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
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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
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
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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
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
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