Riemann zeta function/Record lists: Difference between revisions
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Revision as of 03:38, 12 April 2025
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 | 2 | 3 | 5 | 7 | 12 | 19 | 31 | 41 | 46 | 53 | 72 | 130 | 171 | 224 | 270 | 311 | 764 | 954 | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Zeta integral | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | |||||||||||||
| Zeta gap | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ |
σ = 1/2
| Record type | ||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Absolute zeta peak | ||||||||||||||||||||||||||||
| Absolute zeta integer | ||||||||||||||||||||||||||||
| Real zeta peak | ||||||||||||||||||||||||||||
| Real zeta integer |
σ = 1
| Record type | 1 | 2 | 3 | 5 | 7 | 10 | 12 | 19 | 22 | 31 | 41 | 53 | 72 | 118 | 130 | 171 | 224 | 270 | 494 | 742 | 764 | 1106 | 1236 | 1395 | 1448 | 1578 | 2460 | 2684 | 3395 | 5585 | 7033 | 8269 | 8539 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Absolute zeta peak | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | |||
| Absolute zeta integer | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | |||||||||
| Real zeta peak | |||||||||||||||||||||||||||||||||
| Real zeta integer |
No-threes
| Record type | 1 | 2 | 3 | 4 | 6 | 16 | 21 | 25 | 31 | 37 | 93 | 109 | 124 | 233 | 457 | 581 | 618 | 851 | 1028 | 1121 | 1199 | 1702 | 1963 | 2917 | 3535 | 4880 | 5352 | 8232 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 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 | 71 | 88 | 131 | 245 | 415 | 460 | 630 | 712 | 800 | 1342 | 1473 | 1757 | 2426 | 3624 | 3755 | 4424 | 5136 | 6181 | 8682 | 9805 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Absolute zeta peak | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ||||||
| Absolute zeta integer | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ||||||||
| Real zeta peak | ||||||||||||||||||||||||||||
| Real zeta integer |
σ = 3/2
| Record type | 1 | 2 | 3 | 5 | 7 | 10 | 12 | 19 | 31 | 41 | 53 | 72 | 130 | 171 | 224 | 270 | 494 | 1106 | 1578 | 2460 | 2684 | 3395 | 5585 | 8269 | 8539 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Absolute zeta peak | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | |||||
| Absolute zeta integer | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ||||
| Real zeta peak | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | |||||
| Real zeta integer | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ |
No-threes
| Record type | 1 | 2 | 3 | 6 | 25 | 31 | 37 | 93 | 109 | 124 | 233 | 457 | 581 | 618 | 1199 | 2917 | 3535 | 4880 | 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 | 415 | 712 | 800 | 914 | 1342 | 1757 | 2426 | 3624 | 3755 | 6181 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Absolute zeta peak | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ||||
| Absolute zeta integer | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | |||||
| Real zeta peak | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ||
| Real zeta integer | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ |
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 | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | |||||||
| Real zeta peak (cosine mu peak) | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ||||||
| Real zeta integer (cosine mu 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) | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ |