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Chebyshev function : ウィキペディア英語版
Chebyshev function

In mathematics, the Chebyshev function is either of two related functions. The first Chebyshev function ''ϑ''(''x'') or ''θ''(''x'') is given by
:\vartheta(x)=\sum_ \log p
with the sum extending over all prime numbers ''p'' that are less than or equal to ''x''.
The second Chebyshev function ''ψ''(''x'') is defined similarly, with the sum extending over all prime powers not exceeding ''x'':
: \psi(x) = \sum_\log p=\sum_ \Lambda(n) = \sum_\lfloor\log_p x\rfloor\log p,
where \Lambda is the von Mangoldt function. The Chebyshev functions, especially the second one ''ψ''(''x''), are often used in proofs related to prime numbers, because it is typically simpler to work with them than with the prime-counting function, ''π''(''x'') (See the exact formula, below.) Both Chebyshev functions are asymptotic to ''x'', a statement equivalent to the prime number theorem.
Both functions are named in honour of Pafnuty Chebyshev.
==Relationships==
The second Chebyshev function can be seen to be related to the first by writing it as
:\psi(x)=\sum_ k \log p
where ''k'' is the unique integer such that ''p''''k'' ≤ ''x'' and ''x'' < ''p''''k''+1. The values ''k'' of are given in . A more direct relationship is given by
:\psi(x)=\sum_^\infty \vartheta \left(x^\right).
Note that this last sum has only a finite number of non-vanishing terms, as
:\vartheta \left(x^\right) = 0\textn>\log_2 x\ = \frac,.
The second Chebyshev function is the logarithm of the least common multiple of the integers from 1 to ''n''.
:\operatorname(1,2,\dots, n)=e^.
Values of  \operatorname(1,2,\dots, n)  for the integer variable ''n'' is given at .

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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