Introduction
Analytic number theory is mainly devoted to finding approximate counts
of number theoretical objects in situations where exact counts are out of
reach. Primes, divisors, solutions of Diophantine equations, lattice points
within contours, partitions of integers and ideal classes of algebraic number
fields are some of the objects that have been counted. The prototypical
approximate count in number theory is the Prime Number Theorem (PNT),
stating that
lim
x→+∞
π(x)
x
2
du
log(u)
= 1
where π(x) is the number of primes p x. This was proved independently
in 1896 by Jacques Hadamard and Charles de la Vall´ ee Poussin, building on
ideas of Bernhard Riemann, and applying complex analysis to the Riemann
zeta function
ζ(s) =

n=1
n−s
to establish the result. An asymptotic count like the PNT usually attracts
attention with a view to improve it. As the distribution of prime numbers is
one of the central topics in number theory, much effort has been expended
to obtain improvements to the Prime Number Theorem. We shall prove one
of the weaker ones, to the effect that there exist positive constants c, C, x0
such that
π(x)
x
2
du
log(u)

Cxe−c
log1/10(x)
for x x0.
This is more precise, though also more complicated to state, than the as-
ymptotic form of the PNT.
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