Affiliation:
1. Institute of Geodynamics Sabba S. Ştefănescu of Romanian Academy 19-21 Jean-Louis Calderon St., Bucharest-37, Romania
Abstract
Abstract
Let π(x) be the number of primes not exceeding x. We prove that
π
(
x
)
<
x
log
x
-
1.006789
for
x
≥
e
10
12
, and that for sufficiently large
x
:
x
log
x
-
1
+
(
log
x
)
-
1.5
+
2
(
log
x
)
-
0.5
<
π
(
x
)
<
1
log
x
-
1
-
2
(
log
x
)
-
0.5
-
(
log
x
)
-
1.5
.
We finally prove that for
x
≥
e
10
12
and k = 2, 3,…, 147297098200000, the closed interval [(k – 1)x, kx] contains at least one prime number, i.e. the Bertrand's postulate holds for x and k as above.