Multiple ergodic theorems for arithmetic sets

On the normal number of prime factors of φ(n) subject to certain congruence conditions

An amicable pair [Formula: see text] is a pair of distinct positive integers [Formula: see text] such that [Formula: see text], where [Formula: see text] is the sum of divisors function. In this note, we prove that if [Formula: see text] are amicable and [Formula: see text] is odd, then [Formula: see text], where [Formula: see text] is the number of distinct prime factors.

Let be a positive integer and be the sum of the digits in basis q of the positive integer n. We prove that the quotient has a normal order one, where and are respectively, the number of distinct prime factors and the number of prime factors p of a positive integer n counted with multiplicity such that mod Moreover, we discuss sums of the form where f is a multiplicative function.

We investigate some equations involving the number of divisors d(n); the sum of divisors σ(n); Euler's totient function ϕ(n); the number of distinct prime factors ω(n); and the number of all prime factors (counted with multiplicity) Ω(n). The first part deals with equation f(xy) + f(xz) = f(yz). In the second part, as an analogy to x2 + y2 = z2, we study equation f(x2) + f(y2) = f(z2) and its generalization to higher degrees and more terms. We use just elementary methods and basic facts about the above functions and indicate why and how to discuss this topic in group study sessions or special maths classes of secondary schools in the framework of inquiry based learning.
 Subject Classification: 97F60, 11A25

Let \({q\geqq2}\) be an integer and denote S(n) the sum of the digits in base q of the positive integer n. Our main result is to estimate the sum \({\Sigma_{n\leqq x}\tilde{\omega}(n)}\) where \({\tilde{\omega}(n)}\) is the number of distinct prime factors p of n such that \({S(p) \equiv a \,{\rm mod} \,b \,(a \in \mathbb{Z}, b\geqq 2)}\).

The number of distinct prime factors of integers with missing digits is considered, and both the normal order and large values of the ω function over sets of this type are studied. A conjecture of Mauduit and Sarkozy, on large values of the Ω function over integers whose sum of digits is fixed, is also proved.

The set of the positive integers and positive square—free integers are denoted by IN and IN*, respectively, and we write IN(n) = IN ∩ [1, n], IN* (n) = IN* ∩ [1, n], where [1, n] = {1, 2, ... , n}. The set of primes is denoted by P. The smallest and greatest prime factors of the positive integer n are denoted by p(n) and P(n), respectively. ω(n) denotes the number of distinct prime factors of n, while Ω(n) denotes the number of prime factors of n counted with multiplicity: $$\omega \left( n \right)\, = \,\sum\limits_{p|n} {1,} \,\Omega \left( n \right)\, = \,\sum\limits_{{p^\alpha }\parallel n} {\alpha .} $$ µ(n) denotes the Mobius function.

Let [Formula: see text] be a positive integer, and define [Formula: see text] for [Formula: see text], where [Formula: see text] denotes the number of distinct prime factors of [Formula: see text], and [Formula: see text] represents the total number of prime factors of [Formula: see text] (counted with multiplicity). In this paper, we study these two zeta functions and related arithmetical functions. We show that [Formula: see text] which is similar to the known identity [Formula: see text] equivalent to the Prime Number Theorem. For [Formula: see text], we prove that [Formula: see text] We also raise a hypothesis on the parities of [Formula: see text] which implies the Riemann Hypothesis.

A weighted version of the Erdős–Kac theorem

In this paper we use the Riemann zeta distribution to give a new proof of the Erdös-Kac Central Limit Theorem. That is, if ζ ( s ) = ∑ n ≥ 1 1 n s \zeta (s)=\sum _{n\ge 1} \frac {1}{n^s} , s > 1 , s>1, then we consider the random variable X s X_s with P ( X s = n ) = 1 ζ ( s ) n s , P(X_s=n)=\frac {1}{\zeta (s)n^s}, n ≥ 1. n\ge 1. In an earlier paper, the first author and Adrien Peltzer derived the analog of the Erdös-Kac Central Limit Theorem (CLT) for the number of distinct prime factors, ω ( X s ) , \omega (X_s), of X s , X_s, as s ↘ 1. s\searrow 1. In this paper we show, by means of a Tauberian Theorem, how to obtain the Central Limit Theorem of Erdös-Kac for the uniform distribution from the result for the random variable X s X_s . We also apply the technique to the number of distinct prime divisors of X s X_s that lie in an arithmetic sequence and a local CLT of the type proved by Dixit and Murty [Hardy-Ramanujan J. 43 (2020), 17–23] as well a version of the CLT for irreducible divisors of a monic polynomial over a finite field.

Let ℤ, ℕ, ℚ denote the sets of integers, positive integers and rational numbers, respectively. Solutions (x, y, m, n) of the equation (1) have been investigated in many papers:Let ω(m), ρ(m) denote the number of distinct prime factors and the greatest square free factor of m, respectively. In this note we prove the following results.

We give some topological ergodic theorems inspired by theWiener-Wintnerergodic theorem. These theorems are used to give results for uniquely ergodic transformations and to study unique equilibrium states for shift maps. These latter results give randomL2ergodic theorems for a finite set of commuting measure-preserving transformations.

This book contains a collection of survey papers by leading researchers in ergodic theory, low-dimensional and topological dynamics and it comprises nine chapters on a range of important topics. These include: the role and usefulness of ultrafilters in ergodic theory, topological dynamics and Ramsey theory; topological aspects of kneading theory together with an analogous 2-dimensional theory called pruning; the dynamics of Markov odometers, Bratteli-Vershik diagrams and orbit equivalence of non-singular automorphisms; geometric proofs of Mather's connecting and accelerating theorems; recent results in one dimensional smooth dynamics; periodic points of nonexpansive maps; arithmetic dynamics; the defect of factor maps; entropy theory for actions of countable amenable groups.

We initiate the study of effective pointwise ergodic theorems in resource-bounded settings. Classically, the convergence of the ergodic averages for integrable functions can be arbitrarily slow [Ulrich Krengel, 1978]. In contrast, we show that for a class of PSPACE L¹ functions, and a class of PSPACE computable measure-preserving ergodic transformations, the ergodic average exists and is equal to the space average on every EXP random. We establish a partial converse that PSPACE non-randomness can be characterized as non-convergence of ergodic averages. Further, we prove that there is a class of resource-bounded randoms, viz. SUBEXP-space randoms, on which the corresponding ergodic theorem has an exact converse - a point x is SUBEXP-space random if and only if the corresponding effective ergodic theorem holds for x.

We study the arithmetic analogue of maximal functions on diagonal hypersurfaces. This paper is a natural step following the papers of [13], [14] and [16]. We combine more precise knowledge of oscillatory integrals and exponential sums to generalize the asymptotic formula in Waring’s problem to an approximation formula for the Fourier transform of the solution set of lattice points on hypersurfaces arising in Waring’s problem and apply this result to arithmetic maximal functions and ergodic averages. In sufficiently large dimensions, the approximation formula, l2-maximal theorems and ergodic theorems were previously known. Our contribution is in reducing the dimensional constraint in the approximation formula using recent bounds of Wooley, and improving the range of lp spaces in the maximal and ergodic theorems. We also conjecture the expected range of spaces.