Definitions
Absolute
Based on the absolute value of zeta(s), which is more mathematically convenient to calculate at σ = 0.5.
Real
Based on the real part of zeta(s), which directly follows from the derivation of zeta as an equal tuning badness metric.
Zero-based metrics
Zeta metrics corresponding to regions between zeros on the critical line (σ = 0.5). The Riemann hypothesis is the equivalent to the statement that regions between zeros correspond directly to peaks.
| Record type
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2
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3
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5
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7
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12
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19
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31
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41
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46
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53
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72
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130
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171
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224
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270
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311
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764
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954
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| Zeta integral
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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| Zeta gap
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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σ = 1/2
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| Absolute zeta peak
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| Absolute zeta integer
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| Real zeta peak
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| Real zeta integer
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σ = 1
No-threes
No-twos tritave divisions
σ = 3/2
| Record type
|
1
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2
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3
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5
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7
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10
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12
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19
|
31
|
41
|
53
|
72
|
130
|
171
|
224
|
270
|
494
|
1106
|
1578
|
2460
|
2684
|
3395
|
5585
|
8269
|
8539
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| Absolute zeta peak
|
★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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| Absolute zeta integer
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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| Real zeta peak
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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| Real zeta integer
|
★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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No-threes
| Record type
|
1
|
2
|
3
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6
|
25
|
31
|
37
|
93
|
109
|
124
|
233
|
457
|
581
|
618
|
1199
|
2917
|
3535
|
4880
|
5352
|
6312
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| Absolute zeta peak
|
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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| Absolute zeta integer
|
★
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★
|
★
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★
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★
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★
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★
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★
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★
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★
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★
|
★
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★
|
★
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| Real zeta peak
|
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★
|
★
|
★
|
★
|
★
|
★
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★
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★
|
★
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★
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★
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★
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★
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★
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|
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★
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| Real zeta integer
|
★
|
|
★
|
★
|
|
★
|
★
|
★
|
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★
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★
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★
|
|
★
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|
★
|
★
|
|
★
|
No-twos tritave divisions
| Record type
|
1
|
2
|
4
|
9
|
13
|
17
|
26
|
39
|
88
|
114
|
131
|
415
|
712
|
800
|
914
|
1342
|
1757
|
2426
|
3624
|
3755
|
6181
|
| Absolute zeta peak
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
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★
|
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★
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|
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★
|
★
|
★
|
★
|
★
|
★
|
| Absolute zeta integer
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
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★
|
★
|
★
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|
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★
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★
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★
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★
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★
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★
|
| Real zeta peak
|
★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
|
★
|
★
|
| Real zeta integer
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
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★
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★
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★
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★
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★
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★
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★
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★
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★
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★
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σ = 2
| Record type
|
1
|
2
|
3
|
5
|
7
|
10
|
12
|
19
|
31
|
41
|
53
|
118
|
130
|
171
|
270
|
494
|
1106
|
1578
|
2460
|
2684
|
3395
|
5585
|
6079
|
8269
|
8539
|
| Absolute zeta peak
|
★
|
★
|
★
|
★
|
★
|
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
|
|
★
|
★
|
★
|
★
|
|
★
|
★
|
★
|
| Absolute zeta integer
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
|
★
|
★
|
★
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★
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★
|
★
|
★
|
★
|
|
★
|
|
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★
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|
|
★
|
| Real zeta peak (cosine mu peak)
|
★
|
★
|
★
|
★
|
★
|
|
★
|
★
|
★
|
★
|
★
|
|
★
|
★
|
★
|
|
|
★
|
★
|
★
|
★
|
|
★
|
★
|
|
| Real zeta integer (cosine mu integer)
|
★
|
★
|
|
★
|
★
|
★
|
★
|
|
★
|
★
|
★
|
|
★
|
★
|
★
|
★
|
★
|
|
★
|
|
|
★
|
|
|
★
|
No-threes
| Record type
|
1
|
2
|
3
|
6
|
31
|
37
|
93
|
124
|
233
|
581
|
1199
|
2917
|
3535
|
5352
|
6312
|
| Absolute zeta peak
|
|
★
|
★
|
★
|
★
|
★
|
|
★
|
★
|
★
|
★
|
★
|
|
★
|
|
| Absolute zeta integer
|
★
|
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
|
|
★
|
|
★
|
| Real zeta peak
|
|
★
|
★
|
★
|
★
|
★
|
|
★
|
★
|
★
|
★
|
★
|
|
★
|
|
| Real zeta integer
|
★
|
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
|
|
★
|
|
★
|
No-twos tritave divisions
| Record type
|
1
|
2
|
4
|
9
|
13
|
17
|
26
|
39
|
88
|
114
|
131
|
800
|
914
|
1342
|
1757
|
2426
|
3624
|
3755
|
6181
|
| Absolute zeta peak
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
| Absolute zeta integer
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
|
★
|
★
|
★
|
★
|
|
★
|
★
|
★
|
★
|
|
★
|
| Real zeta peak (cosine mu peak)
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
| Real zeta integer (cosine mu integer)
|
★
|
★
|
★
|
★
|
★
|
★
|
★
|
|
★
|
★
|
★
|
★
|
|
★
|
★
|
|
★
|
|
★
|