A013629 - OEIS

%I #46 Mar 27 2023 22:36:51

%S 14,21,25,30,32,37,40,43,48,49,52,56,59,60,65,67,69,72,75,77,79,82,84,

%T 87,88,92,94,95,98,101,103,105,107,111,111,114,116,118,121,122,124,

%U 127,129,131,133,134,138,139,141,143,146,147,150,150,153,156,157,158,161

%N Floor of imaginary parts of nontrivial zeros of Riemann zeta function.

%D H. M. Edwards, Riemann's Zeta Function, Academic Press, NY, 1974, p. 96.

%D 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.

%H Charles R Greathouse IV, <a href="/A013629/b013629.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Z#zeta_function">Index entries for zeta function</a>.

%F a(n) ~ 2*Pi*n/log n. - _Charles R Greathouse IV_, Jun 30 2011

%F 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

%F a(n) = A092783(n) - 1. - _M. F. Hasler_, Nov 23 2018

%e 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

%t Table[Floor[Im[ZetaZero[n]]], {n, 60}] (* _Alonso del Arte_, Feb 07 2011 *)

%o (Sage)

%o def A013629_list(n):

%o     Z = lcalc.zeros(n)

%o     return [floor(z) for z in Z]

%o A013629_list(50) # _Peter Luschny_, May 02 2014

%o (PARI) lfunzeros(lzeta,100)\1 \\ _Charles R Greathouse IV_, Mar 10 2016

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

%Y 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).

%K nonn

%O 1,1

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

%E Edited by _Daniel Forgues_, Jun 30 2011

%E Definition corrected by _Jonathan Sondow_, Sep 18 2011