole complex plane except for non-positive integers α, in particular, the values of ζ(1, x) is obtained. This definition is equivalent to the Hermite’s integral of ζ(α, x) as α 6= 1, 0,−1, . . .. Moreover, some properties of ζ(1, x) are established and we fi...