arXiv subjects

Noam Kimmel

Publications and source records attributed to Noam Kimmel.

9 recordsLinked to original sources

arXiv2025

Non-vanishing of Poincaré Series on Average

We study when Poincaré series for congruence subgroups do not vanish identically. We show that almost all Poincaré series with suitable parameters do not vanish when either the weight $k$ or the index $m$ varies in a dyadic interval. Crucially, analyzing the problem `on average' over these weights or indices allows us to prove non-vanishing in ranges where the index $m$ is significantly larger than $k^2$ - a range in which proving non-vanishing for individual Poincaré series remains out of reach of current methods.

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arXiv2025

Positive density for consecutive runs of sums of two squares

We study the distribution of consecutive sums of two squares in arithmetic progressions. We show that for any odd squarefree modulus $q$, any two reduced congruence classes $a_1$ and $a_2$ mod $q$, and any $r_1,r_2 \ge 1$, a positive density of sums of two squares begin a chain of $r_1$ consecutive sums of two squares, all of which are $a_1$ mod $q$, followed immediately by a chain of $r_2$ consecutive sums of two squares, all of which are $a_2$ mod $q$. This is an analog of the result of Maynard for the sequence of primes, showing that for any reduced congruence class $a$ mod $q$ and for any $r \ge 1$, a positive density of primes begin a sequence of $r$ consecutive primes, all of which are $a$ mod $q$.

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arXiv2025

Mass equidistribution for Poincaré series of large index

Let $P_{k,m}$ denote the Poincaré series of weight $k$ and index $m$ for the full modular group $\mathrm{SL}_2(\mathbb{Z})$, and let $\{P_{k,m}\}$ be a sequence of Poincaré series for which $m(k)$ satisfies $m(k) / k \rightarrow\infty$ and $m(k) \ll k^{\frac{3}{2} - ε}$. We prove that the $L^2$ mass of such a sequence equidistributes on $\mathrm{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$ with respect to the hyperbolic measure as $k$ goes to infinity. As a consequence, we deduce that the zeros of such a sequence $\{P_{k,m}\}$ become uniformly distributed in $\mathrm{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$ with respect to the hyperbolic measure.

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arXiv2024

Consecutive runs of sums of two squares

We study the distribution of consecutive sums of two squares in arithmetic progressions. If $\{E_n\}_{n \in \mathbb{N}}$ is the sequence of sums of two squares in increasing order, we show that for any modulus $q$ and any congruence classes $a_1,a_2,a_3 \mod q$ which are admissible in the sense that there are solutions to $x^2 + y^2 \equiv a_i \mod q$, there exist infinitely many $n$ with $E_{n+i-1} \equiv a_i \mod q$, for $i = 1,2,3$. We also show that for any $r_1, r_2 \ge 1$, there exist infinitely many $n$ with $E_{n+i-1} \equiv a_1 \mod q$ for $1 \le i \le r_1$ and $E_{n+ i - 1} \equiv a_2 \mod q$ for $r_1 + 1 \le i \le r_1 + r_2$.

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arXiv2024

Artin's Primitive Root Conjecture in Number Fields and For Matrices

In 1927, E. Artin conjectured that all non-square integers $a\neq -1$ are a primitive root of $\mathbb{F}_p$ for infinitely many primes $p$. In 1967, Hooley showed that this conjecture follows from the Generalized Riemann Hypothesis (GRH). In this paper we consider variants of the primitive root conjecture for number fields and for matrices. All results are conditional on GRH. For an algebraic number field $K$ and some element $α\in K$, we examine the order of $α$ modulo various rational primes $p$. We extend previous results of Roskam which only worked for quadratic extensions $K/\mathbb{Q}$ to more general field extensions of higher degree. Specifically, under some constraints on the Galois group of $K/\mathbb{Q}$ and on the element $α\in K$, we show that $α$ is of almost maximal order mod $p$ for almost all rational primes $p$ which factor into primes of degree 2 in $K$. We also consider Artin's primitive root conjecture for matrices. Given a matrix $A\in\text{GL}_n(\mathbb{Q})$, we examine the order of $A\bmod p$ in $\text{GL}_n(\mathbb{F}_p)$ for various primes $p$, which turns out to be equivalent to the number field setting.

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arXiv2024

Asymptotic zeros of Poincaré series

We study the zeros of Poincaré series $P_{k,m}$ for the full modular group. We consider the case where $m \sim αk$ for some constant $α> 0$. We show that in this case a positive proportion of the zeros lie on the line $\frac{1}{2} + it$. We further show that if $α> \frac{\log(2)}{2π}$ then the imaginary axis also contains a positive proportion of zeros. We also give a description for the location of the non-real zeros when $α$ is small.

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arXiv2024

Vanishing of Poincaré series for congruence subgroups

We consider the problem of the vanishing of Poincaré series for congruence subgroups. Denoting by $P_{k,m,N}$ the Poincaré series of weight $k$ and index $m$ for the group $Γ_0(N)$, we show that for certain choices of parameters $k,m,N$, the Poincaré series does not vanish. Our methods improve on previous results of Rankin (1980) and Mozzochi (1989).

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arXiv2023

The Least Common Multiple of a Bivariate Quadratic Sequence

Let $F\in\mathbb{Z}[x,y]$ be some polynomial of degree 2. In this paper we find the asymptotic behaviour of the least common multiple of the values of $F$ up to $N$. More precisely, we consider $ψ_F(N) = \log\left(\text{LCM}_{0<F(x,y)\leq N}\left\lbrace F(x,y)\right\rbrace\right)$ as $N$ tends to infinity. It turns out that there are 4 different possible asymptotic behaviours depending on $F$. For a generic $F$, we show that the function $ψ_F(N)$ has order of magnitude $\frac{N\log\log N}{\sqrt{\log N}}$. We also show that this is the expected order of magnitude according to a suitable random model. However, special polynomials $F$ can have different behaviours, which sometimes deviate from the random model. We give a complete description of the order of magnitude of these possible behaviours, and when each one occurs.

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arXiv2021

Covariance of Error Terms Related to the Dirichlet Eigenvalue Problem

We explore the covariance of error terms coming from Weyl's conjecture regarding the number of Dirichlet eigenvalues up to size $X$. We also consider this problem in short intervals, i.e. the error term of the number of eigenvalues in the window $[X, X+S]$ for some $S(X)$. We look at these error terms for planar domains where the Dirichlet eigenvalues can be explicitly calculated. In these cases, the error term is closely related to the error term from the classical lattice points counting problem of expanding planar domains. We give a formula for the covariance of such error terms, for general planar domains. We also give a formula for the covariance of error terms in short intervals, for sufficiently large intervals. Going back to the Dirichlet eigenvalue problem, we give results regarding the covariance of the error terms in short intervals of 'generic' rectangles. We also explore a specific example, namely we compute the covariance between the error terms of an equilateral triangle and various rectangles.

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