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Abstract We prove that the rational cohomology group$$H^{11}(\overline {\mathcal {M}}_{g,n})$$vanishes unless$$g = 1$$and$$n \geq 11$$. We show furthermore that$$H^k(\overline {\mathcal {M}}_{g,n})$$is pure Hodge–Tate for all even$$k \leq 12$$and deduce that$$\# \overline {\mathcal {M}}_{g,n}(\mathbb {F}_q)$$is surprisingly well approximated by a polynomial inq. In addition, we use$$H^{11}(\overline {\mathcal {M}}_{1,11})$$and its image under Gysin push-forward for tautological maps to produce many new examples of moduli spaces of stable curves with nonvanishing odd cohomology and nontautological algebraic cycle classes in Chow cohomology.
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This content will become publicly available on February 10, 2026
Mean values of multiplicative functions and applications to residue-class distribution
Abstract We provide a uniform bound on the partial sums of multiplicative functions under very general hypotheses. As an application, we give a nearly optimal estimate for the count of$$n \le x$$for which the Alladi–Erdős function$$A(n) = \sum_{p^k \parallel n} k p$$takes values in a given residue class moduloq, whereqvaries uniformly up to a fixed power of$$\log x$$. We establish a similar result for the equidistribution of the Euler totient function$$\phi(n)$$among the coprime residues to the ‘correct’ moduliqthat vary uniformly in a similar range and also quantify the failure of equidistribution of the values of$$\phi(n)$$among the coprime residue classes to the ‘incorrect’ moduli.
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- Award ID(s):
- 2001581
- PAR ID:
- 10612597
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- Proceedings of the Edinburgh Mathematical Society
- ISSN:
- 0013-0915
- Page Range / eLocation ID:
- 1 to 19
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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