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Das Verhalten multiplikativer Funktionen auf gewissen Zahlenfolgen.

Karl-Heinz Indlekofer (1976)

Journal für die reine und angewandte Mathematik

Démonstration de l’équation $\sum_{k = 1}^{\infty} \frac{μ (k)}{k} = 0$

H. von Mangoldt (1898)

Annales scientifiques de l'École Normale Supérieure

Density of solutions to quadratic congruences

Neha Prabhu (2017)

Czechoslovak Mathematical Journal

A classical result in number theory is Dirichlet’s theorem on the density of primes in an arithmetic progression. We prove a similar result for numbers with exactly $k$ prime factors for $k > 1$ . Building upon a proof by E. M. Wright in 1954, we compute the natural density of such numbers where each prime satisfies a congruence condition. As an application, we obtain the density of squarefree $n \leq x$ with $k$ prime factors such that a fixed quadratic equation has exactly $2^{k}$ solutions modulo $n$ .

Développement asymptotique de la somme des inverses d’une fonction arithmétique

Hacène Belbachir, Farid Bencherif (2009)

Annales mathématiques Blaise Pascal

La somme des puissances des inverses de $π (n)$ , $π (x)$ désignant le nombre de nombres premiers n’excédant pas $x$ , a fait l’objet de nombreux travaux. Nous généralisons, dans cet article, les formules asymptotiques obtenues par ces auteurs à toute une classe de fonctions arithmétiques.

Die Wertevertailung Der Anzahl Der Nicht-Isomorphen Abelschen Gruppen Endlicher Ordnung Und Ein Verwandtes Zahlenteoretisches Problem

Ekkehard Krätzel (1982)

Publications de l'Institut Mathématique

Dirichlet series with functional equations and related arithmetical identities

K. Chandrasekharan, H. Joris (1973)

Acta Arithmetica

Discretization of prime counting functions, convexity and the Riemann hypothesis

Emre Alkan (2023)

Czechoslovak Mathematical Journal

We study tails of prime counting functions. Our approach leads to representations with a main term and an error term for the asymptotic size of each tail. It is further shown that the main term is of a specific shape and can be written discretely as a sum involving probabilities of certain events belonging to a perturbed binomial distribution. The limitations of the error term in our representation give us equivalent conditions for various forms of the Riemann hypothesis, for classical type zero-free...