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New equation of prime-zeta? part 2

View part-1. note that by the möbius inversion we have \[Aa=ya*1\] By the dirichlet series : \[\sum_{n=1}^\infty \dfrac{Aa(n)}{n^s}=\left(\sum_{n=1}^\infty \dfrac{ya(n)}{n^s}\right)\left(\sum_{n=1}^\infty \dfrac{1}{n^s}\right)\] \[P(s)=\zeta(s)\sum_{n=1}^\infty \dfrac{ya(n)}{n^s}\] Prime zeta and zeta function being used.

If we can find \(\sum_{n=1}^\infty \dfrac{ya(n)}{n^s}\) then we can maybe have a new equation(or the old one) for the prime zeta. This concludes this note, in part three i will attempt the summation.

Feel free to leave your comments, and reshare if you enjoyed this.

Note by Aareyan Manzoor
10 months, 2 weeks ago

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Extra, might be useful (derived from conversation below):

\[\sum _{ n=1 }^{ \infty } \frac { \mu \left( n \right) }{ n^{ s } } \left( \sum _{ p|n }^{ } \frac { 1 }{ p^{ s } } \right)=\sum _{ n=1 }^{ \infty } \frac { \mu \left( n \right) }{ n^{ s } } \left( \sum _{ p|n }^{ } \frac { 1 }{ p^{ 2s } } \right)-\frac{P(2s)}{\zeta(s)}\] Julian Poon · 10 months, 2 weeks ago

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@Julian Poon I have verified the above numerically with the method shown below. Julian Poon · 10 months, 2 weeks ago

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@Julian Poon More extra, in case my numerical calculation is off: I used desmos to calculate \(\sum _{ p|n }^{ } \frac { 1 }{ p^{ s } }\)

\[\sum _{ p|n }^{ } \frac { 1 }{ p^{ s } }=\sum _{j=1}^n\frac{P_{rime}\left(j\right)}{j^s}\left(\frac{\left|\gcd \left(n,j\right)-1.1-\left|\gcd \left(n,j\right)-1.1\right|\right|}{2\gcd \left(n,j\right)-2.2}+1\right)\]

\[P_{rime}\left(x\right)=G\left(P\left(x\right)\right)\]

\[P\left(x\right)=\left(\sum _{k=1}^{\operatorname{floor}\left(x\right)}\gcd \left(x,k\right)\right)-\left(2x\right)\]

\[G\left(x\right)=\left(\frac{\left|x+0.9-\left|x+0.9\right|\right|}{-2x-1.8}+\frac{\left|-x-1.1-\left|-x-1.1\right|\right|}{2x+1.1\cdot 2}-1\right)\]

\[\mu{(n)}=\sum _{k=1}^{\operatorname{floor}\left(x\right)}\frac{\left|\gcd \left(k,\operatorname{floor}\left(x\right)\right)-1.1-\left|\gcd \left(k,\operatorname{floor}\left(x\right)\right)-1.1\right|\right|}{-2\gcd \left(k,\operatorname{floor}\left(x\right)\right)+2.2}\cos \left(\frac{2\pi k}{\operatorname{floor}\left(x\right)}\right)\]

Essentially, \(P_{rime}(x)\) outputs 1 if x is prime, and outputs 0 otherwise. Julian Poon · 10 months, 2 weeks ago

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@Julian Poon I can't help but notice that the function's name is the letters in your name. :P. gtg, got to do hw. Julian Poon · 10 months, 2 weeks ago

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I think this sum will be very hard to evaluate because it is neither additive nor multiplicative. Here's my progress so far:

The sum can be expressed as

\[\sum _{ n=1 }^{ \infty } \frac { ya(n) }{ n^{ s } } =\sum _{ n=1 }^{ \infty } \frac { \mu \left( n \right) }{ n^{ s } } \left( \sum _{ p|n }^{ } \frac { 1 }{ p^{ s } } \right) -\sum _{ n=1 }^{ \infty } \frac { \mu \left( n \right) \omega \left( n \right) }{ n^{ s } } \]

For the second sum:

\[\mu \omega \ast 1=\begin{cases} 1\text{ if }n=p^a \\ 0 \text{ otherwise}\end{cases}\]

\[\zeta (s)\sum _{ n=1 }^{ \infty } \frac { \mu \left( n \right) \omega \left( n \right) }{ n^{ s } } =\sum _{ a\ge 1 }{ \sum _{ p }{ \frac { 1 }{ { p }^{ as } } } } =\sum _{ p }{ \frac { 1 }{ { p }^{ s }-1 } } \] Julian Poon · 10 months, 2 weeks ago

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@Julian Poon I did a similar thing! I easily worked out summation#1 but got stuck at summation#2(i got something similar but it was school and i didnt have time and focus). Let me show you what i did in summation one:

we want it to be of the form \(p^2 n\) such that n is square free with no prime factor=p.then, \(Ya(p^{2}n)=\mu(n)\). so lets write the summation as \[\sum_{p=prime}\sum_{p\not\mid n} \dfrac{\mu(n)}{(p^2n)^s}=\sum_{p=prime}\dfrac{1}{p^{2s}}\left(\sum_{n=1}^\infty \dfrac{\mu(n)}{n^s}-\dfrac{1}{p}\sum_{n=1}^\infty \dfrac{\mu(n)}{n^s}\right)\\ =\sum_{p=prime}\left(\dfrac{1}{p^{2s}}-\dfrac{1}{p^{2s+1}}\right) \sum_{n=1}^\infty\dfrac{\mu(n)}{n^s}\\ =\dfrac{1}{\zeta(s)} (P(2s)-P(2s+1))\] We derive a beautiful result from this, i will perhaps include this in part 3. Aareyan Manzoor · 10 months, 2 weeks ago

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@Aareyan Manzoor For the first sum, \(s=2\), I am getting \(-0.060460284021\) numercially, summing from \(n=1\) to \(n=100\).

However, with your formula, I am getting \(0.0250697720228\).

Mind if you elaborate on your first step?

Edit: Shouldn't the first equation be

\[\sum_{n=1}\sum_{p\not\mid n} \dfrac{\mu(n)}{(p^2n)^s}\] Julian Poon · 10 months, 2 weeks ago

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@Julian Poon It should be \(p^2\not\mid n\) as i dont want power higher then 2. Aareyan Manzoor · 10 months, 2 weeks ago

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@Aareyan Manzoor I am getting \[\sum_{n=1}\sum_{p\not\mid n} \dfrac{\mu(n)}{(p^2n)^s} = P(2s)\frac{1}{\zeta(s)}-\sum_{n=1}\sum_{p\mid n} \dfrac{\mu(n)}{n^sp^{2s}}\] Julian Poon · 10 months, 2 weeks ago

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@Julian Poon Yes i am getting similar except it should be \[p\not \mid n\] Aareyan Manzoor · 10 months, 2 weeks ago

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@Aareyan Manzoor The frist equation I used \(\not\mid\) while the second one I used \(\mid\). Assuming my way of calculating numerically is correct, I am getting \(-0.0604826745959\), which is close to the original numerical calculation I made. Julian Poon · 10 months, 2 weeks ago

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@Julian Poon Now speaking logically, what do you think? We remember that it has one prime factor squared. So it is p^2times square free number. We the can say n=p^2*k. Mobius dissapears over non square frees so we can just sum over all nin muliples of p. Aareyan Manzoor · 10 months, 2 weeks ago

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@Aareyan Manzoor Well my point is that your equation:

\[\sum_{p=prime}\sum_{p\not\mid n} \dfrac{\mu(n)}{(p^2n)^s}\]

Should be

\[\sum_{n=1}\sum_{p\not\mid n} \dfrac{\mu(n)}{(p^2n)^s}\] Julian Poon · 10 months, 2 weeks ago

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@Julian Poon I think they are the same thing. I am summing over primes you are summing over natural numbers but ebery prime divdes n numbers Aareyan Manzoor · 10 months, 2 weeks ago

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@Aareyan Manzoor So are you saying it should be

\[\sum_{n=prime}\sum_{p\not\mid n} \dfrac{\mu(n)}{(p^2n)^s}\]

?

Or else, what domain are you summing \(n\) over? Julian Poon · 10 months, 2 weeks ago

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@Julian Poon let me make it better \[\sum_{p=prime} \sum_{\text{n is NOT a multiple of p}} \dfrac{\mu(n)}{(p^2n)^s}\] Aareyan Manzoor · 10 months, 2 weeks ago

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@Aareyan Manzoor Ah opps... sorry...

I am getting a different answer though'

\[\sum_{p=prime}\sum_{p\not\mid n} \dfrac{\mu(n)}{(p^2n)^s}=\sum_{p=prime}\dfrac{1}{p^{2s}}\left(\sum_{n=1}^\infty \dfrac{\mu(n)}{n^s}-\sum_{n=1}^\infty \dfrac{\mu(pn)}{(pn)^s}\right)=\sum_{p=prime}\dfrac{1}{p^{2s}}\left(\sum_{n=1}^\infty \dfrac{\mu(n)}{n^s}+\frac{1}{p^s}\sum_{n=1}^\infty \dfrac{\mu(n)}{(n)^s}\right)\]

\[=\frac{1}{\zeta(s)} \left(P(2s)+P(3s)\right)\]

Also, since

\[\sum _{ n=1 }^{ \infty } \frac { \mu \left( n \right) }{ n^{ s } } \left( \sum _{ p|n }^{ } \frac { 1 }{ p^{ s } } \right)=-\sum_{p=prime}\sum_{p\not\mid n} \dfrac{\mu(n)}{(p^2n)^s}\]

\[\sum _{ n=1 }^{ \infty } \frac { \mu \left( n \right) }{ n^{ s } } \left( \sum _{ p|n }^{ } \frac { 1 }{ p^{ s } } \right)=-\frac{1}{\zeta(s)} \left(P(3s)+P(2s)\right)\]

Though I am sure there is problems with this working too since it doesn't match numerically Julian Poon · 10 months, 2 weeks ago

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@Julian Poon in the step where you \[\mu(pn)=-\mu(n)\] you assumed p,n are coprime. try p=2, n=6 we get \[\mu(2*6)=0\neq -\mu(6)\] Aareyan Manzoor · 10 months, 2 weeks ago

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@Aareyan Manzoor Oh yeah

\[\sum_{p\text{ is prime}}\sum_{p\not\mid n} \dfrac{\mu(n)}{(p^2n)^s}=\sum_{p\text{ is prime}}\dfrac{1}{p^{2s}}\left(\prod_{p_i \neq p,p_i \text{ is prime}}\left(1-\frac{1}{p_i^s}\right)\right)=\sum_{p\text{ is prime}}\dfrac{1}{p^{2s}}\left(\frac{\prod_{p_i\text{ is prime}}\left(1-\frac{1}{p_i^s}\right)}{1-\frac{1}{p^s}}\right)\]

\[=\sum_{p\text{ is prime}}\dfrac{1}{p^{2s}}\left(\frac{1}{\zeta(s)}\frac{1}{1-\frac{1}{p^s}}\right)=\frac{1}{\zeta(s)}\sum_{p}\frac{1}{p^{2s}(1-p^{-s})}\]

\[=\frac{1}{\zeta(s)}\left(\sum_p\frac{1}{p^s-1}-P(s)\right)\]

It matches numerically. This equation is disappointing... It gives neither insight into the original problem nor on the evaluation of the sum \(\sum_p\frac{1}{p^s-1}\) Julian Poon · 10 months, 2 weeks ago

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@Julian Poon Oh well..... damn it. Thanks for your help julian poon, evaluating this series has been fun. Aareyan Manzoor · 10 months, 2 weeks ago

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