Prime-counting Function - Other Prime-counting Functions

Other Prime-counting Functions

Other prime-counting functions are also used because they are more convenient to work with. One is Riemann's prime-counting function, usually denoted as or . This has jumps of 1/n for prime powers pn, with it taking a value half-way between the two sides at discontinuities. That added detail is because then it may be defined by an inverse Mellin transform. Formally, we may define by

where p is a prime.

We may also write

where Λ(n) is the von Mangoldt function and

Möbius inversion formula then gives

Knowing the relationship between log of the Riemann zeta function and the von Mangoldt function, and using the Perron formula we have

The Chebyshev function weights primes or prime powers pn by ln(p):

Riemann's prime-counting function has the ordinary generating function:

$\sum_{n=1}^\infty \Pi_0(n)x^n = \sum_{a=2}^\infty \frac{x^{a}}{1-x} - \frac{1}{2}\sum_{a=2}^\infty \sum_{b=2}^\infty \frac{x^{ab}}{1-x} + \frac{1}{3}\sum_{a=2}^\infty \sum_{b=2}^\infty \sum_{c=2}^\infty \frac{x^{abc}}{1 -x} - \frac{1}{4}\sum_{a=2}^\infty \sum_{b=2}^\infty \sum_{c=2}^\infty \sum_{d=2}^\infty \frac{x^{abcd}}{1-x} + \cdots$

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“Empirical science is apt to cloud the sight, and, by the very knowledge of functions and processes, to bereave the student of the manly contemplation of the whole.”
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