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

1,2

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

Table of n, a(n) for n=1..13.

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.

Formula

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

Example

-------------------------------------------------------------------
                                     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.

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

Cf. A002410, A255742.

Sequence in context: A162098 A162099 A307069 * A037389 A037425 A226967

Adjacent sequences: A255736 A255737 A255738 * A255740 A255741 A255742

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