The Number Of The Natural Numbers Coprime To (f (n))

In 1953, G.L.Waston[1] proved the following conclusion by the sieve method:For every real numberα, the positive integers n such that (n, [αn])=1 have a densityδ(α). For every irrationalα,δ(α)= 6/π2. For rationalα=a/q, with (a, q)=1, and q>0,δ(α) depends only on q and has the value which tends to the limit 6/π2 as q→∞.In 1999, during the meeting held in Budapest of which theme was "Paul Erdos and his mathematics", one book named《Some of Paul's favorite problems》was widespread. In the paper[2], Delmer and Deshouillers proved the following result:Suppose that c>0 is a non-integer real number, then the density of the natural numbers such that (n, [nc])= 1 is 6/π2. This conclusion gived a undoubted answer to the problem 134 proposed by Moser, Lambek and Paul Erdos in the above book.Suppose that x is a sufficiently large real number and Nc(x) is the number of the natural numbers up to x such that (n, [nc])=1, then the above theorem can be repeated as: When 0< c< 1, the error term in the above formula can be estimated better. We make f(n)= nc in Theorem 1 of Lambek and Moser's paper[3], then we can get But when c> 1, the paper [2] used the sieve method, so we hardly improve the error term o(x) in the above formula by the method of the paper[3].Then Wenguang Zhai proved the following result by the method of exponential sum: where c>1, (k,λ) is the exponent pair such thatλ+(2c-2)k< 1. When 1< c< 2,η= 1; when c> 2,η= 0.In the first part of this paper, we study the number of the natural numbers coprime to [nc1], [nc2], where 1< c1< c2 are non-integer real numbers. Our main result for this part isTheorem 1 Suppose that x is a sufficiently large real number,1< c1< c2 are non-integer real numbers, and Nc1c2(x) is the number of the natural numbers n up to x such that (n, [nc1])= 1 and (n, [nc2])= 1. Then we have whereδ=δ(c1, c2)> 0 is a constant depending only on c1 and c2, andLet n> 1 be an integer in the second part. An integers a is called regular modulo n if there exists an integer x such that a2x= a(mod n). Letρ(n) denote the number of regular integers a modulo n with 1≤a≤n. Laszlo Toth [11] proved that where B=π2/6≈1.6449.Letγ≥1 be a fixed integer. In the second part of this paper we study the mean value of the function (ρ(n)/φ(n))γ. Our result isTheorem 2 Supposeγ≥1 is a fixed integer, then where Cr is a constant.