On the difference between consecutive squarefree integers
This paper gives further results about the distribution in the arithmetic progressions (modulo a product of two primes) of reducible quadratic polynomials in short intervals , where now . Here we use the Dispersion Method instead of the Large Sieve to get results beyond the classical level , reaching (thus improving also the level of the previous paper, i.e. ), but our new results are different in structure. Then, we make a graphical comparison of the two methods.
Let , be complex-valued multiplicative functions. In the paper the necessary and sufficient conditions are indicated for the convergence in some sense of probability measureas .
A positive integer is called a square-free number if it is not divisible by a perfect square except . Let be an odd prime. For with , the smallest positive integer such that is called the exponent of modulo . If the exponent of modulo is , then is called a primitive root mod . Let be the characteristic function of the square-free primitive roots modulo . In this paper we study the distribution and give an asymptotic formula by using properties of character sums.
Hawkins introduced a probabilistic version of Erathosthenes’ sieve and studied the associated sequence of random “primes” . Using various probabilistic techniques, many authors have obtained sharp results concerning these random “primes”, which are often in agreement with certain classical theorems or conjectures for prime numbers. In this paper, we prove that the number of integers such that is almost surely equivalent to , for a given fixed integer . This is a particular case of a recent...
A natural number is said to be a -integer if , where and is not divisible by the th power of any prime. We study the distribution of such -integers in the Piatetski-Shapiro sequence with . As a corollary, we also obtain similar results for semi--free integers.