A013629 Floor of imaginary parts of nontrivial zeros of Riemann zeta function.

14, 21, 25, 30, 32, 37, 40, 43, 48, 49, 52, 56, 59, 60, 65, 67, 69, 72, 75, 77, 79, 82, 84, 87, 88, 92, 94, 95, 98, 101, 103, 105, 107, 111, 111, 114, 116, 118, 121, 122, 124, 127, 129, 131, 133, 134, 138, 139, 141, 143, 146, 147, 150, 150, 153, 156, 157, 158, 161

Offset: 1

Author

John Morrison (John.Morrison(AT)armltd.co.uk)

Keywords

• nonn

Examples

			The imaginary parts of the first 4 zeros are 14.134725... (A058303), 21.0220396... (A065434), 25.01085758... (A065452), 30.424876... (A065453). Therefore the sequence starts: 14, 21, 25, 30, ..., as does A002410 (rounded values; main entry). But the 5th, 6th and 7th values are 32.935... (A192492), 37.586... (A305741), 40.9187... (A305742), whence a(n) = A002410(n)-1 and A002410 = A092783 (ceiling) for these. - _M. F. Hasler_, Nov 23 2018
		

References

• H. M. Edwards, Riemann's Zeta Function, Academic Press, NY, 1974, p. 96.
• C. B. Haselgrove and J. C. P. Miller, Tables of the Riemann Zeta Function. Royal Society Mathematical Tables, Vol. 6, Cambridge Univ. Press, 1960, p. 58.

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Index entries for zeta function.

Crossrefs

Cf. A002410 (rounded values: main entry), A092783 (ceiling).

Imaginary part of k-th nontrivial zero of Riemann zeta function: A058303 (k=1), A065434 (k=2), A065452 (k=3), A065453 (k=4), A192492 (k=5), A305741 (k=6), A305742 (k=7), A305743 (k=8), A305744 (k=9), A306004 (k=10).

Programs

Mathematica

Table[Floor[Im[ZetaZero[n]]], {n, 60}] (* Alonso del Arte, Feb 07 2011 *)

PARI

lfunzeros(lzeta,100)\1 \\ Charles R Greathouse IV, Mar 10 2016

Sage

def A013629_list(n):
    Z = lcalc.zeros(n)
    return [floor(z) for z in Z]
A013629_list(50) # Peter Luschny, May 02 2014

Formula

a(n) ~ 2*Pi*n/log n. - Charles R Greathouse IV, Jun 30 2011

a(n) ~ (2*Pi*e) * e^(W0(n/e)), where W0 is the principal branch of Lambert's W function. - Hal M. Switkay, Oct 04 2021

a(n) = A092783(n) - 1. - M. F. Hasler, Nov 23 2018

Extensions

Edited by Daniel Forgues, Jun 30 2011

Definition corrected by Jonathan Sondow, Sep 18 2011