A255739 Indices of nontrivial zeros of Riemann zeta function whose imaginary part sets a record for the absolute minimal difference from an integer.

1, 2, 3, 9, 51, 473, 3233, 7657, 7722, 20002, 124170, 126137, 977155

Offset: 1

Author

Omar E. Pol, Mar 17 2015

Keywords

• nonn
• hard
• more

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

Examples

			-------------------------------------------------------------------
                                     Absolute      New
k      Im(rho_k)       A002410(k)   difference   record   n   a(n)
-------------------------------------------------------------------
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.
		

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.
• Index entries for zeta function.

Crossrefs

Cf. A002410, A255742.

Programs

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 *)

Formula

A255742(n) = A002410(a(n)).

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