PROPERTIES OF THE FUNCTION f(x)= x/π(x)

We obtain the asymptotic estimations for ∑k=2nf(k) and ∑k=2n1/f(k), where f(k)=k/π(k), k≥2. We study the expression 2f(x


Introduction.
We denote by π(x) the number of all prime numbers ≤ x.We denote also f (x) = x/π(x) for x ≥ 2. Since π(x) ∼ x/ log x, it follows that f (x) ∼ log x.We could expect that the function f (x) behaves like log x.However, we will see that log x possesses several properties that f (x) does not possess.
Indeed, the function log x is increasing and concave, while f (x) does not have these properties.Denoting by p n the nth prime number, we remark that f (p n )−f (p n −1) = p n /n − (p n − 1)/(n − 1) = (n − p n )/n(n − 1) < 0, so the function f is not increasing.
As shown also in [3], the function f is not concave because for x 1 = p n − 1 and The following fact was proved in [1]: for a, b > 0 and x sufficiently large.A property of the function log is given by Stirling's formula asserting that n! ∼ n n e −n √ 2nπ , that is,
By means of a similar method we now prove the following theorem.
Theorem 2.2.For fixed m ≥ 1 the relation Proof.In [2], the following relation was used:

.13)
With the notation from the proof of Theorem 2.1, we have that is, In view of (2.5) and of the fact that It easily follows from (2.7) that S(n) = m i=1 q i / log i n + O(n/ log m+1 n) and (2.17) Comparing the above relation with (2.16), we get (2.18) Consequently q i = i! and the proof is finished.

An inequality for the function f (x).
We have shown in the introduction that the function f is not concave.In particular, it follows neither that f (x + y) ≥ f (x) nor that f (x + y) ≥ f (y).However, we can prove the following theorem.
Proof.In [3], it was proved that In view of these inequalities, it follows that for x, y ≥ 59 it suffices to prove that Consequence.If the inequality from (4.1) is false for some integers x ≥ y ≥ 2, then f (x) > f (y) and f (x) > f (x + y).
Remark that for x = y the statement of Theorem 3.1 reduces to π(2x) ≤ 2π(x).This is just Landau's theorem, that is a special case of the Hardy-Littlewood conjecture.