Related papers: Tree-lattice zeta functions and class numbers

We give a new and short proof of the Mallows-Sloane upper bound for self-dual codes. We formulate a version of Greene's theorem for normalized weight enumerators. We relate normalized rank-generating polynomials to two-variable zeta…

An extension of two finite trigonometric series is studied to derive closed form formulae involving the Hurwitz-Lerch zeta function. The trigonometric series involves angles with a geometric series involving the powers of 3. These closed…

We use the theory of arithmetic quotients of the Bruhat-Tits tree developed by Serre and others to obtain Dirichlet-style theorems for Diophantine approximation on global function fields. This approach allows us to find sharp values for the…

We define generalised zeta functions associated to indefinite quadratic forms of signature (g-1,1) -- and more generally, to complex symmetric matrices whose imaginary part has signature (g-1,1) -- and we investigate their properties. These…

We define zeta-functions of weight lattices of compact connected semisimple Lie groups. If the group is simply-connected, these zeta-functions coincide with ordinary zeta-functions of root systems of associated Lie algebras. In this paper…

For the space $Lat_n$ of all the lattices in a $p$-adic $n$-dimensional linear space we obtain an analog of matrix beta-functions; this beta-function has a degeneration to the Tamagawa zeta-function. We propose an analog of Berezin kernels…

We consider a family of isometric extensions of the full shift on p symbols (for p a prime) parametrized by a probability space. Using Heath-Brown's work on the Artin conjecture, it is shown that for all but two primes p the set of limit…

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…

We introduce new zeta functions related to an endomorphism $\phi$ of a discrete group $\Gamma$. They are of two types: counting numbers of fixed ($\rho\sim \rho\circ\phi^n$) irreducible representations for iterations of $\phi$ from an…

Let K be a function field with constant field k and let "infinity" be a fixed place of K. Let C be the Dedekind domain consisting of all those elements of K which are integral outside "infinity". The group G=GL_2(C) is important for a…

Let K be a p-adic field. We explore Igusa's p-adic zeta function, which is associated to a K-analytic function on an open and compact subset of K^n. First we deduce a formula for an important coefficient in the Laurent series of this…

We establish three independent results on groups acting on trees. The first implies that a compactly generated locally compact group which acts continuously on a locally finite tree with nilpotent local action and no global fixed point is…

Using a remainder theorem for valuations of a field, we give a new perspective on the norm function of a global field. We define the Euler totient function of a global field and recover the essential analytical properties of the classical…

It is well known that the distribution of the prime numbers plays a central role in number theory. It has been known, since Riemann's memoir in 1860, that the distribution of prime numbers can be described by the zero-free region of the…

We study the asymptotic distribution of integers sharing the same rooted-tree structure that encodes their complete prime factorization tower. For each tree we derive an explicit density formula depending only on a pair $(m,k)$, the density…

Let f be a regular function on a nonsingular complex algebraic variety of dimension d. We prove a formula for the motivic zeta function of f in terms of an embedded resolution. This formula is over the Grothendieck ring itself, and…

A new realization of the conformal algebra is studied which mimics the behaviour of a statistical system on a discrete albeit infinite lattice. The two-point function is found from the requirement that it transforms covariantly under this…

In this paper we introduce a new infinite set of transcendental integrals. Each of them is expressed by corresponding value of the function $|\zf|^{-2}$. Such a property is another argument about universality of the Riemann zeta-function…

Motivated by the connection to the pair correlation of the Riemann zeros, we investigate the second derivative of the logarithm of the Riemann zeta function, in particular the zeros of this function. Theorem 1 gives a zero-free region.…

We introduce motivic zeta functions for matroids. These zeta functions are defined as sums over the lattice points of Bergman fans, and in the realizable case, they coincide with the motivic Igusa zeta functions of hyperplane arrangements.…