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# Sum to infinity of recipricol of squares watch

1. (Original post by Gaz031)
People who are doing / have just finished A-Levels (including myself) won't really know much about the riemann zeta function (sp?)
I have read a few books on The Riemann Hypothese and the Riemann Zeta Function.
2. (Original post by davros)
But it wouldn't do them any harm to learn!

Anyone who can find the value of ζ(3) will get their own little piece of fame

Thank the lord I bought The Music of the Primes. Hopefully I'll be able to read up on this.

This thread's really making me look forward to doing Mathematics (amongst girls other things ) at university! Thanks, you guys. Anything else you can recommend to us budding students?
3. (Original post by Newton)
A very neat proof is done by using the Maclaurin series, by considering the zeros of sinx

0=x-((x^3)/3!)+((x^5)/5!)+...

((x^2)/3!)+((x^4)/5!)+...=1

Let (x^2)=w

=>((w/3!)+((w^2)/5!)+...=1

Now you know that the zeros of sinx occur at w=n(Pi), nER, applying the root linear coefficient theorem you know that the sum of the roots equals the coefficient of the leading term

=>(1/((Pi)^2))+(1/(4((Pi)^2))+(1/(9((Pi)^2))+...=(1/3!)=(1/6)

If you know rearrange the above result

ζ(2)=(((Pi)^2)/6) Q. E. D.

Another neat proof which I quite like was done by Simmons in 1992 using Beukers's integral.

Newton.
Well as I pointed out in post #9 whilst the identity

sinx = x [1-x^2/(pi^2)] [1-x^2/(4pi^2)] [1-x^2/(9pi^2)] ...

is true, I don't see that one can simply say that the roots of function add up to minus the coefficient of x in its power series.

For example e^x has no roots in the complex plane yet these roots should add up to -1 by the above reasoning as

e^x = 1 + x + x^2/2! + x^3/3! + ...
4. (Original post by JohnSPals)
Thank the lord I bought The Music of the Primes. Hopefully I'll be able to read up on this.

This thread's really making me look forward to doing Mathematics at university! Thanks, you guys. Anything else you can recommend to us budding students?
Ditto to all of the above!
5. (Original post by RichE)
Well as I pointed out in post #9 whilst the identity

sinx = x [1-x^2/(pi^2)] [1-x^2/(4pi^2)] [1-x^2/(9pi^2)] ...

is true, I don't see that one can simply say that the roots of function add up to minus the coefficient of x in its power series.

For example e^x has no roots in the complex plane yet these roots should add up to -1 by the above reasoning as

e^x = 1 + x + x^2/2! + x^3/3! + ...
Mathworld seems to agree (its almost a direct quote infact ), but to me it seems wrong. The series used is not even sin(x) anymore, and half the zeroes are missed (all the negative ones) aswell as x = 0, which screws everything up when taking the sum of the reciprocals.

All of the 5 references on google come straight from mathworld, so the whole thing seems a bit dubious really.
6. I've just remembered I got asked this at my Cambridge interview! I can't remember it though, sorry
7. (Original post by JamesF)
Mathworld seems to agree (its almost a direct quote infact ), but to me it seems wrong. The series used is not even sin(x) anymore, and half the zeroes are missed (all the negative ones) aswell as x = 0, which screws everything up when taking the sum of the reciprocals.

All of the 5 references on google come straight from mathworld, so the whole thing seems a bit dubious really.
I only know that formula for polynomials and the exp(x) function seems to be a counter-example.
8. (Original post by JohnSPals)
Thank the lord I bought The Music of the Primes. Hopefully I'll be able to read up on this.

This thread's really making me look forward to doing Mathematics (amongst girls other things ) at university! Thanks, you guys. Anything else you can recommend to us budding students?
Try "From Here to Infinity" (Ian Stewart) for a good overview of various maths topics. It's also useful to read a few biographical works so you're not just trying to digest purely academic stuff - e.g. Hardy's "A Mathematician's Apology", or Paul Hoffman's "The Man Who Loved Only Numbers".
9. (Original post by davros)
Try "From Here to Infinity" (Ian Stewart) for a good overview of various maths topics. It's also useful to read a few biographical works so you're not just trying to digest purely academic stuff - e.g. Hardy's "A Mathematician's Apology", or Paul Hoffman's "The Man Who Loved Only Numbers".
I've already read From Here to Infinity, and wasn't a massive fan. I was looking into The Man Who Loved Only Numbers, so I will probably get that soon. Thanks
10. (Original post by JohnSPals)
I've already read From Here to Infinity, and wasn't a massive fan. I was looking into The Man Who Loved Only Numbers, so I will probably get that soon. Thanks
I felt that "From Here To Infinity" got a bit too technical (lie groups and topology in particular went right over my head ).
I was reading a few pages of TMWLON and I really want to get it now
11. (Original post by SsEe)
I felt that "From Here To Infinity" got a bit too technical (lie groups and topology in particular went right over my head ).
I was reading a few pages of TMWLON and I really want to get it now
I fully agree with that. The only topic that I even remember (that is, any meaningful stuff) from the book was the stuff on the lottery. Because I could easily follow what was going on. Why did he even bother with the chapter(s?) on knots - that was impossible to follow!
12. This is my first post. I hope I'm not posting out of place.

I've read 4 books on the Riemann Hypothesis and Music of the Primes was the best. Prime Obsession is also an excellent read. Most of the Titchmarsh book went way over my head (decided to go back to it next year) and Dr Riemann's Zeros didn't seem to go into enough detail at all.

The Man Who Loved Only Numbers is a brilliant book. If you like books that are more inspiring that instructional then Complexity (Roger Lewin) is also a nice one to read.
13. (Original post by AMathStudent)
This is my first post.
Welcome to the forum

I've read 4 books on the Riemann Hypothesis and Music of the Primes was the best. Prime Obsession is also an excellent read. Most of the Titchmarsh book went way over my head (decided to go back to it next year) and Dr Riemann's Zeros didn't seem to go into enough detail at all.

The Man Who Loved Only Numbers is a brilliant book. If you like books that are more inspiring that instructional then Complexity (Roger Lewin) is also a nice one to read.
Thanks for the recommendations. I think i'm going to read prime obsession sometime this summer.

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