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ON LITTLEWOOD’S PROOF OF THE PRIME NUMBER THEOREM

  • ALEKSANDER SIMONIČ (a1)

Abstract

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 nonvanishing of Riemann’s zeta-function on the one-line. Our approach goes through the theory of almost periodic functions and is self-contained.

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[1] Besicovitch, A. S., Almost Periodic Functions (Dover, New York, 1955).
[2] Corduneanu, C., Almost Periodic Functions (Chelsea, New York, 1989).
[3] Hardy, G. H., Ramanujan. Twelve Lectures on Subjects Suggested by his Life and Work (Cambridge University Press, Cambridge, UK–Macmillan, New York, 1940).
[4] Hardy, G. H. and Littlewood, J. E., ‘Contributions to the theory of the Riemann zeta-function and the theory of the distribution of primes’, Acta Math. 41(1) (1916), 119196.
[5] Ingham, A. E., The Distribution of Prime Numbers, Cambridge Mathematical Library (Cambridge University Press, Cambridge, 1990).
[6] Kaczorowski, J. and Perelli, A., ‘On the prime number theorem for the Selberg class’, Arch. Math. (Basel) 80(3) (2003), 255263.
[7] Littlewood, J. E., ‘The quickest proof of the prime number theorem’, Acta Arith. 18 (1971), 8386.
[8] Titchmarsh, E. C., The Theory of Functions (Oxford University Press, Oxford, 1958).
[9] Titchmarsh, E. C., The Theory of the Riemann Zeta-Function, 2nd edn (The Clarendon Press–Oxford University Press, New York, 1986).
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ON LITTLEWOOD’S PROOF OF THE PRIME NUMBER THEOREM

  • ALEKSANDER SIMONIČ (a1)

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