Karl-Heinz Indlekofer

A New Method in Probabilistic Number Theory

Probabilistic number theory can be described as the result of the fusion of probability theory and asymptotic estimates, where the integral of a random variable is replaced by the arithmetical mean-value. In this context, divisibility by a prime p is an event A p, and the A p are statistically independent of one another, where the underlying ”measure” is given by the arithmetical mean-value (or asymptotic density).

A Mean-Value Theorem for Multiplicative Functions on the Set of Shifted Primes

Let f be a complex-valued multiplicative function, let p denote a prime and let π(x) be the number of primes not exceeding x. Further put

A Comparative Result for Multiplicative Functions

{m_p}\left( f \right): = \mathop {\lim }\limits_{x \to \infty } \frac{1}{{\pi \left( x \right)}}\sum\limits_{p \le x} {f\left( {p + 1} \right)} ,\quad M\left( f \right): = \mathop {\lim }\limits_{x \to \infty } \frac{1}{{\pi \left( x \right)}}\sum\limits_{n \le x} {f\left( n \right)}

Generalized Renewal Processes

and suppose that

Nondegenerate Groups of Regular Points

\mathop {\lim sub}\limits_{x \to \infty } \frac{1}{x}\sum\limits_{n \le x} {{{\left| {f\left( n \right)} \right|}^2}} < \infty ,\quad \sum\limits_{p \le x} {{{\left| {f\left( p \right)} \right|}^2}} \ll x{\left( {\ln x} \right)^{ - \varrho }}

Asymptotically Quasi-inverse Functions

with some ϱ > 0. For such functions we prove: If there is a Dirichlet character χ d such that the mean-value M(fχ d ) exists and is different from zero, then the mean-value m p (f) exists. If the mean-value M(|f|) exists, then the same is true for the mean-value m p ( |f| ).