Chebyshev Functions and Inclusion-Exclusion

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Authors

  • Department of Pure Mathematics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, England ,GB

Abstract

The Chebyshev functions θ and ψ are defined by
θ(x)=∑log p, ψ(x)=∑log p
where the sums are taken over primes and prime powers respectively. Clearly θ(x)≤ψ(x) and it is not difficult, using Chebyshev's theorem in a weak form, to prove that
0≤ψ(x)-θ(x)≤θ(x1/2)log x/log 2
  ≤π(x1/2)(log x)2/2 log 2
  ≤4x1/2log x/log 2.

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Published

1991-12-01

How to Cite

Cook, R. J. (1991). Chebyshev Functions and Inclusion-Exclusion. The Journal of the Indian Mathematical Society, 57(1-4), 109–116. Retrieved from https://informaticsjournals.co.in/index.php/jims/article/view/21920