Riemann zeta function/Record lists: Difference between revisions
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Revision as of 20:15, 11 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 these correspond directly to peaks.
| Record type | ||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 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 | ||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 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-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
For σ = 2, the record lists for real zeta are identical to those for absolute zeta out to 8539edo.
| 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) | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ | ★ |