Arithmetic function: Difference between revisions

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Revision as of 15:38, 17 January 2025

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''ϑ''(''x'') and ''ψ''(''x''), the [[Chebyshev function]]s, are defined as sums of the natural logarithms of the primes not exceeding ''x''.

<math display="block">\vartheta(x)=\sum_{p\le x} \logln p,</math>

<math display="block"> \psi(x) = \sum_{p^k\le x} \logln p.</math>

The second Chebyshev function ''ψ''(''x'') is the summation function of the von Mangoldt function just below.

=== Λ(''n'') – von Mangoldt function ===

'''[[von Mangoldt function|Λ(''n'')]]''', the von Mangoldt function, is 0 unless the argument ''n'' is a prime power {{math|''p''^''k''}}, in which case it is the natural loglogarithm of the prime ''p'':

<math display="block">\Lambda(n) = \begin{cases}

\logln p &\text{if } n = 2,3,4,5,7,8,9,11,13,16,\ldots=p^k \text{ is a prime power}\\

0&\text{if } n=1,6,10,12,14,15,18,20,21,\dots \;\;\;\;\text{ is not a prime power}.

\end{cases}</math>