Tootfinder

On the connection between zero-free regions and the error term in the Prime Number Theorem
Frederik Broucke
https://arxiv.org/abs/2507.13780 https://

On the connection between zero-free regions and the error term in the Prime Number Theorem
We provide for a wide class of zero-free regions an upper bound for the error term in the Prime Number Theorem, refining works of Pintz (1980), Johnston (2024), and Révész (2024). Our method does not only apply to the Riemann zeta function, but to general Beurling zeta functions. Next we construct Beurling zeta functions having infinitely many zeros on a prescribed contour, and none to the right, for a wide class of such contours. We also deduce an oscillation result for the corresponding err…

A note on the singularity conjecture for infinite covolume discrete subgroups
Dongryul M. Kim, Andrew Zimmer
https://arxiv.org/abs/2508.05756 https://arxiv…

A note on the singularity conjecture for infinite covolume discrete subgroups
We consider random walks on semisimple Lie groups where the support of the step distribution generates (as a group) a Zariski dense discrete subgroup of infinite covolume. When the semisimple Lie group has property (T), we show that the stationary measure on the Furstenberg boundary is singular to the Lebesgue measure class. This result does not require any condition on the moment or symmetry of the step distribution. When the semisimple Lie group has rank one and the step distribution has a fi…

Singularity of Furstenberg measure for infinite covolume discrete subroups in higher rank
Homin Lee, Wouter Van LimBeek, Giulio Tiozzo
https://arxiv.org/abs/2508.06329 https://

Singularity of Furstenberg measure for infinite covolume discrete subroups in higher rank
We consider symmetric random walks on discrete, Zariski-dense subgroups $Γ$ of a semisimple Lie group $G$ with Property (T). We prove that if $Γ$ has infinite covolume, then the associated hitting measure on the Furstenberg boundary of $G$ is singular. This is in contrast to Furstenberg's discretization of Brownian motion to lattices, and it is the first result of this type when $G$ has higher rank.

On the characteristic form of $\mathfrak{g}$-valued zero-curvature representations
Jirina Jahnova
https://arxiv.org/abs/2508.01224 https://arxiv.org/pdf/25…

On the characteristic form of $\mathfrak{g}$-valued zero-curvature representations
We study $\mathfrak{g}$-valued zero-curvature representations (ZCRs) for partial differential equations in two independent variables from the perspective of their extension to the entire infinite jet space, focusing on their characteristic elements. Since conservation laws -- more precisely, conserved currents -- and their generating functions for a given equation are precisely the $\mathbb{R}$-valued ZCRs and their characteristic elements, a natural question arises: to what extent can results …

Fujita-Zariski decompositions on some product threefolds
Mihai Fulger, Victor Lozovanu
https://arxiv.org/abs/2506.17956 https://arxiv…

Fujita-Zariski decompositions on some product threefolds
We use explicit blow-ups and computations of birational Fujita-Zariski decompositions to determine generic infinitesimal Newton-Okounkov bodies for box-product ample polarizations on three classes of spaces: product between a curve and the projective plane, products of three curves, and the product between a curve and a Jacobian surface. In particular we compute the volume of many big but non-nef divisors on blow-ups of these threefolds.

Bodhisattvas are embodiments of inspiring qualities, avatars of the possible. Imagine thousands of bodhisattvas walking together down Fifth Avenue and into the Village, past the Stonewall Inn. Imagine the infinite variety of their forms. Imagine their celebration!
https://salrandolph.substack.com/p/…

Purmutation--invariant Niven numbers
Hui-Ling Wu, S. Y. Lou
https://arxiv.org/abs/2508.01611 https://arxiv.org/pdf/2508.01611

Purmutation--invariant Niven numbers
This paper introduces permutation-invariant Niven numbers--a novel class of Niven numbers where all digit permutations (with leading zeros automatically ignored) must retain the Niven property. We demonstrate that there exist infinitely many such numbers and that their magnitude is unbounded. Furthermore, we present an exhaustive search method for identifying permutation--invariant Niven numbers.

Number Field Analogue of Jacobi Theta Relation And Zeros of Dedekind zeta function on Re$(s)=1/2$
Diksha Rani Bansal, Bibekananda Maji
https://arxiv.org/abs/2507.18121 https://

Number Field Analogue of Jacobi Theta Relation And Zeros of Dedekind zeta function on Re$(s)=1/2$
In 1914, Hardy proved that there are infinitely many non-trivial zeros of the Riemann zeta function $ζ(s)$ on the critical line Re$(s)=1/2$ using the Jacobi theta relation. In this paper, we first establish a number field analogue of the Jacobi theta relation and as an application, we show the existence of infinitely many non-trivial zeros of the Dedekind zeta function $ζ_\mathbb{F}(s)$ on Re$(s)=1/2$, for any number field $\mathbb{F}$. Quite interestingly, we also prove that the Jacobi theta…

A Microlocal Theory for Zariski-Constructible Sheaves on Rigid Analytic Varieties
Tong Zhou
https://arxiv.org/abs/2507.18604 https://arxiv.org/pdf/2507.186…

A Microlocal Theory for Zariski-Constructible Sheaves on Rigid Analytic Varieties
We develop a microlocal theory, in the sense of Kashiwara-Schapira, for Zariski-constructible sheaves on rigid analytic varieties. We define and study monodromic sheaves, the monodromic Fourier transform, specialisation, microlocalisation, micro-hom, and singular support in this context. Some questions and conjectures are formulated in the end. The appendix contains infinity-categorical characterisations of monodromic sheaves.

Approximation of the number of nonunimodular zeros of a self-reciprocal polynomial
Dragan Stankov
https://arxiv.org/abs/2509.01015 https://arxiv.org/pdf/25…

Approximation of the number of nonunimodular zeros of a self-reciprocal polynomial
We introduce the ratio of the number of roots, not equal to 1 in modulus, of a reciprocal polynomial $R_d(x)$ to its degree $d$. For some sequences of reciprocal polynomials we show that the ratio has a limit $L$ when $d$ tends to infinity. Each of these sequences is defined using a two-variable polynomial $P(x,y)$ so that $R_d(x) = P(x,x^n)$. We present a few methods for approximation of the limit ratio, some of them we originally developed. In a previous paper we have calculated the e…