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Abstract We obtain conditional upper bounds for negative discrete moments of the derivative of the Riemann zeta‐function averaged over a subfamily of zeros of the zeta function that is expected to be arbitrarily close to full density inside the set of all zeros. For , our bounds for the ‐th moments are expected to be almost optimal. Assuming a conjecture about the maximum size of the argument of the zeta function on the critical line, we obtain upper bounds for these negative moments of the same strength while summing over a larger subfamily of zeta zeros.
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Quadratic enrichment of the logarithmic derivative of the zeta function
We define an enrichment of the logarithmic derivative of the zeta function of a variety over a finite field to a power series with coefficients in the Grothendieck–Witt group. We show that this enrichment is related to the topology of the real points of a lift. For cellular schemes over a field, we prove a rationality result for this enriched logarithmic derivative of the zeta function as an analogue of part of the Weil conjectures. We also compute several examples, including toric varieties, and show that the enrichment is a motivic measure.
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- PAR ID:
- 10546429
- Publisher / Repository:
- American Mathematical Society (AMS)
- Date Published:
- Journal Name:
- Transactions of the American Mathematical Society, Series B
- Volume:
- 11
- Issue:
- 33
- ISSN:
- 2330-0000
- Format(s):
- Medium: X Size: p. 1183-1225
- Size(s):
- p. 1183-1225
- Sponsoring Org:
- National Science Foundation
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