Chebyshev Functions and Inclusion-Exclusion
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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.
Abstract
The Chebyshev functions θ and ψ are defined byθ(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
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Copyright (c) 1991 R. J. Cook
This work is licensed under a Creative Commons Attribution 4.0 International License.