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

1,1

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.

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

Example

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

Mathematica

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

Prog

(Sage)
def A013629_list(n):
    Z = lcalc.zeros(n)
    return [floor(z) for z in Z]
A013629_list(50) # Peter Luschny, May 02 2014
(PARI) lfunzeros(lzeta, 100)\1 \\ Charles R Greathouse IV, Mar 10 2016

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).

Sequence in context: A330137 A230095 A133185 * A234802 A162780 A002410

Adjacent sequences: A013626 A013627 A013628 * A013630 A013631 A013632

Keyword

nonn

Author

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

Extensions

Edited by Daniel Forgues, Jun 30 2011

Definition corrected by Jonathan Sondow, Sep 18 2011

Status

approved