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. xvii

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