Related papers: Average $r$-rank Artin Conjecture

Let $K$ be a number field, $k\geq 2$ an integer, $(K^*)^k$ the $k$-fold direct product of $K^*$ with coordinatewise multiplication, and $\Gamma$ a finitely generated subgroup of rank $r$ of $(K^*)^k$. Further, let $H(\alpha )$ denote the…

A set of natural numbers $A$ is called primitive if no element of $A$ divides any other. Let $\Omega(n)$ be the number of prime divisors of $n$ counted with multiplicity. Let $f_z(A) = \sum_{a \in A}\frac{z^{\Omega(a)}}{a (\log a)^z}$,…

Given a hypergraph $\Gamma=(\Omega,\mathcal{X})$ and a sequence $\mathbf{p} = (p_\omega)_{\omega\in \Omega}$ of values in $(0,1)$, let $\Omega_{\mathbf{p}}$ be the random subset of $\Omega$ obtained by keeping every vertex $\omega$…

Let $G$ be a multiplicative subgroup of the prime field $\mathbb F_p$ of size $|G|> p^{1-\kappa}$ and $r$ an arbitrarily fixed positive integer. Assuming $\kappa=\kappa(r)>0$ and $p$ large enough, it is shown that any proportional subset…

Let $S(n)$ denote the least primary factor in the primary decomposition of the multiplicative group $M_n = (\Bbb Z/n\Bbb Z)^\times$. We give an asymptotic formula, with order of magnitude $x/(\log x)^{1/2}$, for the counting function of…

For certain primes $p$, the average digit in the expansion of $1/p$ was found to have a deviation from random behaviour related to the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-p})$ (Girstmair 1994). In this short…

Using a sieve-theoretic argument, we show that almost all gaps $(p_n, p_{n+1})$ between consecutive primes $p_n, p_{n+1}$ contain a natural number $m$ whose least prime factor $p(m)$ is at least the length $p_{n+1} - p_n$ of the gap,…

Given a prime $p$ and $q$ a power of $p$, we study the statistics of $p$-ranks of Artin--Schreier covers of given genus defined over $\mathbb{F}_q$, in the large $q$-limit. We refer to this problem as the geometric problem. We also study an…

The main results in this thesis deal with the representation growth of certain classes of groups. In chapter $1$ we present the required preliminary theory. In chapter $2$ we introduce the Congruence Subgroup Problem for an algebraic group…

A new derivation of the classic asymptotic expansion of the n-th prime is presented. A fast algorithm for the computation of its terms is also given, which will be an improvement of that by Salvy (1994). Realistic bounds for the error with…

For $p > 1$ let a function $\varphi_p(x) = x^2/2$ if $|x|\le 1$ and $\varphi_p(x) = 1/p|x|^p -1/p + 1/2$ if $|x| > 1$. For a random variable $\xi$ let $\tau_{\varphi_p}(\xi)$ denote $\inf\{c\ge 0 :\; \forall_{\lambda\in\mathbb{R}}\;…

We determine the epimorphisms $A \to W$ from the Artin group $A$ of type $\Gamma$ onto the Coxeter group $W$ of type $\Gamma$, in case $\Gamma$ is an irreducible Coxeter graph of spherical type, and we prove that the kernel of the standard…

The arithmetic average of the first $n$ primes, $\bar p_n = {1\over n} \sum_{i=1}^n p_i$, exhibits very many interesting and subtle properties. Since the transformation from $p_n \to \bar p_n$ is extremely easy to invert, $p_n = n\bar p_n -…

Let K/F be an arbitrary Galois extension of number fields and r be a representation of Gal(K/F) into GSp(4,C). Let E_16 be the elemetary abelian group of order 16 and C_5 the cyclic group of order 5. If the image of r in the projective…

Given an elliptic curve E/Q and a prime p at which E has good reduction, let e_p be the exponent of the group E_p(F_p) of F_p-rational points on the reduction of E modulo p. Under the Generalized Riemann Hypothesis (GRH) for the Dedekind…

We construct for $d\geq 2$ and $\epsilon>0$ a $d$-generated $p$-group $\Gamma$, which in an asymptotic sense behaves almost like a $d$-generated free pro-$p$-group. We show that a subgroup of index $p^n$ needs $(d-\epsilon)p^n$ generators,…

The centrepiece of this paper is a normal form for primitive elements which facilitates the use of induction arguments to prove properties of primitive elements. The normal form arises from an elementary algorithm for constructing a…

Let M=M(n,q) be the average of the orders of the elements in the finite unitary group U(n,q). The asymptotic estimate log(M)=nlog(q)-log(n) +o(log n) is proved.

The main goal of this note is to suggest an algebraic approach to the quasi-isometric classification of partially commutative groups (alias right-angled Artin groups). More precisely, we conjecture that if the partially commutative groups…

In this article we study the arithmetic mean value $\Sigma(n)$ of the square roots of the first $n$ integers. For this quantity, we develop an asymptotic expression, and derive a formula for its integer part which has been conjectured…