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Zeta Function

by sparsh12
Tags: function, zeta
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sparsh12
#1
Feb24-12, 01:20 PM
P: 13
I don't know anything of complex analysis or analytic number theory or analytic continuation. But i read about zeta function and riemann hypothesis over wikipedia, clay institute's website and few other sources. I started with original zeta function



and then for complex s of form a+ib, where a and b are real,it would be



Then i did few things and it became,



It can be observed that above relation is actually,



and if reimann hypothesis is true, first term diverges in above equation, which would in turn mean second term must tend to -∞.

Now my questions are:
1)Am i on right path? I have plans to start real and complex analysis soon. Would above progress be useful?
2)Has anyone around got any idea of proving second term tending to -∞ without assuming riemann hypothesis true? Won't this method help prove reimann hypothesis true? As far as i understand, all solutions of above zeta function satisfies riemann zeta function(one of analytic continuation).

If i talked nonsense above, please rectify me.
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Wizlem
#2
Feb24-12, 02:35 PM
P: 35
The summation formula for the zeta function is only valid for real part of s greater than 1. Where the non-trivial zeros lye is not in this region hence your problem. Showing that the second term diverges wouldn't be so helpful either since it would probably diverge for any a<1 and not just a = 1/2. I like your enthusiasm of the problem though. I've had some interest of it in the past and have this book:

http://www.amazon.com/Riemanns-Zeta-...0115338&sr=8-2

I haven't had the patience to get very far in it so I wouldn't be helpful answering questions. Maybe others with more knowledge can recommend some resources for you to look into. Go learn more of analytic continuation though, it is interesting stuff.
DonAntonio
#3
Feb25-12, 08:24 AM
P: 606
Quote Quote by sparsh12 View Post
I don't know anything of complex analysis or analytic number theory or analytic continuation. But i read about zeta function and riemann hypothesis over wikipedia, clay institute's website and few other sources. I started with original zeta function



and then for complex s of form a+ib, where a and b are real,it would be



Then i did few things and it became,



It can be observed that above relation is actually,



and if reimann hypothesis is true, first term diverges in above equation, which would in turn mean second term must tend to -∞.

Now my questions are:
1)Am i on right path? I have plans to start real and complex analysis soon. Would above progress be useful?
2)Has anyone around got any idea of proving second term tending to -∞ without assuming riemann hypothesis true? Won't this method help prove reimann hypothesis true? As far as i understand, all solutions of above zeta function satisfies riemann zeta function(one of analytic continuation).

If i talked nonsense above, please rectify me.


I agree with the other answerer about the kudos to you for your interest, but in general I'm afraid you won't get very far away without a robust basis in basic number theory and complex analysis.

The (Euler-)Riemann Zeta Function and all the stuff around it are, imho, one of the most fascinating and beautiful of all subjects in mathematics, but it is not elementary stuff.

You could try some of the basic theory books in the bibliography in the book given by the other answerer (Edward's) and begin learning that.

Tonio

Tonio


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