OFFSET
1,2
COMMENTS
We consider here the imaginary part of 1/2 + iy = z, for which Zeta(z) is a zero.
No more terms below 600000. - Robert G. Wilson v, Sep 30 2015
Is there an Im(rho_k) that is also a positive integer? Is there a minimum gap between an Im(rho_k) and a positive integer? At present it is not known whether this sequence is finite or infinite. - Omar E. Pol, Oct 13 2015
No more terms below 2001052. - Amiram Eldar, Aug 10 2023
LINKS
Andrew M. Odlyzko, Tables of zeros of the Riemann zeta function.
Andrew M. Odlyzko, On the distribution of spacings between zeros of the zeta function.
EXAMPLE
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Absolute New
k Im(rho_k) A002410(k) difference record n a(n)
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1 14.134725142 > 14 0.134725142 Yes 1 1
2 21.022039639 > 21 0.022039639 Yes 2 2
3 25.010857580 > 25 0.010857580 Yes 3 3
4 30.424876126 > 30 0.424876126 Not
5 32.935061588 < 33 0.064938412 Not
6 37.586178159 < 38 0.413821841 Not
7 40.918719012 < 41 0.081280988 Not
8 43.327073281 > 43 0.327073281 Not
9 48.005150881 > 48 0.005150881 Yes 4 9
10 49.773832478 < 50 0.226167522 Not
...
where rho_k is the k-th nontrivial zero of Riemann zeta function.
We computed more digits of Im(rho_k), but in the above table only 9 digits beyond the decimal point appear.
MATHEMATICA
mn = Infinity; k = 1; lst = {}; While[k < 2501, a = N[ Abs[ Im[ ZetaZero[
k]] - Round[ Im[ ZetaZero[ k]] ]], 32]; If[a < mn, AppendTo[lst, k];
Print[k]; mn = a]; k++]; lst (* Robert G. Wilson v, Sep 29 2015 *)
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Omar E. Pol, Mar 17 2015
EXTENSIONS
a(6)-a(10) from Robert G. Wilson v, Sep 29 2015
a(11)-a(12) from Robert G. Wilson v, Sep 30 2015
a(13) using Odlyzko's tables added by Amiram Eldar, Aug 10 2023
STATUS
approved