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A non-trivial infinity of zeroes with real part one-half

  1. Jun 22, 2009 #1
    Other than the Riemann Zeta function, what equation has a non-trivial infinity of zeroes with real part one-half?
     
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  3. Jun 23, 2009 #2
    T(a) = a*(a-1)/2
    T(1/2 + b*i) has an imaginary part equal to zero

    I(a) = a*(a-i)/2
    I(b +(1/2)*i) also has imaginary part equal to zero

    Let c = a+b
    Then N(a+bi) =I(b+a +(a)i) - T(a+(b+a)i) has a zero when a = 1/2
     
    Last edited: Jun 23, 2009
  4. Jun 23, 2009 #3

    Office_Shredder

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    What equation? How about Re(z) - 1/2 = 0

    Maybe you're looking for holomorphic functions
     
  5. Jun 23, 2009 #4
    How about f(z)=0?
     
  6. Jun 23, 2009 #5
    Perhaps holomorphic functions agree with what is meant by "non-trivial" in the definition of the Riemann Zeta function. In this latter regard the above examples appear trivial.

    Also, by "real part" I implied that there was an "imaginary part." Is the Riemann-equivalent "alternating Zeta function" (a real, generalized alternating harmonic series) holomorphic?
     
  7. Jun 23, 2009 #6

    Hurkyl

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    No. "Trivial zero of the Riemann zeta function" is simply a synonym for "negative even integer". We choose to call them the "trivial" zeroes because it's very easy to prove they are zeroes, and they are irrelevant to the Riemann hypothesis.

    (I highly doubt you have ever see the word "nontrivial" in any definition of the Riemann Zeta function)

    Office_Shredder mentioned "holomorphic functions" because he was trying to guess what the heck you meant by "equation".
     
  8. Jun 26, 2009 #7
    Of course, the RZF is just a simple function, the infinite series of th inverse intergers raised to the s power argument of the RZF. Nontrivial zeros are nowhere mentioned there, but they have to be mentioned in a definition of the RH, because the RH is just a conjecture about the RZF when the argument s now becomes a complex z = a + bi. Then Euler found the connection with the primes, which makes everything interesting. When we talk about the RH, you have to specify that you're talking about the nontrivial zeros.
     
  9. Jun 26, 2009 #8

    HallsofIvy

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    I'm wondering what in the world a "trivial infinity" would be!
     
  10. Jun 26, 2009 #9
    Sorry, it should read "infinite number," not "infinity."
     
  11. Jun 27, 2009 #10
    Perhaps this is what you're looking for:

    Consider the function sin(z). This has zeroes at all values n*pi, where n is an integer. Now rotate by 90 deg to get sin(iz). This has zeroes at at all values n*pi*i, where again n is any integer. Finally, translate by 1/2, to get sin(i(z - (1/2))). This last function has zeroes at all values (1/2) + n*pi*i, which have real part 1/2, as desired.
     
  12. Jun 27, 2009 #11
    Thanks, Petek.

    I had earlier in this thread misinterpreted "non-trivial" as representing the seeming "random" distribution of primes on the line Re(z)=1/2. The sine function you mention generates answers with much more simple functions. Despite my initial confusion, yours is an elegant answer, although not akin to the class of function I was looking for.

    Perhaps sequences like prime numbers could be found through such functions as I suggest. Perhaps I'm barking up the wrong tree.
     
  13. Jun 27, 2009 #12
    You might want to look at the Weierstrass factorization theorem, which allows one to construct functions that have zeroes at any sequence {a_n} that satisfies certain growth conditions.
     
  14. Jun 27, 2009 #13
    The Weierstrass factorization theorem may introduce the tools needed to find functions in the complex plane similar to that described in the Riemann hypothesis, and generate series of numbers relating unique properties to the harmonic series.

    Weierstrass' elementary factors seem to enable this. I would not be surprised in Riemann used a form of this theorem if it were available to him. This concept is very close to what I had envisioned.
     
  15. Jun 28, 2009 #14
    Yes, Riemann knew a product expansion of the zeta function in terms of its zeroes. It's now called the Hadamard product.
     
  16. Jul 23, 2009 #15
    the question is that someone can proof that 'at leat the 40 % of zeros have real part 1/2' is there an argument to generalize this to 100 % , how can we really know that the 40 50 % of zeros have real part 1/2 if there is an infinite amount of them.
     
  17. Jul 23, 2009 #16
    To reply to the second part of your post, define N(T) to be the number of zeros s + it of the zeta function such that 0 < s < 1 and 0 < t [tex]\leq[/tex] T and define [tex]N_0[/tex](T) to be the number of zeros such that 0 < t [tex]\leq[/tex] T. In other words, N(T) is the number of zeros in the critical strip with imaginary part > 0 and [tex]\leq[/tex] T. [tex]N_0[/tex] is the number of zeros on the corresponding critical line. The Riemann Hypothesis asserts that [tex]N_0[/tex](T) = N(T). What has been proved is that

    [tex]N_0[/tex] > [tex]\frac{2}{5}[/tex]N(T)

    for sufficiently large T. This is what's meant when saying that at least 40% of the (non-trivial) zeros have real part [tex]\frac{1}{2}[/tex].

    Hope that helps.
     
    Last edited: Jul 23, 2009
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