In this note we examine Littlewood's proof of the prime number theorem. We show that this can be extended to provide an equivalence between the prime number theorem and the non-vanishing of Riemann's zeta-function on the one-line. Our approach goes through the theory of almost periodic functions and is self-contained.
Refinements to the prime number theorem for arithmetic progressions
Some explicit estimates for the error term in the prime number theorem
Higher Correlations of Divisor Sums Related to Primes II: Variations of the error term in the prime number theorem
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