A255742 Integers setting a record for the absolute minimal difference from the imaginary part of a nontrivial zero of the Riemann zeta function.

14, 21, 25, 48, 146, 776, 3764, 7847, 7904, 18048, 90930, 92219, 587741

Offset: 1

Author

Omar E. Pol, Mar 16 2015

Keywords

• nonn
• hard
• more

Comments

We consider here the imaginary part of 1/2 + i*y = z, for which Zeta(z) is a zero.

No more terms below the 600000th nontrivial zero of the Riemann zeta function. - Robert G. Wilson v, Sep 30 2015

Is there an Im(rho_k) that is also an 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

Examples

			-------------------------------------------------------------------
                                     Absolute      New
k      Im(rho_k)       A002410(k)   difference   record   n   a(n)
-------------------------------------------------------------------
1    14.134725142    >    14        0.134725142    Yes    1    14
2    21.022039639    >    21        0.022039639    Yes    2    21
3    25.010857580    >    25        0.010857580    Yes    3    25
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    48
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 after 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, A013629, A092783, A255739.

Formula

a(n) = A002410(A255739(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