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Thread: Riemann Zeta function.

  1. #1 Riemann Zeta function. 
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    Does someone know if for the Riemann Zeta function the following has been proven:

    1) Re(Zeta(s)) has no zeros for Re(s) >1

    2) The minima for Re (Zeta(s)) in the strip 1/2<Re(s)<1 does not tend to minus infinity as Im(s) tends to infinity.

    3) Has the zeros of Zeta on the critical line been plotted against zero number ({Im(s) | Zeta(s) = 0 } against the natural numbers) and a function fitted to the result?


    It also matters what isn't there - Tao Te Ching interpreted.
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  3. #2  
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    I'm not sure about your first question, but I'd be surprised if it were true. I feel it shouldn't be too hard to display a case where Re(zeta(s)) < 0.

    The answer to your second question is no. The zeta function enjoys a certain "universality" property which states that it approximates holomorphic functions very well. Check http://en.wikipedia.org/wiki/Zeta_function_universality for details. In particular, for any e > 0, any complex number w, and any compact subset U of 1/2 < Re(s) < 1 with connected complement, the result gives that the measure of real t which satisfy |zeta(s+it) - w| < e uniformly for s in U is nonzero, and the set of such t is unbounded. So, letting w approach negative infinity, we see that there always exists some 1/2 < Re(s) < 1 with arbitrarily large imaginary part and, say, |zeta(s) - w| < 1, and this implies liminf(zeta(s)) = -infinity as Im(s) goes to infinity.

    I don't quite understand your third question, but it's well known that the number of zeros in the strip with imaginary part between 0 and T is asymptotic to (T/2pi)log(T/2pi). (I think this is right, I may have some of the constants messed up). And we know a positive proportion (at least 40%, I seem to remember) of the zeros lie on the line, so the same number of zeros on the line of imginary part between 0 and T is on the order of TlogT.

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    Edit: I might not agree with my first sentence anymore... but I don't have solid evidence either way.


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    Where can I find a list of known zeros?

    I think it may be usefull to plot the Im(s) against the zeros given numbers (counted in sequence: 1= 1st zero, 2 = 2nd zero etc.) Then you get a discrete graph and maybe a real function is fittable to this.

    I have read that the Riemann Hypothesis is true if Zeta(s) approximates itself. But the Universality Theorem has the only conditions that g(s) must be non-vanishing, continuous, and analytic on the disc interior of |s| < r (see Voronin Universality Theorem, www.mathworld.wolfram.com).

    Is'nt these conditions easily provable for Zeta(s)?
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  5. #4  
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    It took em a second to see this: Voronin only gives the uniform approximation if the function is nonvanishing. So you cannot use it to approximate zeta in discs about its zeros. I'm assuming that approximating zeta on these discs is included in the "zeta approximates itself iff RH is true" statement. That's an unintuitive (and neat) statement to me: you can approximate zeta uniformly about a zero by zeta somewhere in half of the strip iff zeta doesn't vanish anywhere in that half of the strip.

    I bet you could just google "list of zeros of riemann zeta function". Actually, you might try searching for Mike Rubenstein--he does a lot of statistical analyses with zeros of L-functions, and so he may have some data available to the public.

    Again, the number of zeros with imaginary part up to T is on the order of TlogT, so this says that the n-th zero has imaginary part on the order of n/log n. You can fit myriad real functions to this, but it's probably more interesting and useful to develop an asymptotic expansion. It also seems like it's more useful to think of things as "how many zeros are there up to imaginary part T" just because it's easy to calculate this: you just integrate the logarithmic derivative of zeta on boxes of height T in the strip.

    There's a lot of work being done right now analyzing the spacing of the imaginary parts of the zeros on the line. Statistical analyses show that they exhibit distributions similar to those of eigenvalues in certain matrix groups. (I believe the zeros of the zeta look like eigenvalues for unitary groups, but it's been a while since I thought about this.) You'd probably be interested in this: search for applications of random matrix theory to the Riemann zeta function and to families of L-functions.
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    :?

    I found too many Mike Rubensteins. Can you give me any other Google-bait (keywords) ?
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  7. #6  
    Forum Professor serpicojr's Avatar
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    Try adding math or number theory or Riemann zeta function or random matrix theory to you query.
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  8. #7  
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    And try the names Sarnak and Snaith.
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