2
$\begingroup$

since we know that the number of Riemann zeros on the interval $ (0,E) $ is given by $ N(E) = \frac{1}{\pi}\operatorname{Arg}\xi(1/2+iE) $

is then possible to get the inverse function $ N(E)^{-1}$ so with this inverse we can evaluate the Riemann zeros $ \rho $ ??

i mean the Riemann zeros are the inverse function of $\arg\xi(1/2+ix) $

  • 3
    I'm not entirely sure what you are asking but it seems to be related to what I'm about to say. It is entirely possible to calculate actual zeros of the Riemann-zeta function (up to some given height in the complex plane) and verify that they all(up to that height) lie on the critical line. This has been studied extensively, notably, by Odlyzko: http://www.dtc.umn.edu/~odlyzko/ All the information you would need to implement some algorithms and do some calculations yourself are in this well known and cheap book: http://www.amazon.com/Riemanns-Zeta-Function-Harold-Edwards/dp/04864174092012-04-28
  • 1
    How are we to understand a formula that has $E$ on one side but no $x$, and $x$ on the other side but no $E$? Is $E$ related to $x$ in some way you aren't disclosing?2012-04-30
  • 0
    The inverse function would be denoted $N^{-1}(E)$. If by "get" you mean "obtain explicit formula for," then the answer is no, we have no such formula for obtaining the $n$th zero on the upper critical line.2012-04-30
  • 0
    and if we plot $ N(E) $ and invert it numerically could we then obtain the zeros ?? ,2012-04-30
  • 0
    Sounds like it, yes. Not sure about how that fits into the scheme of things concerning the efficiency of numerical computation of the zeros though.2012-04-30
  • 0
    $N$ is a function from the (positive) reals to the (non-negative) integers. It is infinity-to-one so there is no inverse function. The inverse image $N^{-1}(n)$ of an integer $n$ would be an interval of real numbers, not a single real number. Taking all this into account, what are you asking?2012-04-30
  • 0
    $ N(E) $ is an staircase function, the inverse of an staircase function is another staircase function , simply take the reflection over the line $ y=x $ so i believe that the ivnerse of $ N(E) $ could be evaluated.2012-04-30

1 Answers 1