Vol. 21, No. 3, 1967

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 290: 1  2
Vol. 289: 1  2
Vol. 288: 1  2
Vol. 287: 1  2
Vol. 286: 1  2
Vol. 285: 1  2
Vol. 284: 1  2
Vol. 283: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Editorial Board
Officers
Special Issues
Submission Guidelines
Submission Form
Subscriptions
Contacts
Author Index
To Appear
 
ISSN: 0030-8730
A refinement of Selberg’s asymptotic equation

Veikko Nevanlinna

Vol. 21 (1967), No. 3, 537–540
Abstract

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),
(A)

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
Milestones
Received: 26 May 1966
Published: 1 June 1967
Authors
Veikko Nevanlinna