We present a number of examples to illustrate the use of small quotient dessins as substitutes for their often much larger and more complicated Galois (minimal regular) covers. In doing so we employ several useful group-theoretic techniques, such as the Frobenius character formula for counting triples in a finite group, pointing out some common traps and misconceptions associated with them. Alt...

This paper investigates a generalization the classical Hurwitz zeta function. It is shown that many of the properties exhibited by this special function extends to class of functions called hypergeometric Hurwitz zeta functions, including their analytic continuation to the complex plane and a pre-functional equation satisfied by them. As an application, a formula for moments of hypergeometric H...

In this paper, we study the Markoff-Hurwitz equations x0 + ...+x 2 n = ax0 · · ·xn. The variety V defined by this equation admits a group of automorphisms A ∼= Z/2∗· · ·∗Z/2 (an n+1 fold free product). For a solution P on this variety, we consider the number NP (t) of points Q in the A-orbit of P with logarithmic height h(Q) less than t. We show that if a is rational, and P is a non-trivial rat...

For distinct primes / and p , the Iwasawa invariant XT stabilizes in the cyclotomic Zp -tower over a complex abelian base field. We calculate this stable invariant for p = 3 and various / and K. Our motivation was to search for a formula of Riemann-Hurwitz type for XT that might hold in a p-extension. From our numerical results, it is clear that no formula of such a simple kind can hold. In the...

Between 1968 and 1978 four negative income tax experiments were conducted; they were designed to measure labor supply and earnings. The experiments were not designed to measure the effects of government programs on such demographic behavior as marital dissolution, fertility, family composition, or the decision to marry or remarry. Nevertheless, the data from the experiments have been used to an...

We show that the Hurwitz zeta function, ζ(ν, a), and the Legendre chi function, χν(z), defined by ζ(ν, a) = ∞ ∑ k=0 1 (k + a)ν , 0 < a ≤ 1, Re ν > 1, and χν(z) = ∞ ∑ k=0 z2k+1 (2k + 1)ν , |z| ≤ 1, Re ν > 1 with ν = 2, 3, 4, . . . , respectively, form a discrete Fourier transform pair. Many formulae involving the values of these functions at rational arguments, most of them unknown, are obtained...

The main result proved in the paper is the computation of the explicit equations defining the Hurwitz schemes of coverings with punctures as subschemes of the Sato infinite Grassmannian. As an application, we characterize the existence of certain linear series on a smooth curve in terms of soliton equations.

Consider the polynomial tr (A + tB)m in t for positive hermitian matrices A and B with m ∈ N. The Bessis-Moussa-Villani conjecture (in the equivalent form of Lieb and Seiringer) states that this polynomial has nonnegative coefficients only. We prove that they are at least asymptotically positive, for the nontrivial case of AB 6= 0. More precisely, we show —once complex-analytically, once combin...

In this paper, which is a companion paper to W , starting from the Euler integral which appears in a generalization of Jensen’s formula, we shall give a closed form for the integral of log Γ 1± t . This enables us to locate the genesis of two new functions A1/a and C1/a considered by Srivastava and Choi. We consider the closely related function A(a) and the Hurwitz zeta function, which render t...