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<math display="block">\Pi(x) = \sum_{p^k\le x}\frac{1}{k}.</math>
'''[[Chebyshev function|''θ''(''x'')]]''' and '''''ψ''(''x'')''', the [[Chebyshev function|Chebyshev functions]], are defined as sums of the natural logarithms of the primes not exceeding ''x''.
<math display="block">\vartheta(x)=\sum_{p\le x} \log p,</math>
<math display="block"> \psi(x) = \sum_{p^k\le x} \log p.</math>
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