Zeros of Partial Sums of the Riemann Zeta Function
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Bibliographic record
Abstract
The semiperiodic behavior of the zeta function ζ(s) and its partial sums ζN(s) as a function of the imaginary coordinate has been long established. In fact, the zeros of a ζN(s), when reduced into imaginary periods derived from primes less than or equal to N, establish regular patterns. We show that these zeros can be embedded as a dense set in the period of a surface in ℝk+1, where k is the number of primes in the expansion. This enables us, for example, to establish the lower bound for the real parts of zeros of ζN(s) for prime N and justifies the use of methods of calculus to find expressions for the bounding curves for sets of reduced zeros in ℂ.
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Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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