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

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

Author

Jean-François Alcover, Mar 09 2011

Keywords

• nonn

Comments

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

Examples

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

References

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

Links

• 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

Crossrefs

Cf. A002410, A013629, A092783.

Programs

Mathematica

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 *)

PARI

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

Sage

# 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

Formula

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)