arXiv subjects
Publications and source records attributed to Martin Čech.
We study the first moment of primitive quadratic Dirichlet $L$-functions. Assuming the Riemann hypothesis and the generalized Lindel\"of hypothesis, we obtain an asymptotic formula at the central point with error $O(X^{1/4+\epsilon})$, and identify lower-order terms of size $X^{1/3}\log X$.
We study the one-level density of low-lying zeros in the family of Maass form $L$-functions of prime level $N$ tending to infinity. Generalizing the influential work of Iwaniec, Luo and Sarnak to this context, Alpoge et al. have proven the Katz-Sarnak prediction for test functions whose Fourier transform is supported in $(-\frac32,\frac32)$. In this paper, we extend the unconditional admissible support to $(-\frac{15}8,\frac{15}8)$. The key tools in our approach are analytic estimates for integrals appearing in the Kutznetsov trace formula, as well as a reduction to bounds on Dirichlet polynomials, which eventually are obtained from the large sieve and the fourth moment bound for Dirichlet $L$-functions. Assuming the Grand Density Conjecture, we extend the admissible support to $(-2,2)$. In addition, we show that the same techniques also allow for an unconditional improvement of the admissible support in the corresponding family of $L$-functions attached to holomorphic forms.
The increasing complexity of natural disaster incidents demands innovative technological solutions to support first responders in their efforts. This paper introduces the TRIFFID system, a comprehensive technical framework that integrates unmanned ground and aerial vehicles with advanced artificial intelligence functionalities to enhance disaster response capabilities across wildfires, urban floods, and post-earthquake search and rescue missions. By leveraging state-of-the-art autonomous navigation, semantic perception, and human-robot interaction technologies, TRIFFID provides a sophisticated system composed of the following key components: hybrid robotic platform, centralized ground station, custom communication infrastructure, and smartphone application. The defined research and development activities demonstrate how deep neural networks, knowledge graphs, and multimodal information fusion can enable robots to autonomously navigate and analyze disaster environments, reducing personnel risks and accelerating response times. The proposed system enhances emergency response teams by providing advanced mission planning, safety monitoring, and adaptive task execution capabilities. Moreover, it ensures real-time situational awareness and operational support in complex and risky situations, facilitating rapid and precise information collection and coordinated actions.
Mollifiers are used in a variety of contexts, for instance to study the non-vanishing of $L$-functions. In this paper, we study the general question of finding optimal mollifiers and provide criteria to identify them provided the corresponding mollified moments can be computed. As an application, we study the non-vanishing of central values of Dirichlet $L$-functions. In particular we show that the Michel-Vanderkam mollifier is optimal in a wide class of balanced two-piece mollifiers as well as provide a new proof that the Iwaniec-Sarnak mollifier is optimal in a wide class of one-piece mollifiers.
We devise heuristics using multiple Dirichlet series to predict asymptotic formulas for shifted moments of (1) the family of Dirichlet $L$-functions of all even primitive characters of conductor $\leq Q$, with $Q$ a parameter tending to infinity, (2) the family of quadratic Dirichlet $L$-functions, (3) the family of quadratic twists of an $L$-function associated to a fixed Hecke eigencuspform for the full modular group, and (4) the family of quadratic twists of an $L$-function of a fixed arbitrary elliptic curve over $\mathbb{Q}$ that has a non-square conductor. For each of these families, the resulting predictions agree with the predictions of the recipe developed by Conrey, Farmer, Keating, Rubinstein, and Snaith, except for (4), where the recipe requires a slight modification due to a correlation between the Dirichlet coefficients and the root number of the corresponding $L$-functions. We find a one-to-one correspondence between the residues from the multiple Dirichlet series analysis and the terms from the recipe prediction.
We consider the multiple Dirichlet series associated to the $k$th moment of real Dirichlet $L$-functions, and prove that it has a meromorphic continuation to a specific region in $\mathbb{C}^{k+1}$, which is conditional under the generalized Lindel\"of hypothesis for $k\geq 5$. As a corollary, we obtain asymptotic formulas for the first three moments with a power-saving error term, and detect the 0- and 1-swap terms in related problems for any $k$ (conditionally under the Generalized Lindel\"of Hypothesis), recovering the recent results of Conrey and Rodgers on long Dirichlet polynomials. The advantage of our method is its simplicity, since we don't need to modify the multiple Dirichlet series to obtain its meromorphic continuation. As a result, we obtain the asymptotic formulas directly in the form as they appear in the recipe predictions of Conrey, Farmer, Keating, Rubinstein and Snaith.
Conrey, Farmer and Zirnbauer introduced a recipe to find asymptotic formulas for the sum of ratios of products of shifted L-functions. These ratios conjectures are very powerful and can be used to determine many statistics of L-functions, including moments or statistics about the distribution of zeros. We consider the family of real Dirichlet characters, and use multiple Dirichlet series to prove the ratios conjectures with one shift in the numerator and denominator in some range of the shifts. This range can be improved by extending the family to include non-primitive characters. All of the results are conditional under the Generalized Riemann hypothesis. This extended range is good enough to enable us to compute an asymptotic formula for the sum of shifted logarithmic derivatives for some range of the shift.
We study non-vanishing of Dirichlet $L$-functions at the central point under the unlikely assumption that there exists an exceptional Dirichlet character. In particular we prove that if $ψ$ is a real primitive character modulo $D \in \mathbb{N}$ with $L(1, ψ) \ll (\log D)^{-25-\varepsilon}$, then, for any prime $q \in [D^{300}, D^{O(1)}]$, one has $L(1/2, χ) \neq 0$ for almost all Dirichlet characters $χ\pmod{q}$.
We study the double character sum $\sum\limits_{\substack{m\leq X,\\m\odd}}\sum\limits_{\substack{n\leq Y,\\n\odd}}\leg mn$ and its smoothly weighted counterpart. An asymptotic formula with power saving error term was obtained by Conrey, Farmer and Soundararajan by applying the Poisson summation formula. The result is interesting, because the main term involves a non-smooth function. In this paper, we apply the inverse Mellin transform twice and study the resulting double integral that involves a double Dirichlet series. This method has two advantages -- it leads to a better error term, and the surprising main term naturally arises from three residues of the double Dirichlet series.
The aim of this article is to study (additively) indecomposable algebraic integers $\mathcal O_K$ of biquadratic number fields $K$ and universal totally positive quadratic forms with coefficients in $\mathcal O_K$. There are given sufficient conditions for an indecomposable element of a quadratic subfield to remain indecomposable in the biquadratic number field $K$. Furthermore, estimates are proven which enable algorithmization of the method of escalation over $K$. These are used to prove, over two particular biquadratic number fields $\mathbb{Q}(\sqrt{2}, \sqrt{3})$ and $\mathbb{Q}(\sqrt{6}, \sqrt{19})$, a lower bound on the number of variables of a universal quadratic forms, verifying Kitaoka's conjecture.