Abstract
AbstractWe prove that the Riemann hypothesis is equivalent to the condition
$\int _{2}^x\left (\pi (t)-\operatorname {\textrm {li}}(t)\right )\textrm {d}t<0$
for all
$x>2$
. Here,
$\pi (t)$
is the prime-counting function and
$\operatorname {\textrm {li}}(t)$
is the logarithmic integral. This makes explicit a claim of Pintz. Moreover, we prove an analogous result for the Chebyshev function
$\theta (t)$
and discuss the extent to which one can make related claims unconditionally.
Publisher
Canadian Mathematical Society