Related papers: Cotlar's ergodic theorem along the prime numbers
We present a new, elementary, dynamical proof of the prime number theorem.
Our goal in the present paper is to give a new ergodic proof of a well-known Veech's result, build upon our previous works.
We establish results with an arithmetic flavor that generalize the polynomial multidimensional Szemeredi theorem and related multiple recurrence and convergence results in ergodic theory. For instance, we show that in all these statements…
In this paper, we prove certain theorems about three consecutive primes.
In this expository article, we describe the recent approach, motivated by ergodic theory, towards detecting arithmetic patterns in the primes, and in particular establishing that the primes contain arbitrarily long arithmetic progressions.…
We consider weighted ergodic averages indexed by primes, where the weight depends on the prime, and is a "trace function" coming from algebraic geometry. We obtain extensions the classical mean-ergodic and pointwise ergodic theorems, as…
We prove the constructive version of Birkhoff's ergodic theorem following Vyugin but trying to separate and state explicitly the combinatorial statement on which this proof is based. We pose some questions related to this statement (and the…
We give a short combinatorial proof of the classical pointwise ergodic theorem for probability measure preserving $\mathbb{Z}$-actions. Our approach reduces the theorem to a tiling problem: tightly tile each orbit by intervals with desired…
In this paper, we establish a theorem on the distribution of primes in quadratic progressions on average.
Dirichlet's theorem on arithmetic progressions called as Dirichlet prime number theorem is a classical result in number theory. Atle Selberg\cite{Selberg} gave an elementary proof of this theorem. In this article we give an alternative…
In this paper, we prove a theorem on the distribution of primes in cubic progressions on average.
We survey the classical results on the prime number theorem
This work is meant to demonstrate new class of prime numbers -- cyclic prime numbers, that can be derived from any prime number at certain numeric systems. Cyclic prime numbers are also related to the cyclic numbers and full reptend prime…
The pentagonal number theorem is extended to the sequence of the number of integer partitions with all parts equal. The new pentagonal number theorem implies that the distribution of the primes is just a specific detail of the application…
In the first part of the paper the natural scheme for proving noncommutative individual ergodic theorems for multiple sequences is described and applied to obtain results on unrestricted convergence of multiaverages. In the second part…
The purpose of this study is to show how to get a necessary criterion for prime numbers with the help of special matrices. My special interest lies in the empirical research of these matrices and their patterns, structures and symmetries.…
In this paper, we establish some theorems on the distribution of primes in higher-order progressions on average.
Most of the assertions in the theory of well ordered sets are quite simple. However, one of its central statements, Zermelo's theorem, stands out of this rule, for its well-known proofs are rather complicated. The aim of the current paper…
The following paper follows on from work by Kamae, and gives a rigorous proof of the Ergodic Theorem, using nonstandard analysis.
The theorem presented in this paper allows the creation of large prime numbers (of order up to o(n^2)) given a table of all primes up to n.