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A329742
Indices n of Riemann zeta zeros for successive records of the normalized delta defined as d(n) = (z(n+1)-z(n))*(log(z(n)/(2Pi))/(2Pi)) where z(n) is the imaginary part of the n-th Riemann zero.
3
1, 3, 5, 8, 14, 25, 33, 64, 126, 213, 256, 379, 1704, 1935, 2292, 8571, 10942, 12347, 13298, 15323, 36719, 46589, 103715, 185013, 880694, 1493008, 3206674, 12534781, 14145077, 22653912, 24246374, 33742399, 65336924, 298466597, 566415148, 1938289664, 2122614029, 4020755339, 4219726754, 16265396008, 17003807756
OFFSET
1,2
COMMENTS
No more records up to n = 103800788359.
d(17003807756) = 4.3018209763411.
Successive records occur when gaps between two successive zeros are large.
Recent record of normalized delta computed by Hiary at 2011 occurs for n=436677148707320393224019748290912 where d(n) = 5.77979.
Conjectural next term: 77528045597.
Indices of zeros for successive minimal records of the normalized delta see A328656.
LINKS
Ghaith Ayesh Hiary, Fast methods to compute the Riemann zeta function, Ann. of Math. (2) 174 (2011), no. 2, 891-946. MR 2831110 (2012g:11154).
EXAMPLE
n | a(n) | d(n)
---+---------+---------
1 | 1 | 0.88871
2 | 3 | 1.19034
3 | 5 | 1.22634
4 | 8 | 1.43763
5 | 14 | 1.54672
6 | 25 | 1.55244
7 | 33 | 1.74300
8 | 64 | 1.83656
9 | 126 | 1.95400
10 | 213 | 1.95626
11 | 256 | 1.99205
12 | 379 | 2.20138
13 | 1704 | 2.20198
14 | 1935 | 2.45843
15 | 2292 | 2.46772
16 | 8571 | 2.48347
17 | 10942 | 2.50594
18 | 12347 | 2.50648
19 | 13298 | 2.52517
20 | 15323 | 2.67728
21 | 36719 | 2.76188
22 | 46589 | 2.80523
23 | 103715 | 2.83121
24 | 185013 | 3.11058
25 | 880694 | 3.21426
26 | 1493008 | 3.30347
MATHEMATICA
prec = 30; max = 0; aa = {}; Do[kk = N[Im[(ZetaZero[n + 1] - ZetaZero[n])], prec] (Log[N[Im[ZetaZero[n]], prec]/(2 Pi)]/(2 Pi));
If[kk > max, max = kk; AppendTo[aa, n]], {n, 1, 2000000}]; aa
CROSSREFS
KEYWORD
nonn
AUTHOR
Artur Jasinski, Nov 20 2019
EXTENSIONS
a(27)-a(41) computed by David Platt, Jan 03 2020
STATUS
approved