Vol. 21, No. 3, 1967

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A refinement of Selberg’s asymptotic equation

Veikko Nevanlinna

Vol. 21 (1967), No. 3, 537–540

The elementary proofs of the prime number theorem are essentially based on asymptotic equations of the form

x  x-
f(x)logx+  1 f(t )dψ(t) = O(x),

where f(x) is some function concerning the primes, ψ(x) is Tchebychev’6 function and the limits in the integral—as throughout in this paper—are taken from 1—to x+. This paper gives an elementary method for refining the right hand side of (A).

This method is based on the lemma of Tatuzawa and Iseki [2], and, assuming the prime number theorem, on an estimation of remainder integral which is more accurate than earlier ones.

Mathematical Subject Classification
Primary: 10.42
Received: 26 May 1966
Published: 1 June 1967
Veikko Nevanlinna