Number theory -- probabilistic, heuristic, and computational approaches
In this paper (with more than 100 bibliographical items) the author surveys some areas of number theory, to which the author has given important contributions, and he gives some numerical results about large twin primes. The author deals with -- the beginnings of probabilistic number theory (Theorem of Erdős and Kac, Kubilius model) -- ``Approximation of independence'', convergence to the normal distribution: \[ {1\over x} \cdot \# \left\{ n\leq x; f(n) - \sum_{p\leq x} {{f(p)}\over p} \leq z \cdot \left( \sum_{p\leq x} {{f^2(p)}\over p} \right)^{{1\over 2}} \right\} \Longrightarrow {1\over{\sqrt{2\pi}}} \cdot \int_{-\infty}^z e^{ - {1\over 2} w^2} dw, \] under suitable assumptions on the real-valued additive function \(f\). -- Uniform integrability: spaces of arithmetical functions, generalization of Delange's, Elliott's and Halász' mean-value theorems. -- Polyadic numbers, a first attempt of an integration theory for arithmetical functions (E. V. Novoselov, Schwarz and Spilker). -- Pseudomeasures on \(\mathbb N\) and the Stone-Čech compactification. In this section the author explains his theory of integration based on an application of the Stone-Čech compactification, and shows its wide usability. -- Primes play a game of chance: here a table of the counting function for primes and twin primes is given in short intervals up to \(10^{15}\). -- The last section (Computational Results, together with Járai) gives again numerical results on large twin primes and Sophie-Germain primes. The author, expert in the topics mentioned, gives many results, hints to the bibliography and valuable explanations.
Asymptotic results on arithmetical functions
measure and integration on the set of positive integers
scientific article; zbMATH DE number 2093573
scientific article