Tootfinder

Secondary terms in the counting functions of quartic fields II
Arul Shankar, Jacob Tsimerman
https://arxiv.org/abs/2508.08527 https://arxiv.org/pdf/2508.08…

Secondary terms in the counting functions of quartic fields II
We determine the smoothed counts of $S_4$-quartic fields with bounded discriminant, satisfying any finite specified set of local conditions, as the sum of two main terms with a power saving error term. We also prove an analogous result for quartic rings (weighted by the number of cubic resolvents), deducing as a consequence that the Shintani zeta functions associated to the prehomogeneous vector space $\mathbb{C}^2\otimes\mathrm{Sym}^2(\mathbb{C}^3)$ have at most a simple pole at $s=5/6$.

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…

Three-dimensional numerical study on hydrogen bubble growth at electrode
Wei Qin, Tian Long, Jacob Maarek, St\'ephane Zaleski
https://arxiv.org/abs/2507.15582 https://

Three-dimensional numerical study on hydrogen bubble growth at electrode
3D direct numerical simulation of electrolysis is applied to investigate the growth and detachment of bubbles at electrodes. The moving gas-liquid interface is modeled employing the VOF-based method. To ensure the accuracy of the simulations, a mesh-independence study has been performed. The simulations include the growth phase of the bubbles followed by their detachment from the electrode surface and the results are validated with analytical models and experimental data. The bubble g…