The error term in the Prime Number Theorem

The Riemann hypothesis is equivalent to the following statement.

For every positive $\epsilon$, the number $\pi (x)$ of prime numbers $\leq x$ is

\begin{displaymath}
Li (x) + O(x^{1/2 + \epsilon}).
\end{displaymath}

Here $Li$ is the ``Logarithmic integral'' function, defined by

\begin{displaymath}
Li(x) := \int_0^x \frac{dt}{\log t},
\end{displaymath}

the integral being evaluated in principal value in the neighbourhood of $x=1$.

Roughly speaking, it means that the first half of the digits of the n-th prime are those of $Li^{-1}(n)$.




Back to the main index for The Riemann Hypothesis.