Reference request: a formula for the prime-counting function
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Let $pi(n)$ denote the prime counting function, which returns the number of primes less than or equal to $n$. When one asks WolframAlpha for $pi(10,000)$ (or any suitably small number), it displays the result, along with several formulas it uses to calculate this result. The final listed formula caught my eye - WolframAlpha claims that $$pi(n) = -sum_{k=1}^{log_2(n)}mu(k)sum_{l=2}^{lfloor sqrt[k]{n} rfloor} leftlfloor frac{sqrt[k]{n}}{l}rightrfloor mu(l)Omega(l)$$ where $mu(k)$ is the Mobius function and $Omega(l)$ is the function that gives the number of prime factors counting multiplicities in $l$.
Question: Does this formula have a name? Where is it's correctness proven? Alternatively, I'd be interested in a proof of it's correctness.
number-theory reference-request prime-numbers
asked yesterday
KReiser
8,92711233
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up vote
1
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Let $pi(n)$ denote the prime counting function, which returns the number of primes less than or equal to $n$. When one asks WolframAlpha for $pi(10,000)$ (or any suitably small number), it displays the result, along with several formulas it uses to calculate this result. The final listed formula caught my eye - WolframAlpha claims that $$pi(n) = -sum_{k=1}^{log_2(n)}mu(k)sum_{l=2}^{lfloor sqrt[k]{n} rfloor} leftlfloor frac{sqrt[k]{n}}{l}rightrfloor mu(l)Omega(l)$$ where $mu(k)$ is the Mobius function and $Omega(l)$ is the function that gives the number of prime factors counting multiplicities in $l$.
Question: Does this formula have a name? Where is it's correctness proven? Alternatively, I'd be interested in a proof of it's correctness.
number-theory reference-request prime-numbers
asked yesterday
KReiser
8,92711233
Looks like it tries to build the legendre formula in a highly inneficient way. Interesting formula, I'll dig into it as soon as I find some time. The values used to sum the thing reminds me of what I used for this mathoverflow.net/a/300060/118898
– Collag3n
yesterday
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $pi(n)$ denote the prime counting function, which returns the number of primes less than or equal to $n$. When one asks WolframAlpha for $pi(10,000)$ (or any suitably small number), it displays the result, along with several formulas it uses to calculate this result. The final listed formula caught my eye - WolframAlpha claims that $$pi(n) = -sum_{k=1}^{log_2(n)}mu(k)sum_{l=2}^{lfloor sqrt[k]{n} rfloor} leftlfloor frac{sqrt[k]{n}}{l}rightrfloor mu(l)Omega(l)$$ where $mu(k)$ is the Mobius function and $Omega(l)$ is the function that gives the number of prime factors counting multiplicities in $l$.
Question: Does this formula have a name? Where is it's correctness proven? Alternatively, I'd be interested in a proof of it's correctness.
number-theory reference-request prime-numbers
asked yesterday
KReiser
8,92711233
Let $pi(n)$ denote the prime counting function, which returns the number of primes less than or equal to $n$. When one asks WolframAlpha for $pi(10,000)$ (or any suitably small number), it displays the result, along with several formulas it uses to calculate this result. The final listed formula caught my eye - WolframAlpha claims that $$pi(n) = -sum_{k=1}^{log_2(n)}mu(k)sum_{l=2}^{lfloor sqrt[k]{n} rfloor} leftlfloor frac{sqrt[k]{n}}{l}rightrfloor mu(l)Omega(l)$$ where $mu(k)$ is the Mobius function and $Omega(l)$ is the function that gives the number of prime factors counting multiplicities in $l$.
Question: Does this formula have a name? Where is it's correctness proven? Alternatively, I'd be interested in a proof of it's correctness.
number-theory reference-request prime-numbers
number-theory reference-request prime-numbers
asked yesterday
KReiser
8,92711233
asked yesterday
KReiser
8,92711233
asked yesterday
KReiser
8,92711233
asked yesterday
KReiser
8,92711233
asked yesterday
KReiser
8,92711233
8,92711233
Looks like it tries to build the legendre formula in a highly inneficient way. Interesting formula, I'll dig into it as soon as I find some time. The values used to sum the thing reminds me of what I used for this mathoverflow.net/a/300060/118898
– Collag3n
yesterday
add a comment |
Looks like it tries to build the legendre formula in a highly inneficient way. Interesting formula, I'll dig into it as soon as I find some time. The values used to sum the thing reminds me of what I used for this mathoverflow.net/a/300060/118898
– Collag3n
yesterday
Looks like it tries to build the legendre formula in a highly inneficient way. Interesting formula, I'll dig into it as soon as I find some time. The values used to sum the thing reminds me of what I used for this mathoverflow.net/a/300060/118898
– Collag3n
yesterday
Looks like it tries to build the legendre formula in a highly inneficient way. Interesting formula, I'll dig into it as soon as I find some time. The values used to sum the thing reminds me of what I used for this mathoverflow.net/a/300060/118898
– Collag3n
yesterday
add a comment |
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Looks like it tries to build the legendre formula in a highly inneficient way. Interesting formula, I'll dig into it as soon as I find some time. The values used to sum the thing reminds me of what I used for this mathoverflow.net/a/300060/118898
– Collag3n
yesterday