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Cohen, DIA and Katz, TM (1984)

Prime Numbers and the First Digit Phenomenon

Journal of Number Theory 18(3), pp. 261-268.

ISSN/ISBN: 0022-314X DOI: 10.1016/0022-314X(84)90061-1

Abstract: The set of primes which have lead digit 1 does not have relative natural density in the prime numbers. However, Bombieri has shown that this set does have relative Zeta density equal to log102. This means that a prime chosen at random (w.r.t. the Zeta distribution) will have lead digit 1 with the determined probability. Here the question, Is this a special property of Zeta density or a more universal property of primes? is answered. It is shown that for any generalization of relative natural density (obeying a few basic assumptions) if a value is assigned to the relative density of primes of lead digit 1 then this value is always log102. Another density which does converge on this set is also exhibited. Additionally the relative densities of primes beginning with any specified string of digits are found.

@article{, title = "Prime numbers and the first digit phenomenon", journal = "Journal of Number Theory", volume = "18", number = "3", pages = "261--268", year = "1984", issn = "0022-314X", doi = "10.1016/0022-314X(84)90061-1", url = "", author = "Daniel I.A Cohen and Talbot M Katz" }

Reference Type: Journal Article

Subject Area(s): Number Theory