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Equivariant Grothendieck-Riemann-Roch and localization in operational K-theory
(Mathematical Sciences Publishers, 2021-01-01)
We produce a Grothendieck transformation from bivariant operational K-theory to Chow, with a Riemann-Roch formula that generalizes classical Grothendieck-Verdier-Riemann-Roch. We also produce Grothendieck transformations and Riemann-Roch formulas that generalize the classical Adams-Riemann-Roch and equivariant localization theorems. As applications, we exhibit a projective toric variety X whose equivariant K-theory of vector bundles does not surject onto its ordinary K-theory, and describe the operational K-theory of spherical varieties in terms of fixed-point data. In an appendix, Vezzosi studies operational K-theory of derived schemes and constructs a Grothendieck transformation from bivariant algebraic K-theory of relatively perfect complexes to bivariant operational K-theory.
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Bounding zeta on the 1-line under the partial Riemann hypothesis
(Cambridge University Press, 2024-10-01)
We provide explicit bounds for the Riemann zeta-function on the line, assuming that the Riemann hypothesis holds up to height T. In particular, we improve some bounds in finite regions for the logarithmic derivative and the reciprocal of the Riemann zeta-function.
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Remarks on a formula of Ramanujan
(Cambridge University Press, 2025-08-01)
Assuming an averaged form of Mertens’ conjecture and that the ordinates of the non-trivial zeros of the Riemann zeta function are linearly independent over the rationals, we analyse the finer structure of the terms in a well-known formula of Ramanujan.
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Explicit conditional bounds for $\zeta(s)$ at the edge of the critical strip
(European Mathematical Society, 2026-06-19)
In this paper, we obtain explicit bounds for the real part of the logarithmic derivative of the Riemann zeta-function on the line \operatorname{Re}s=1 , assuming the Riemann hypothesis. The proof combines the Guinand–Weil explicit formula with extremal bandlimited majorants and minorants for the Poisson kernel. As an application, we revisit the classical estimates of Littlewood for the modulus of the Riemann zeta-function and of its reciprocal on the line \operatorname{Re}{s}=1 , and derive a slight refinement of the bounds of Lamzouri, Li, and Soundararajan. In addition, we establish an explicit bound for the modulus of the logarithmic derivative of the Riemann zeta-function on the line \operatorname{Re}{s}=1 under the Riemann hypothesis, improving the lower-order term in a result of Chirre, Hagen, and Simonič.
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Explicit conditional bounds for $ζ(s)$ at the edge of the critical strip
(Cornell University, 2026-02-05)
In this paper, we obtain explicit bounds for the real part of the logarithmic derivative of the Riemann zeta-function on the line $\re s=1$, assuming the Riemann hypothesis. The proof combines the Guinand--Weil explicit formula with extremal bandlimited majorants and minorants for the Poisson kernel. As an application, we revisit the classical estimates of Littlewood for the modulus of the Riemann zeta-function and of its reciprocal on the line $\re{s}=1$, and derive a slight refinement of the bounds of Lamzouri, Li, and Soundararajan. In addition, we establish an explicit bound for the modulus of the logarithmic derivative of the Riemann zeta-function on the line $\re{s}=1$ under the Riemann hypothesis, improving the lower-order term in a result of Chirre, Valås, and Simonič.
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