Mathematika最新文献

We construct families of optimal Hardy-weights for a subcritical linear second-order elliptic operator $��(�P�,�B�)�$ with degenerate mixed boundary conditions. By an optimal Hardy-weight for a subcritical operator we mean a nonzero nonnegative weight function W such that $��(�P�-�W�,�B�)�$ is critical, and null-critical with respect to W. Our results rely on a recently developed criticality theory for positive solutions of the corresponding mixed boundary value problem.

We prove a dimension-free stability result for polydisc slicing due to Oleszkiewicz and Pełczyński. Intriguingly, compared to the real case, there is an additional asymptotic maximizer. In addition to Fourier-analytic bounds, we crucially rely on a self-improving feature of polydisc slicing, established via probabilistic arguments.

Let $��\lambda �\left(�n�\right)�$ denote the exponent of the multiplicative group modulo n. We show that when q is odd, each coprime residue class modulo q is hit equally often by $��\lambda �\left(�n�\right)�$ as n varies. Under the stronger assumption that $��gcd�(�q�,�6�)�=�1�$, we prove that equidistribution persists throughout a Siegel–Walfisz-type range of uniformity. By similar methods we show that $��\lambda �\left(�n�\right)�$ obeys Benford's leading digit law with respect to natural density. Moreover, if we assume Generalized Riemann Hypothesis, then Benford's law holds for the order of a mod n, for any fixed integer $��a�\notin �\{�0�,�\pm �1�\}�$.

For a prime number p and integer x with $��gcd�(�x�,�p�)�=�1�$, let $�\overline{x}$ denote the multiplicative inverse of x modulo p. In this paper, we are interested in the problem of distribution modulo p of the sequence

For any fixed $��A�>�1�$ and for any sufficiently large integer N, there exists a prime number p with

For any fixed positive , there exists a positive constant c such that the following holds: for any sufficiently large integer N there is a prime number $��p�>�N�$ such that

Given a norm ν on $�R^2$, the set of ν-Dirichlet improvable numbers $�{\mathrm{DI}}_{\nu }$ was defined and studied in the papers (Andersen and Duke, Acta Arith. 198 (2021) 37–75 and Kleinbock and Rao, Internat. Math. Res. Notices 2022 (2022) 5617–5657). When ν is the supremum norm, $��{\mathrm{DI}}_{\nu }�=�\mathrm{BA}�\cup �Q�$, where $�\mathrm{BA}$ is the set of badly approximable numbers. Each of the sets $�{\mathrm{DI}}_{\nu }$, like $�\mathrm{BA}$, is of measure zero and satisfies the winning property of Schmidt. Hence for every norm ν, $��\mathrm{BA}�\cap �{\mathrm{DI}}_{\nu }�$ is winning and thus has full Hausdorff dimension. In this article, we prove the following dichotomy phenomenon: either $��\mathrm{BA}�\subset �{\mathrm{DI}}_{\nu }�$ or else $��\mathrm{BA}�\setminus �{\mathrm{DI}}_{\nu }�$

We establish lower bounds for the discrete 2kth moment of the derivative of the Riemann zeta function at nontrivial zeros for all under the Riemann hypothesis and the assumption that all zeros of $��\zeta �\left(�s�\right)�$ are simple.

We study the moments of L-functions associated with primitive cusp forms, in the weight aspect. In particular, we obtain an asymptotic formula for the twisted moments of a long Dirichlet polynomial with modular coefficients. This result, which is conditional on the Generalized Lindelöf Hypothesis, agrees with the prediction of the recipe by Conrey, Farmer, Keating, Rubinstein and Snaith.

If Z is an open subscheme of $��Spec�Z�$, X is a sufficiently nice Z-model of a smooth curve over $�Q$, and p is a closed point of Z, the Chabauty–Kim method leads to the construction of locally analytic functions on $��X�\left(�Z_p�\right)�$ which vanish on $��X�\left(�Z�\right)�$; we call such functions “Kim functions”. At least in broad outline, the method generalizes readily to higher dimensions. In fact, in some sense, the surface M_{0, 5} should be easier than the previously studied curve $��M_{�0�,�4�}�=�P^1�\setminus ��\{�0�,�1�,�\infty �\}��$ since its points are closely related to those of M_{0, 4}, yet they face a further condition to integrality. This is mirrored by a certain weight advantage we encounter, because of which, M_{0, 5} possesses new Kim functions not coming from M_{0, 4}. Here we focus on the case “$��$

We investigate the question whether a scattered compact topological space K such that $��C�\left(�K�\right)�$ has a norming Markushevich basis (M-basis, for short) must be Eberlein. This question originates from the recent solution, due to Hájek, Todorčević and the authors, to an open problem from the 1990s, due to Godefroy. Our prime tool consists in proving that $��C�\left(��[�0�,�{\omega }_1�]��\right)�$ does not embed in a Banach space with a norming M-basis, thereby generalising a result due to Alexandrov and Plichko. Subsequently, we give sufficient conditions on a compact K for $��C�\left(�K�\right)�$ not to embed in a Banach space with a norming M-basis. Examples of such conditions are that K is a zero-dimensional compact space with a P-point, or a compact tree of height at least $��{\omega }_1�+�1�$. In particular, this allows us to answer the said question in the case when K is a tree and to obtain a rather general result for Valdivia compacta. Finally, we give some structural results for scattered compact trees; in particular, we prove that scattered trees of height less than ω₂ are Valdivia.