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Consider the group ${SL}_{2} (O_{K})$ over the ring of algebraic integers of a number field $K$ . Define the height of a matrix to be the maximum over all the conjugates of its entries in absolute value. Let ${SL}_{2} (O_{K}, t)$ be the number of matrices in ${SL}_{2} (O_{K})$ with height bounded by $t$ . We determine the asymptotic behaviour of ${SL}_{2} (O_{K}, t)$ as $t$ goes to infinity including an error term, ${SL}_{2} (O_{K}, t) = C t^{2 n} + O (t^{2 n - η})$ with $n$ being the degree of $K$ . The constant $C$ involves the discriminant of $K$ , an integral depending only on the signature of $K$ , and the value of the Dedekind zeta function...

Counting irreducible polynomials over finite fields

Qichun Wang, Haibin Kan (2010)

Czechoslovak Mathematical Journal

In this paper we generalize the method used to prove the Prime Number Theorem to deal with finite fields, and prove the following theorem: $π (x) = \frac{q}{q - 1} \frac{x}{{log}_{q} x} + \frac{q}{{(q - 1)}^{2}} \frac{x}{{log}_{q}^{2} x} + O (\frac{x}{{log}_{q}^{3} x}), x = q^{n} \to \infty$ where $π (x)$ denotes the number of monic irreducible polynomials in $F_{q} [t]$ with norm $\leq x$ .

Counting monic irreducible polynomials $P$ in $𝔽_{q} [X]$ for which order of $X (mod P)$ is odd

Christian Ballot (2007)

Journal de Théorie des Nombres de Bordeaux

Hasse showed the existence and computed the Dirichlet density of the set of primes $p$ for which the order of $2 (mod p)$ is odd; it is $7 / 24$ . Here we mimic successfully Hasse’s method to compute the density $δ_{q}$ of monic irreducibles $P$ in $𝔽_{q} [X]$ for which the order of $X (mod P)$ is odd. But on the way, we are also led to a new and elementary proof of these densities. More observations are made, and averages are considered, in particular, an average of the $δ_{p}$ ’s as $p$ varies through all rational primes.

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Correction to the paper "On a kind of uniform distribution of values of multiplicative functions in residue classes", Acta Arithmetica 31(1976), pp. 291-294

Correction to the paper "On power-free numbers and polynomials. II".

Correction to the paper: On the sum-of-Divisors Function of the numbers of Fermat and Ferentinou-Nicolacopoulou

Correction to the paper "The divisor problem for (k, r)-integers. II".

Correction to the paper "The number of k-free and k-ary divisors of m which are prime to n".

Corrections to "Asymptotic behaviour of some infinite products involving prime numbers" (Acta Arith. 75 (1996), 339-350)

Corrélation de suites arithmétiques

Correlations of representation functions of binary quadratic forms

Corrigendum to my paper "On twin almost primes" and an addendum on Selberg's sieve

Corrigendum to “On short sums of certain multiplicative functions”.

Corrigendum to "Stability aspects of arithmetic functions, II" (Acta Arith. 139 (2009), 131-146)

Corrigendum to the paper "Additive problems with prime numbers of special type" (Acta Arith. 96 (2000), 53-88)

Corrigendum to the paper "On the probability that n and f(n) are relatively prime"

Corrigendum zur Arbeit „Über die reellen Nullstellen der Dirichletschen L-Reihen"

Counting invertible matrices and uniform distribution

Counting irreducible polynomials over finite fields

Counting monic irreducible polynomials $P$ in $𝔽_{q} [X]$ for which order of $X (mod P)$ is odd

Counting $p$ -groups and nilpotent groups

Counting solutions of decomposable form equations

Covering abelian groups with cyclic subsets.

Currently displaying 301 – 320 of 1791