Can someone explain zeros and zeta function for Riemann Hypothesis? (Yr13)

[2710]

Hi,

I'm Yr 13 and just wanted to do some further reading/exploring.

So i understand that the zeta function is something to do with summing up like this:

1/ (1^s) + 1/(2^s) etc etc

Now, I just want to know what are non-trivial zeros and trivial zeros? I just want to be able to understand this concept. And that graph where all non-trivial zeros lie on the x=1/2 line. How is that graph plotted? Btw, im only Yr 13, so layman terms please >__>

Thanks!

PS: Also, IF EVER one zero was found OFF of the critical line...would the e-commerce community collapse? Coz I heard that many companies use prime numbers to construct the encryption of their security. Or is that just the RSA Number Challenge? Hmm not so sure.

[ramsey2879]

See http://www.physicsforums.com/showthread.php?t=322847&highlight=Riemann+Hypothesis Also the imaginary part (which is n* the square root of minus 1 or "i") + 1/2 = s in this case. Even though the square root of -1 is imaginary, it can be multiplied in algebraic equations and the like. The answer will be either real or have an imaginary component B*i. In the Riemann Hypothesis the sum of the real parts and the imaginary parts for non-trival zeros will both equal zero.

[ramsey2879]

I inadvertently posted the wrong link which was a followup on the thread I wanted to cite. It is http://www.physicsforums.com/showthread.php?p=2235996#post2235996 The two components which sum to zero are the real component which corresponds to the x axis and the imaginary component which corresponds to the "y" axis. If you draw a unit circle on such a plane, and draw a radius at the angles 120 degrees or 240 degrees the x and y component (which correspond to the sine and cosine ) of the meeting point with the circle correspond respectively to the real and imaginary components of a complex number which if cube would be equal to 1, other points on the unit circle when raised to the power of three would give other points on the unit circle. I'm not sure if they would be again spaced at +/- 120 degrees, but the cube roots of -1 are also -1 or imaginary points spaced 120 degrees on the unit circle from -1.

[2710]

Thanks ! :D

[chascomm]

Hi,

I'm Yr 13 and just wanted to do some further reading/exploring.

So i understand that the zeta function is something to do with summing up like this:

1/ (1^s) + 1/(2^s) etc etc

Now, I just want to know what are non-trivial zeros and trivial zeros? I just want to be able to understand this concept. And that graph where all non-trivial zeros lie on the x=1/2 line. How is that graph plotted? Btw, im only Yr 13, so layman terms please >__>

Thanks!

PS: Also, IF EVER one zero was found OFF of the critical line...would the e-commerce community collapse? Coz I heard that many companies use prime numbers to construct the encryption of their security. Or is that just the RSA Number Challenge? Hmm not so sure.
REPLY. I am not a grammarian but have been in conn special subject of Topology at oxon in the past. I am permitted ethically a shot at this. Have you read DR PENROSE BOOK ON ROAD TO REALITY? He mentions this subject. I am not satisfied myself that it would answer your quest but there is a section on (1 minus third root of z) empowered minus 1. Terrible for yourstage but take note. Suppose the minus one root had a particular route associated with it so that all such paths went through it to yield a power view for higher logic ie an iterative implication. Could it not be that you yourself would become a philosopher of the moment and show that a Riemann Zeta function is on offshoot of this line of reasoning? I am afraid I do not have status myself to say much about this view of grammarians but you should continue to look around at new books on the subject. Iassume you have read Dr Riemanns Zeroes by Dr Sabach;a famous writer known to Dily Telegraph. They get wind sometimes.

[2710]

REPLY. I am not a grammarian but have been in conn special subject of Topology at oxon in the past. I am permitted ethically a shot at this. Have you read DR PENROSE BOOK ON ROAD TO REALITY? He mentions this subject. I am not satisfied myself that it would answer your quest but there is a section on (1 minus third root of z) empowered minus 1. Terrible for yourstage but take note. Suppose the minus one root had a particular route associated with it so that all such paths went through it to yield a power view for higher logic ie an iterative implication. Could it not be that you yourself would become a philosopher of the moment and show that a Riemann Zeta function is on offshoot of this line of reasoning? I am afraid I do not have status myself to say much about this view of grammarians but you should continue to look around at new books on the subject. Iassume you have read Dr Riemanns Zeroes by Dr Sabach;a famous writer known to Dily Telegraph. They get wind sometimes.

Um...sorry lol, I dont really understand your post. The only book Ive read so far is Marcus Du Sautoy's Music of the Primes. Ive heard about Road to reality and how notoriously difficult it is to read (bearing in mind its length) Lol. But I will definitely give more books a shot.

Thanks

[2710]

Ok so I've read all of it. All I understand now is how non-trivial zeros converge at your starting point when you follow the rules. So, where is the critical 'strip' in these graphs? And how do these link to the primes? Apparently when you choose a correct value for T, it will bring you back to where you started, but what has any of this got to do with primes?

Apparently the first zero happens at t=14.134725 .......um...not a prime number... yeh sorry for being stupid, but I cant find it anywhere on the internet. On wikipedia they just go on to talk about it in another language (ie, im stupid).

Thanks

[CRGreathouse]

The number of primes up to x can be approximated, as Gauss guessed as a child, by x / log x. A better estimate turns out to be the logarithmic integral Li(x), which is the integral of 1/log(t) up to x.

Riemann's explicit formula is a way to transform this estimate into an exact value by considering not only the logarithmic integral of x, but the logarithmic integrals of x raised to the power of the zeros of the zeta function. Try this page for an overview of that:
http://www.math.ucsb.edu/~stopple/explicit.html