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A329742
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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.
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3
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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
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MATHEMATICA
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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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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a(27)-a(41) computed by David Platt, Jan 03 2020
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STATUS
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approved
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