$operatorname{Li}(x)$ vs $x/log(x)$ vs ${pi}(x)$ - terminology in PNT for arithmetic progressions
Wikipedia describes the prime number theorem for arithmetic progressions in the following terms:
$pi_{n,a}(x)sim frac{1}{varphi(n)}operatorname{Li}(x)$
Other sources use $frac{x}{log(x)}$ in place of $operatorname{Li}(x)$.
Is it not more accurate to express the theorem (for arithmetic progressions) in terms of the prime counting function itself? Ie:
$pi_{n,a}(x)sim frac{1}{varphi(n)}{pi}(x)$
As such, either $operatorname{Li}(x)$ or $frac{x}{log(x)}$ is implicit? I understand it may be less useful in a practical sense, eg. when calculating a large value.
elementary-number-theory
edited Dec 9 at 11:14
Henrik
5,95692030
asked Dec 9 at 11:11
Molonglo
255
add a comment |
Wikipedia describes the prime number theorem for arithmetic progressions in the following terms:
$pi_{n,a}(x)sim frac{1}{varphi(n)}operatorname{Li}(x)$
Other sources use $frac{x}{log(x)}$ in place of $operatorname{Li}(x)$.
Is it not more accurate to express the theorem (for arithmetic progressions) in terms of the prime counting function itself? Ie:
$pi_{n,a}(x)sim frac{1}{varphi(n)}{pi}(x)$
As such, either $operatorname{Li}(x)$ or $frac{x}{log(x)}$ is implicit? I understand it may be less useful in a practical sense, eg. when calculating a large value.
elementary-number-theory
edited Dec 9 at 11:14
Henrik
5,95692030
asked Dec 9 at 11:11
Molonglo
255
add a comment |
Wikipedia describes the prime number theorem for arithmetic progressions in the following terms:
$pi_{n,a}(x)sim frac{1}{varphi(n)}operatorname{Li}(x)$
Other sources use $frac{x}{log(x)}$ in place of $operatorname{Li}(x)$.
Is it not more accurate to express the theorem (for arithmetic progressions) in terms of the prime counting function itself? Ie:
$pi_{n,a}(x)sim frac{1}{varphi(n)}{pi}(x)$
As such, either $operatorname{Li}(x)$ or $frac{x}{log(x)}$ is implicit? I understand it may be less useful in a practical sense, eg. when calculating a large value.
elementary-number-theory
edited Dec 9 at 11:14
Henrik
5,95692030
asked Dec 9 at 11:11
Molonglo
255
Wikipedia describes the prime number theorem for arithmetic progressions in the following terms:
$pi_{n,a}(x)sim frac{1}{varphi(n)}operatorname{Li}(x)$
Other sources use $frac{x}{log(x)}$ in place of $operatorname{Li}(x)$.
Is it not more accurate to express the theorem (for arithmetic progressions) in terms of the prime counting function itself? Ie:
$pi_{n,a}(x)sim frac{1}{varphi(n)}{pi}(x)$
As such, either $operatorname{Li}(x)$ or $frac{x}{log(x)}$ is implicit? I understand it may be less useful in a practical sense, eg. when calculating a large value.
elementary-number-theory
elementary-number-theory
edited Dec 9 at 11:14
Henrik
5,95692030
asked Dec 9 at 11:11
Molonglo
255
edited Dec 9 at 11:14
Henrik
5,95692030
asked Dec 9 at 11:11
Molonglo
255
edited Dec 9 at 11:14
Henrik
5,95692030
edited Dec 9 at 11:14
Henrik
5,95692030
edited Dec 9 at 11:14
Henrik
5,95692030
5,95692030
asked Dec 9 at 11:11
Molonglo
255
asked Dec 9 at 11:11
Molonglo
255
asked Dec 9 at 11:11
Molonglo
255
255
add a comment |
add a comment |
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