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PRIME NUMBER THEOREM EQUIVALENCES AND NON-EQUIVALENCES

  • Harold G. Diamond (a1) and Wen-Bin Zhang (a2)

Abstract

There are several formulas in classical prime number theory that are said to be “equivalent” to the Prime Number Theorem. For Beurling generalized numbers, not all such implications hold unconditionally. Here we investigate conditions under which the Beurling version of these relations do or do not hold.

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1. Bateman, P. T. and Diamond, H. G., Asymptotic distribution of Beurling’s generalized prime numbers. In Studies in Number Theory (Mathematical Association of America, Studies in Mathematics 6 ) (ed. LeVeque, W. J.), Prentice-Hall (Englewood Cliffs, NJ, 1969), 152210.
2. Bateman, P. T. and Diamond, H. G., Analytic Number Theory: An Introductory Course (Monographs in Number Theory 1 ), World Scientific (Singapore, 2004). Reprinted with minor changes 2009.
3. DeBruyne, G., Diamond, H. G. and Vindas, J., M (x) = o (x) estimates for Beurling numbers. J. Théor. Nombres Bordeaux (to appear).
4. Diamond, H. G. and Zhang, W. B., A PNT equivalence for Beurling numbers. Funct. Approx. Comment. Math. 46 2012, 225234.
5. Diamond, H. G. and Zhang, W. B., Beurling Generalized Numbers (Mathematical Surveys and Monographs 213 ), American Mathematical Society (Providence, RI, 2016).
6. Hardy, G. H., Divergent Series, Clarendon Press (Oxford, 1949).
7. Montgomery, H. L. and Vaughan, R. C., Multiplicative Number Theory: I. Classical Theory (Cambridge Studies in Advanced Mathematics 97 ), Cambridge University Press (Cambridge, 2007).
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