The Weak Gram Law for Hecke $L$-functions

Hardy Z-函数的新判别式与修正的Gram定律

A New Discriminant for the Hardy Z-Function and the Corrected Gram's law

本文引入源于代数几何的变分框架分析Hardy Z-函数，定义了度量连续零点实数性的新判别式$Δ_n(\overline{a})$，发现修正的Gram定律与G-B-G排斥关系，为黎曼假设提供了新的合理性论证。

On Benford's Law for multiplicative functions

We provide a criterion to determine whether a real multiplicative function is a strong Benford sequence. The criterion implies that the $k$-divisor functions, where $k \neq 10^j$, and Hecke eigenvalues of newforms, such as Ramanujan tau function, are strong Benford. Moreover, we deduce from the criterion that the collection of multiplicative functions which are not strong Benford forms a group under pointwise multiplication. In contrast to earlier work, our approach is based on Halász's Theorem.

Non-abelian Abel's theorems and quaternionic rotation

In order to compute with $l$--adic sheaves or crystals on a line over $\mathbb{F} _q$ a low-technology alternative to the traditional computation with the Hecke operators on the automorphic side could be helpful. A program which has evolved over the years in our discussions with M. Kontsevich centers around the concept that, in the geometric case, there must exist certain multiplication laws on the Galois--representation side that could be thought of as precursors of the automorphic lifts: non-abelian Abel's theorems, and their restrictions to diagonal, Clausen identities. To a varying extent,…

Conductor zeta function for the GL(2) universal family

We obtain a Weyl law with power savings for the universal families of cuspidal automorphic representations, ordered by analytic conductor, of $\mathrm{GL}_2$ over $\mathbb{Q}$, as well as for Hecke characters over any number field. The method proceeds by establishing the requisite analytic properties of the underlying conductor zeta function.

Central limit theorems for elliptic curves and modular forms with smooth weight functions

The second and third-named authors (arXiv:1705.04115) established a Central Limit Theorem for the error term in the Sato-Tate law for families of modular forms. This method was adapted to families of elliptic curves in by the first and second-named authors (arXiv:1705.09229). In this context, a Central Limit Theorem was established only under a strong hypothesis going beyond the Riemann Hypothesis. In the present paper, we consider a smoothed version of the Sato-Tate conjecture, which allows us to overcome several limitations. In particular, for the smoothed version, we are able to establish a…

正交和辛$L$ -函数族中对数的分布

The distribution of the logarithm in an orthogonal and a symplectic family of $L$-functions

我们考虑在正交族${L(s,f)}_{f \in H_k}$中的中心值$\log L(1/2)$的对数，其中$H_k$是$SL_2(\mathbb{Z})$的权重$k$ Hecke-本征尖点形式的集合，以及在辛族${L(s,χ_{8d})}_{d \asymp D}$中的中心值$\log L(1/2)$的对数，其中$χ_{8d}$是与基本判别式$8d$相关联的实数。无条件地，我们证明了这两个分布是渐近有界以上的高斯分布，在第一种情况下的平均值$-\frac{1}{2} \log \log k$和方差$\log \log k$，并在第二种情况下的平均值$\frac{1}{2}\log \log D$和方差$\log \log D$。假设这两个家庭的黎曼和零密度假设，我们得到了充分的正常的法律在这两个家庭，证实了一个猜想的基廷和Snaith。

算术函数与Kloosterman和的混合矩

Mixed moments of arithmetic functions and Kloosterman sums

本文推广Nair--Tenenbaum估计至Kloosterman和的混合矩，证明其渐近公式，并给出Hecke特征值混合矩上界、大Kloosterman值频率及谱计数方差下界。