On a sum involving squarefull numbers, II

A positive integer is square-full if each prime factor occurs to the second power or higher. Each square-full number can be written uniquely as a square times the cube of a square-free number. The perfect squares make up more than three-quarters of the sequence {si} of square-full numbers, so that a pair of consecutive square-full numbers is a pair of consecutive squares at least half the time, with

In 2022, Bergelson and Richter gave a new dynamical generalization of the prime number theorem by establishing an ergodic theorem along the number of prime factors of integers. They also showed that this generalization holds as well if the integers are restricted to be squarefree. In this paper, we present the concept of invariant averages under multiplications for arithmetic functions. Utilizing the properties of these invariant averages, we derive several ergodic theorems over squarefree numbers and squarefull numbers. These theorems have significant connections with the Erdős–Kac theorem, the Bergelson–Richter theorem, and the Loyd theorem.

We give upper bounds for character sums over squarefree and squarefull numbers sharper than the prior known in the literature. As an application, we study the distribution of squarefull numbers in arithmetic progressions and sharpen (without restriction on the modulus) the recent results obtained by Liu and Zhang.

There are many results on the distribution of square-full and cube-full numbers. In this article the distribution of these numbers are studied in more detail. Suchk-full numbers (k=2,3) are considered which are at the same time 1-free (1≥k+2). At first an asymptotic result is given for the numberNk,1(x) ofk-full and 1-free numbers not exceedingx. Then the distribution of these numbers in short intervals is investigated. We obtain different estimations of the differenceNk,1(x+h)−Nk,1(x) in the casesk=2, 1=4,5,6,7,1≥8 andk=3, 1=5,6,7, 1≥8.

A positive integer n is called a square-full number if p2 divides n whenever p is a prime divisor of n. In this paper we study the distribution of square-full numbers in arithmetic progressions by using the properties of Riemann zeta functions and Dirichlet L-functions.

Square-free and square-full numbers in Piatetski–Shapiro subsequences

In this paper, we study squarefull numbers in arithmetic progressions. We find the least such squarefull number by Dirichlet's hyperbola method as well as Burgess bound on character sums. We also obtain a best possible almost all result via a large sieve inequality of Heath-Brown on real characters.

A problem of Erdös on sums of two squarefull numbers

A natural number n is said to be squarefull if p|n implies p2|n for primes p. The set of all squarefull numbers is not much more dense in the natural numbers than the set of perfect squares but their additive properties may be rather different. We are more precise only in the case of sums of two such integers as this is the problem with which we are concerned here. Let U(x) be the number of integers not exceeding x and representable as the sum of two integer squares. Then, according to a theorem of Landau [4],as x tends to infinity.

We answer a number of questions of Erdős on the existence of arithmetic progressions in [Formula: see text]-full numbers (i.e. integers with the property that every prime divisor necessarily occurs to at least the [Formula: see text]th power). Further, we deduce a variety of arithmetic constraints upon such progressions, under the assumption of the [Formula: see text]-conjecture of Masser and Oesterlé.

In In this paper, the sixth mean value of the generalized quadratic Gauss sums considered by Y.F. He and W.P. Zhang is further studied when module is an arbitrary square-full number. The precise calculation formula is obtained. This improves the existed research result by averting the restriction that the module being an odd square-full number.

An innovation by D. R. Heath-Brown on square-full numbers in short intervals is applied to establish an asymptotic formula for the number of cube-full numbers in the interval

Square-full numbers with an even number of prime factors

For any positive real numbers $x$ and $y$, we shall consider several asymptotic formulas for sums of the modified squarefull numbers, namely $$ S_{k}(x,y):=\sum _{n\leq y}\biggl (\sum _{q\leq x}\sum _{d|(n,q)}df\biggl (\frac {q}{d}\bigg )\bigg )^{k} \quad

It is shown that the number of square-full numbers in the interval $$[x,x + x^{\frac{1}{2} + \theta } ]$$ is asymptotically equal to $$\frac{1}{2} \cdot \frac{{\zeta \left( {\frac{3}{2}} \right)}}{{\zeta (3)}}x^\theta$$ for everyθ in the range 1/6>θ⩾0.14254, which extends P.Shiu's range 1/6>θ⩾0.1526.