A092783 -id:A092783 - cp's OEIS frontend

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

Omar E. Pol, Mar 16 2015

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

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

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.

Cf. A002410, A013629, A092783, A255739.

a(n) = A002410(A255739(n)).

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

A135297 Number of Riemann zeta function zeros on the critical line, less than n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 8, 9, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 13, 14, 14, 14, 14, 14, 15, 15, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 20, 20, 21, 21, 21, 22, 22, 23, 23, 23, 24, 25, 25, 25, 25, 26, 26, 27, 28, 28, 28, 29, 29

Offset: 1

Jean-François Alcover, Mar 09 2011

nonn

This sequence is just the cumulative distribution of the zeros.

Apart from differing singularities, the beginning of this sequence agrees with the zeta zero counting functions (RiemannSiegelTheta(n) + im(log(zeta(1/2 + i*n))))/Pi + 1 and (sign(im(zeta(1/2 + i*n))) - 1)/2 + floor(n/(2*Pi)*log(n/(2*Pi*e)) + 7/8) + 1, but disagrees later. The first deviations are seen in the continuous counting function at locations of zeta zeros with indices A153815. See also A282793 and A282794. - Mats Granvik, Feb 21 2017

			The first nontrivial zero is 1/2 + 14.1347...*i; hence, a(15)=1.

H. M. Edwards, Riemann's Zeta Function, Dover Publications, New York, 1974 (ISBN 978-0-486-41740-0)

T. D. Noe, Table of n, a(n) for n = 1..10000
Mats Granvik, Mathematics Stackexchange
Andrew Guinand, A summation formula in the theory of prime numbers, page 111
Andrew Guinand, A summation formula in the theory of prime numbers, Proc. London Math. Soc. (1948) s2-50 (1): 107-119, see page 111.
Raymond Manzoni, Riemann Zeta function - number of zeros, Mathematics Stackexchange, 2013.
Eric W. Weisstein, MathWorld: Riemann Zeta Function Zeros
Wikipedia, Riemann Zeta Function
Index entries for sequences related to zeta function

Cf. A002410, A013629, A092783.

nn = 100; t = Table[0, {nn}]; k = 1; While[z = Im[ZetaZero[k]]; z < nn, k++; t[[Ceiling[z] ;; nn]]++]
With[{zz=Ceiling[Im[N[ZetaZero[Range[30]]]]]},Table[If[MemberQ[zz,n],1,0],{n,Max[zz]}]]//Accumulate (* Harvey P. Dale, Aug 15 2017 *)

a(n) = #lfunzeros(L, n) \\ Felix Fröhlich, Jun 10 2019

# This function makes sure no zeros are missed.
def A135297_list(n):
    Z = lcalc.zeros(n)
    R = []; pos = 1; count = 0
    for z in Z:
        while pos < z:
            R.append(count)
            pos += 1
        count += 1
    return R
A135297_list(30) # Peter Luschny, May 02 2014

a(n) ~ n log (n/(2*Pi*e)) / (2*Pi). - Charles R Greathouse IV, Mar 11 2011, corrected by Hal M. Switkay, Oct 03 2021
From Mats Granvik, May 13 2017: (Start)
a(n) ~ im(LogGamma(1/4 + i*n/2))/Pi - n/(2*Pi)*log(Pi) + im(log(zeta(1/2 + i*n)))/Pi + 1.
a(n) ~ floor(im(LogGamma(1/4 + i*n/2))/Pi - n/(2*Pi)*log(Pi) + 1) + (sign(im(zeta (1/2 + i*n))) - 1)/2 + 1.
a(n) ~ (RiemannSiegelTheta(n) + im(log(zeta(1/2 + i*n))))/Pi + 1.
a(n) ~ (floor(RiemannSiegelTheta(n)/Pi + 1)) + (sign(im(zeta(1/2 + i*n))) - 1)/2 + 1.
a(n) ~ n/(2*Pi)*log(n/(2*Pi*e)) + 7/8 + (im(log(zeta(1/2 + i*n))))/Pi - 1 - O(n^(-1)) + 1.
a(n) ~ floor(n/(2*Pi)*log(n/(2*Pi*e)) + 7/8) + (sign(im(zeta(1/2 + i*n))) - 1)/2 + 1.
See A286707 for exact relations.
(End)

A106635 a(n) = round(2*Im(z(n))/Pi - 4), where z(n) is the n-th zero of the Riemann zeta function on the critical line (with a positive imaginary part).

Original entry on oeis.org

5, 9, 12, 15, 17, 20, 22, 24, 27, 28, 30, 32, 34, 35, 37, 39, 40, 42, 44, 45, 47, 49, 50, 52, 53, 55, 56, 57, 59, 61, 62, 63, 64, 67, 67, 69, 70, 72, 73, 74, 75, 77, 78, 79, 81, 82, 84, 85, 86, 87, 89, 90, 92, 92, 93, 95, 96, 97, 99, 100, 101, 102, 104, 104, 106

Offset: 0

Roger L. Bagula, May 11 2005

nonn

Previous name: Rational approximations of Zeta zeros as an integer sequence.

The average error in the approximation is low (-0.0559729) for the first 30 zeta zeros. The idea is that the imaginary part of the zeta zero is a bad rational approximation of the type: 4/(a(n)+4) to give b(n) = 2*Pi*(a(n)+4)/4.

Cf. A002410, A013629, A092783, A106636.

a[n_] := Round[Im[ZetaZero[n]]*2/Pi - 4]; Array[a, 70] (* Amiram Eldar, Jun 07 2025 *)

Name edited and data corrected and extended by Amiram Eldar, Jun 07 2025

A106636 a(n) = round(2*Im(z(n))/Pi), where z(n) is the n-th zero of the Riemann zeta function on the critical line (with a positive imaginary part).

Original entry on oeis.org

9, 13, 16, 19, 21, 24, 26, 28, 31, 32, 34, 36, 38, 39, 41, 43, 44, 46, 48, 49, 51, 53, 54, 56, 57, 59, 60, 61, 63, 65, 66, 67, 68, 71, 71, 73, 74, 76, 77, 78, 79, 81, 82, 83, 85, 86, 88, 89, 90, 91, 93, 94, 96, 96, 97, 99, 100, 101, 103, 104, 105, 106, 108, 108

Offset: 0

Roger L. Bagula, May 11 2005

nonn

Previous name: Pair rational approximations of Zeta zeros as an integer sequence.

Cf. A002410, A013629, A092783, A106635.

a[n_] := Round[Im[ZetaZero[n]]*2/Pi]; Array[a, 70] (* Amiram Eldar, Jun 07 2025 *)

Name edited and data corrected and extended by Amiram Eldar, Jun 07 2025

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

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A135297 Number of Riemann zeta function zeros on the critical line, less than n.

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A106635 a(n) = round(2*Im(z(n))/Pi - 4), where z(n) is the n-th zero of the Riemann zeta function on the critical line (with a positive imaginary part).

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A106636 a(n) = round(2*Im(z(n))/Pi), where z(n) is the n-th zero of the Riemann zeta function on the critical line (with a positive imaginary part).

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Mathematica

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