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Motivation for the Basel problem
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I realized that I know of several ways how to prove that $sum_n=1^inftyfrac1n^2=fracpi^26$, but I have no idea why I would want to know the
answer in the first place.
Answers I have found by myself:
- the probability of two integers chosen at random to be prime to each other is $frac6pi^2$. The proof is
understandable by a smart undergraduate. This is the kind of motivation I am looking for. - by the inverse square law of sound, a line of cars blowing their horns at a car stopped in a one lane road will sound $fracpi^26$ louder than a single car. Or similarly, intensity of traffic lights on a long road at night,...
- destructive testing of $n$ wooden beams will break on average $H_n=1+frac12+frac13+ldots+frac1n$ beams with a variance of $H_n-sum_k=1^nfrac1k^2$.
- $zeta(2)=fracpi^26$ is all over quantum mechanics. The simplest, most understandable example I have found is Johnson-Nyquist noise. Still, there is a lot
of physics background required to figure it out, so this does not satisfy me much.
Are there any other good reason why to compute $zeta(2)$ ? Why were Mengoli, Euler and other mathematicians from the Enlightenment interested in the answer ? Any application in physics, chemistry, economics,... ? As far as I am concerned, the closer to reality, the better.
Thanks in advance for any help.
sequences-and-series soft-question riemann-zeta applications
asked Aug 1 at 11:31
Dan Odare
162
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up vote
3
down vote
favorite
I realized that I know of several ways how to prove that $sum_n=1^inftyfrac1n^2=fracpi^26$, but I have no idea why I would want to know the
answer in the first place.
Answers I have found by myself:
- the probability of two integers chosen at random to be prime to each other is $frac6pi^2$. The proof is
understandable by a smart undergraduate. This is the kind of motivation I am looking for. - by the inverse square law of sound, a line of cars blowing their horns at a car stopped in a one lane road will sound $fracpi^26$ louder than a single car. Or similarly, intensity of traffic lights on a long road at night,...
- destructive testing of $n$ wooden beams will break on average $H_n=1+frac12+frac13+ldots+frac1n$ beams with a variance of $H_n-sum_k=1^nfrac1k^2$.
- $zeta(2)=fracpi^26$ is all over quantum mechanics. The simplest, most understandable example I have found is Johnson-Nyquist noise. Still, there is a lot
of physics background required to figure it out, so this does not satisfy me much.
Are there any other good reason why to compute $zeta(2)$ ? Why were Mengoli, Euler and other mathematicians from the Enlightenment interested in the answer ? Any application in physics, chemistry, economics,... ? As far as I am concerned, the closer to reality, the better.
Thanks in advance for any help.
sequences-and-series soft-question riemann-zeta applications
asked Aug 1 at 11:31
Dan Odare
162
I don't know the history but I suspect Euler and others wanted to know the value of $zeta(2)$ out of mere curiosity - they knew the series converged. Euler knew the relationship between $zeta$ and the primes. Of course not the quantum mechanics.
– Ethan Bolker
Aug 1 at 11:35
After the geometric series and the Leibniz series for $piover2$ and $log 2$ this is the next simplest series you could think of. It is natural to come up with this problem.
– Christian Blatter
Aug 5 at 13:06
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
I realized that I know of several ways how to prove that $sum_n=1^inftyfrac1n^2=fracpi^26$, but I have no idea why I would want to know the
answer in the first place.
Answers I have found by myself:
- the probability of two integers chosen at random to be prime to each other is $frac6pi^2$. The proof is
understandable by a smart undergraduate. This is the kind of motivation I am looking for. - by the inverse square law of sound, a line of cars blowing their horns at a car stopped in a one lane road will sound $fracpi^26$ louder than a single car. Or similarly, intensity of traffic lights on a long road at night,...
- destructive testing of $n$ wooden beams will break on average $H_n=1+frac12+frac13+ldots+frac1n$ beams with a variance of $H_n-sum_k=1^nfrac1k^2$.
- $zeta(2)=fracpi^26$ is all over quantum mechanics. The simplest, most understandable example I have found is Johnson-Nyquist noise. Still, there is a lot
of physics background required to figure it out, so this does not satisfy me much.
Are there any other good reason why to compute $zeta(2)$ ? Why were Mengoli, Euler and other mathematicians from the Enlightenment interested in the answer ? Any application in physics, chemistry, economics,... ? As far as I am concerned, the closer to reality, the better.
Thanks in advance for any help.
sequences-and-series soft-question riemann-zeta applications
asked Aug 1 at 11:31
Dan Odare
162
I realized that I know of several ways how to prove that $sum_n=1^inftyfrac1n^2=fracpi^26$, but I have no idea why I would want to know the
answer in the first place.
Answers I have found by myself:
- the probability of two integers chosen at random to be prime to each other is $frac6pi^2$. The proof is
understandable by a smart undergraduate. This is the kind of motivation I am looking for. - by the inverse square law of sound, a line of cars blowing their horns at a car stopped in a one lane road will sound $fracpi^26$ louder than a single car. Or similarly, intensity of traffic lights on a long road at night,...
- destructive testing of $n$ wooden beams will break on average $H_n=1+frac12+frac13+ldots+frac1n$ beams with a variance of $H_n-sum_k=1^nfrac1k^2$.
- $zeta(2)=fracpi^26$ is all over quantum mechanics. The simplest, most understandable example I have found is Johnson-Nyquist noise. Still, there is a lot
of physics background required to figure it out, so this does not satisfy me much.
Are there any other good reason why to compute $zeta(2)$ ? Why were Mengoli, Euler and other mathematicians from the Enlightenment interested in the answer ? Any application in physics, chemistry, economics,... ? As far as I am concerned, the closer to reality, the better.
Thanks in advance for any help.
sequences-and-series soft-question riemann-zeta applications
asked Aug 1 at 11:31
Dan Odare
162
edited Aug 1 at 13:40
asked Aug 1 at 11:31
Dan Odare
162
asked Aug 1 at 11:31
Dan Odare
162
asked Aug 1 at 11:31
Dan Odare
162
162
I don't know the history but I suspect Euler and others wanted to know the value of $zeta(2)$ out of mere curiosity - they knew the series converged. Euler knew the relationship between $zeta$ and the primes. Of course not the quantum mechanics.
– Ethan Bolker
Aug 1 at 11:35
After the geometric series and the Leibniz series for $piover2$ and $log 2$ this is the next simplest series you could think of. It is natural to come up with this problem.
– Christian Blatter
Aug 5 at 13:06
add a comment |Â
I don't know the history but I suspect Euler and others wanted to know the value of $zeta(2)$ out of mere curiosity - they knew the series converged. Euler knew the relationship between $zeta$ and the primes. Of course not the quantum mechanics.
– Ethan Bolker
Aug 1 at 11:35
After the geometric series and the Leibniz series for $piover2$ and $log 2$ this is the next simplest series you could think of. It is natural to come up with this problem.
– Christian Blatter
Aug 5 at 13:06
I don't know the history but I suspect Euler and others wanted to know the value of $zeta(2)$ out of mere curiosity - they knew the series converged. Euler knew the relationship between $zeta$ and the primes. Of course not the quantum mechanics.
– Ethan Bolker
Aug 1 at 11:35
I don't know the history but I suspect Euler and others wanted to know the value of $zeta(2)$ out of mere curiosity - they knew the series converged. Euler knew the relationship between $zeta$ and the primes. Of course not the quantum mechanics.
– Ethan Bolker
Aug 1 at 11:35
After the geometric series and the Leibniz series for $piover2$ and $log 2$ this is the next simplest series you could think of. It is natural to come up with this problem.
– Christian Blatter
Aug 5 at 13:06
After the geometric series and the Leibniz series for $piover2$ and $log 2$ this is the next simplest series you could think of. It is natural to come up with this problem.
– Christian Blatter
Aug 5 at 13:06
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
1
down vote
I know two other problem involving this identity.
$1.$ The probability of integers chosen at random to be square-free is $frac6pi^2$. The proof is similar to the first problem you mentioned.
$2.$ Parisi conjecture, which is not a conjecture anymore. It's about finding minimum weighted matching in bipartite graphs. You may find a script about that here.
But most of all I recommend to read this article by Raymond Ayoub about Euler and $zeta$-function.
answered Aug 5 at 12:57
Leila
3,37942755
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
I know two other problem involving this identity.
$1.$ The probability of integers chosen at random to be square-free is $frac6pi^2$. The proof is similar to the first problem you mentioned.
$2.$ Parisi conjecture, which is not a conjecture anymore. It's about finding minimum weighted matching in bipartite graphs. You may find a script about that here.
But most of all I recommend to read this article by Raymond Ayoub about Euler and $zeta$-function.
answered Aug 5 at 12:57
Leila
3,37942755
add a comment |Â
up vote
1
down vote
I know two other problem involving this identity.
$1.$ The probability of integers chosen at random to be square-free is $frac6pi^2$. The proof is similar to the first problem you mentioned.
$2.$ Parisi conjecture, which is not a conjecture anymore. It's about finding minimum weighted matching in bipartite graphs. You may find a script about that here.
But most of all I recommend to read this article by Raymond Ayoub about Euler and $zeta$-function.
answered Aug 5 at 12:57
Leila
3,37942755
add a comment |Â
up vote
1
down vote
up vote
1
down vote
I know two other problem involving this identity.
$1.$ The probability of integers chosen at random to be square-free is $frac6pi^2$. The proof is similar to the first problem you mentioned.
$2.$ Parisi conjecture, which is not a conjecture anymore. It's about finding minimum weighted matching in bipartite graphs. You may find a script about that here.
But most of all I recommend to read this article by Raymond Ayoub about Euler and $zeta$-function.
answered Aug 5 at 12:57
Leila
3,37942755
I know two other problem involving this identity.
$1.$ The probability of integers chosen at random to be square-free is $frac6pi^2$. The proof is similar to the first problem you mentioned.
$2.$ Parisi conjecture, which is not a conjecture anymore. It's about finding minimum weighted matching in bipartite graphs. You may find a script about that here.
But most of all I recommend to read this article by Raymond Ayoub about Euler and $zeta$-function.
answered Aug 5 at 12:57
Leila
3,37942755
answered Aug 5 at 12:57
Leila
3,37942755
answered Aug 5 at 12:57
Leila
3,37942755
answered Aug 5 at 12:57
Leila
3,37942755
3,37942755
add a comment |Â
add a comment |Â
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I don't know the history but I suspect Euler and others wanted to know the value of $zeta(2)$ out of mere curiosity - they knew the series converged. Euler knew the relationship between $zeta$ and the primes. Of course not the quantum mechanics.
– Ethan Bolker
Aug 1 at 11:35
After the geometric series and the Leibniz series for $piover2$ and $log 2$ this is the next simplest series you could think of. It is natural to come up with this problem.
– Christian Blatter
Aug 5 at 13:06