Morten Skarsholm Risager
We define the counting function for non-analytic (Maass) newforms of Hecke congruence groups_Gamma_0(M)_. We then calculate the three main terms of this counting function and give necessary and sufficient conditions on M for this counting function to have the same shape as if it were counting eigenvalues related to a cocompact group.
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Moment of derivatives of L-functions for two distinct newforms
本文针对两个不同尖点新形式的二次扭曲乘积 L 函数，研究其导数之矩，并给出无条件的渐近公式。
Breaking Universality in the Lower Order Terms in the 1-level and 2-level Density of Holomorphic Cusp Newforms
本文研究全纯尖点新形式族低位零点密度，识别加权1层与2层密度中依赖层数算术结构的低阶修正项，并证明不同素因子增长速率破坏主项普适性。
On the Pair Correlation of Zeros of $L$-Functions for Non-CM Newforms in Shifted Ranges
本文研究非CM新形式平移辅助$L$-函数零点的配对相关，证明无条件渐近结果并提出简单性假设，提供宏观信息以对比原始$L$-函数的微观配对相关。
Building on the work of Iwaniec, Luo and Sarnak, we use the $n$-level density to bound the probability of vanishing to order at least $r$ at the central point for families of cuspidal newforms of prime level $N \to \infty$, split by sign. There are three methods to improve bounds on the order of vanishing: optimizing the test functions, increasing the support, and increasing the $n$-level density studied. Previous work determined the optimal test functions for the $1$ and $2$-level densities in certain support ranges, leading to marginal improvements in bounds and making it not a productive av…
We relate the study of Landau-Siegel zeros to the ranks of Jacobians $J_0(q)$ of modular curves for large primes $q$. By a conjecture of Brumer-Murty, the rank should be equal to half of the dimension. Equivalently, almost all newforms of weight two and level $q$ have analytic rank $\leq 1$. We show that either Landau-Siegel zeros do not exist, or that almost all such newforms have analytic rank $\leq 2$. In particular, almost all odd newforms have analytic rank equal to one. Additionally, for a sparse set of primes $q$ we show the rank of $J_0(q)$ is asymptotically equal to the rank predicted…
Extending the support of $1$- and $2$-level densities for cusp form $L$-functions under square-root cancellation hypotheses
本文在平方根抵消假设下，通过提出二维类比并证明素数上经典指数和的平方根抵消，扩展了尖点新形式 $1$-级和 $2$-级密度测试函数的支撑集，使其主项收敛于随机矩阵理论预测值，并推广至任意 $n$-级密度。
We consider the Rankin-Selberg L-functions associated with a fixed modular form of full level and holomorphic cuspidal newforms of large even weight, fixed level and fixed primitive nebentypus. We compute the second moment of this family in fairly general ranges, and obtain an asymptotic formula with a power saving error term. A special case treats the fourth moment of L-functions associated with holomorphic cusp forms.
We prove an asymptotic formula expressing the regularized fourth moment of all newform Eisenstein series of primitive nebentypus as a mean value of $L$-functions. This is an analogue of the recent work of Djanković and Khan in the spectral aspect.
In this paper we consider weakly holomorphic modular forms (i.e. those meromorphic modular forms for which poles only possibly occur at the cusps) of weight $2-k\in 2\Z$ for the full modular group $\SL_2(\Z)$. The space has a distinguished set of generators $f_{m,2-k}$. Such weakly holomorphic modular forms have been classified in terms of finitely many Eisenstein series, the unique weight 12 newform $Δ$, and certain Faber polynomials in the modular invariant $j(z)$, the Hauptmodul for $\SL_2(\Z)$. We employ the theory of harmonic weak Maass forms and (non-holomorphic) Maass-Poincaré series in…
Centered Moments of Weighted One-Level Densities of $GL(2)$ $L$-Functions
本文研究了$GL(2)$ $L$-函数加权一级密度的中心矩，通过推广加权迹公式并分析交叉项，揭示了对称性依赖于权重的选择，并提出了一种组合技巧以促进与随机矩阵理论的比较。
We determine the asymptotic density $δ_k$ of the set of ordered $k$-tuples $(n_1,...,n_k)\in \N^k, k\ge 2$, such that there exists no prime power $p^a$, $a\ge 1$, appearing in the canonical factorization of each $n_i$, $1\le i\le k$, and deduce asymptotic formulae with error terms regarding this problem and analogous ones. We give numerical approximations of the constants $δ_k$ and improve the error term of formula (1.2) due to {\sc E. Cohen}. We point out that our treatment, based on certain inversion functions, is applicable also in case $k=1$ in order to establish asymptotic formulae with e…
On the moments of one-level densities in families of holomorphic cusp forms in the level aspect
研究大素数水平下模形式族中1级密度矩的矩，扩展了传统范围并改进了L-函数在中心点处的消失阶数界。
The $n^{th}$ centered moments of a large orthogonal family of automorphic $L$-functions
本文研究了与level $q$全纯Hecke newforms相关联的大型正交族$L$-函数的one level densities的第$n$次中心矩，验证了Katz-Sarnak猜想，并解决了幻影超大项和组合问题。
Tree asymptotic densities in number theory
本文研究了数论中通过根树结构编码素因数分解的整数渐近分布，推导了树密度公式并推广了素数ζ函数。
Let $0\le α\le β\le 1$. For any finite set $B\subset\mathbb{N}$, we show that there exists a set $A\subset\mathbb{N}$ such that $\underline{d}(A+B) = α$ and $\bar{d}(A+B) = β$, where $\underline{d}(A+ B)$ and $\bar{d}(A+B)$ are the lower and upper asymptotic densities of the set $A+B$, respectively. This partially answers a question by Faisant et al. A theorem involving the so-called highly sparse sets was proved in the previous arXiv version of this note; however, as pointed out by Sai Teja Somu, the proof of the theorem was flawed. The theorem is now an open question.
On a question of Ruzsa about densities of sumsets
本文否证了Ruzsa关于和集密度的一个猜想，证明不存在常数使得下密度与上密度及渐近密度满足特定不等式。
Iwasawa invariants of modular forms with $a_p=0$
本文针对满足 $a_p=0$ 的尖点新形式，建立了 Mazur-Tate 元素与带符号 $p$-进 $L$-函数的岩泽不变量之间的关系，并由此确定临界值的 $p$-进赋值及同余模形式不变量的联系，提供了计算带符号岩泽不变量的渐近方法。
The Katz-Sarnak density conjecture states that the scaling limits of the distributions of zeros of families of automorphic $L$-functions agree with the scaling limits of eigenvalue distributions of classical subgroups of the unitary groups $U(N)$. This conjecture is often tested by way of computing particular statistics, such as the one-level density, which evaluates a test function with compactly supported Fourier transform at normalized zeros near the central point. Iwaniec, Luo, and Sarnak studied the one-level densities of cuspidal newforms of weight $k$ and level $N$. They showed in the l…
Chebyshev was the first to observe a bias in the distribution of primes in residue classes. The general phenomenon is that if $a$ is a nonsquare\mod q and $b$ is a square\mod q, then there tend to be more primes congruent to $a\mod q$ than $b\mod q$ in initial intervals of the positive integers; more succinctly, there is a tendency for $π(x;q,a)$ to exceed $π(x;q,b)$. Rubinstein and Sarnak defined $δ(q;a,b)$ to be the logarithmic density of the set of positive real numbers $x$ for which this inequality holds; intuitively, $δ(q;a,b)$ is the "probability" that $π(x;q,a) > π(x;q,b)$ when $x$ is "…
A canonical construction of signed $p$-adic $L$-functions for non-ordinary modular forms of weight $\leq p+1$
本文针对权重≤p+1的p-非普通模形式，通过无需p进霍奇理论的显式对数型矩阵分解无界p进L函数，构造出一对有界p进L函数，还计算了相关Iwasawa不变量的渐近公式并推广了已有结果。
Determining optimal test functions for $2$-level densities
本文研究2层密度中使正交、辛或酉对称密度积分最小的测试函数，在乘积形式与Fourier支集限制下显式求出最优ϕ，并强化newform秩比例估计。
Densities of arithmetic Hecke triangle group orbits
研究\(q = 4\)和\(q = 6\)时算术赫克三角形群\(\Gamma_q\)轨道的渐近密度，运用初等数论技术及相关函数性质与同余条件给出新证明，发现\(q = 4\)时互质整数对集合的划分特性。
We consider in this paper the set of transfer times between two measurable subsets of positive measures in an ergodic probability measure-preserving system of a countable abelian group. If the lower asymptotic density of the transfer times is small, then we prove this set must be either periodic or Sturmian. Our results can be viewed as ergodic-theoretical extensions of some classical sumset theorems in compact abelian groups due to Kneser. Our proofs are based on a correspondence principle for action sets which was developed previously by the first two authors.
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