Prime-Counting Function in terms of Eulerian Logarithmic Integral

Theorem

Let $\map \pi x$ denote the prime-counting function of a number $x$.

Let $\map \Li x$ denote the Eulerian logarithmic integral of $x$:

$\map \Li x := \ds \int_2^x \dfrac {\d t} {\ln t}$

Then:

$\map \pi x = \map \Li x + \map \OO {x \map \exp {-c \sqrt {\ln x} } }$

where:

$\OO$ is the big-O notation

$c$ is some constant.

Riemann Hypothesis Holds

If the Riemann Hypothesis holds, then:

$\map \pi x = \map \Li x + \map \OO {\sqrt x \ln x}$

Proof

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Sources

• 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $0,5$