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Chebyshev Functions and Inclusion-Exclusion
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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.
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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