International Journal of Number Theory

We show that the sum of the traces of Frobenius elements of Artin $L$-functions in a family of $G$-fields satisfies the Gaussian distribution under certain counting conjectures. We prove the counting conjectures for $S_4$ and $S_5$-fields. We also prove a central limit theorem for the $L$-functions of modular forms on congruence subgroups ${\mathrm{\Gamma }}_0(N)$ as $N\to \infty$.

The Calkin–Wilf tree is an infinite binary tree whose vertices are the positive rational numbers. Each number occurs in the tree exactly once and in the form $a/b$, where $a$ and $b$ are relatively prime positive integers. In this paper, certain subsemigroups of the modular group are used to construct similar trees in the set ${{𝒟}}_0$ of positive complex numbers. Associated to each subsemigroup is a forest of trees that partitions ${{𝒟}}_0$. The fundamental domain and the set of cusps of the subsemigroup are defined and computed.

In 2007, Zhi-Wei Sun defined a covering number to be a positive integer $L$ such that there exists a covering system of the integers where the moduli are distinct divisors of $L$ greater than 1. A covering number $L$ is called primitive if no proper divisor of $L$ is a covering number. Sun constructed an infinite set ${ℒ}$ of primitive covering numbers, and he conjectured that every primitive covering number must satisfy a certain condition. In this paper, for a given $L\in {ℒ}$, we derive a formula that gives the exact number of coverings that have $L$ as the least common multiple of the set $M$ of moduli, under certain restrictions on $M$. Additionally, we disprove Sun’s conjecture by constructing an infinite set of primitive covering numbers that do not satisfy his primitive covering number condition.

This paper presents the theory of holomorphic vector-valued modular forms from a geometric perspective. More precisely, we define certain holomorphic vector bundles on the modular orbifold of generalized elliptic curves whose sections are vector-valued modular forms. This perspective simplifies the theory, and it clarifies the role that exponents of representations of ${\mathrm{\text{SL}}}_2(Z)$ play in the holomorphic theory of vector-valued modular forms. Further, it allows one to use standard techniques in algebraic geometry to deduce free-module theorems and dimension formulae (deduced previously by other authors using different techniques), by identifying the modular orbifold with the weighted projective line $P(4,6)$.

Let $E_{S,D}$ be a set defined by digit restrictions. In this paper, we determine the Assouad dimension of $E_{S,D}$. As an application, we construct a set of different Hausdorff, lower box, upper box, and Assouad dimensions. Also, we prove that the set $E_{S,D}$ is either quasisymmetrically thick or thin. In addition, we show that $E_{S,D}$ is quasisymmetrically minimal provided that ${dim}_H{E}_{S,D}=1$.

Let $f\in {𝔽}_q[t][x]$ be a square-free polynomial where ${𝔽}_q$ is a field of $q$ elements. We view $f$ as a polynomial in the variable $x$ with coefficients in the ring ${𝔽}_q[t]$. We study square-free values of $f$ in sparse subsets of ${𝔽}_q[t]$ which are given by a linear condition. The motivation for our study is an analogue problem of representing square-free integers by integer polynomials, where it is conjectured that setting aside some simple exceptional cases, a square-free polynomial $f\in \mathbb{Z}[x]$ takes infinitely many square-free values. Let $\kappa \in \mathbb{N}$ be co-prime to $q$, and let ${\gamma }_1,\dots ,{\gamma }_{\kappa -1},{\gamma }_{\kappa +1}\dots ,{\gamma }_m\in {𝔽}_q$. A consequence of the main result we show is that if $q$ is sufficiently large with respect to ${deg}_{x}f,{deg}_{t}f$ and $m$, then there exist ${\gamma }_0,{\gamma }_{\kappa }\in {𝔽}_q$ such that $f(t,{\sum }_{i=0}^m{\gamma }_i{t}^i)$ is square-free. Moreover, as $q\to \infty$, the last is true for almost all ${\gamma }_0,{\gamma }_{\kappa }\in {𝔽}_q$. The main result shows that a similar result holds also for other cases. We then generalize the results to multivariate polynomials.

In recent years, the congruence

It is proved among other things that every prime $p\ne 3$ of the form $8k+3$ can be represented as $p=a^2+bc$ where $a,b,c$ are positive integers satisfying $b<c<\sqrt{p}$ and the number of such representations is odd.

Assuming the Generalized Riemann Hypothesis, we design a deterministic algorithm that, given a prime $p$ and positive integer $m=o(p^{1/2}{(\mathrm{\text{log}}p)}^{-4})$, outputs an elliptic curve $E$ over the finite field ${𝔽}_p$ for which the cardinality of $E({𝔽}_p)$ is divisible by $m$. The running time of the algorithm is $m{p}^{1/2+o(1)}$, and this leads to more efficient constructions of rational functions over ${𝔽}_p$ whose image is small relative to $p$. We also give an unconditional version of the algorithm that works for almost all primes $p$, and give a probabilistic algorithm with subexponential time complexity.

A number is said to be $y$-friable if it has no prime factor greater than $y$. In this paper, we prove a central limit theorem on average for the distribution of divisors of $y$-friable numbers less than $x$, for all $(x,y)$ satisfying $2\le y\le {\mathrm{\text{e}}}^{(logx)/{(loglogx)}^{1+\epsilon }}$. This was previously known under the additional constraint $y\ge {\mathrm{\text{e}}}^{{(loglogx)}^{5/3+\epsilon }}$, by work of Basquin. Our argument relies on the two-variable saddle-point method.

Let $M_k^{♯}(5)$ be the space of weakly holomorphic modular forms of weight $k$ and level $5$ that are holomorphic away from the cusp at $\infty$. We study a canonical basis for $M_k^{♯}(5)$ and the locations of zeros of this basis in a fundamental domain. We give a lower bound for the number of zeros lying on a lower boundary arc of the fundamental domain.

We investigate the divisors $d$ of the numbers $P(n)$ for various polynomials $P\in \mathbb{Z}[x]$ such that $d\equiv 1(\mathrm{\text{mod}}n)$. We obtain the complete classification of such divisors for a class of polynomials, in particular for $P(x)=x^4+1$. We also construct a fast algorithm which provides all such factorizations up to a given limit for another class, for example for $P(x)=2x^4+1$. We use these results to find all the divisors $d=2^{m}k+1$ of numbers $2^{4m}+1$ and $2^{4m+1}+1$. For the numbers $2^{4m}+1$ the complete classification of such divisors is provided while for the numbers $2^{4m+1}+1$ the given classification is proved to be exhaustive only for $m\le 1000$.

We study ${\nu }_k(n)$, the number of partitions of $n$ into $k$ part sizes, and find numerous arithmetic progressions where ${\nu }_2$ and ${\nu }_3$ take on values divisible by 2 and 4. Expanding an earlier work, we show ${\nu }_2(An+B)\equiv 0(\mathrm{\text{mod}}4)$ for ($A,B$) = (36,30), (72,42), (252,114), (196,70), and likely many other progressions for which our method should easily generalize. Of some independent interest, we prove that the overpartition function $\overline{p}(n)\equiv 0(\mathrm{\text{mod}}16)$ in the first three progressions (the fourth is known), and thereby show that ${\nu }_3(An+B)\equiv 0(\mathrm{\text{mod}}2)$ in each of these progressions as well, and discuss the relationship between these congruences in more generality. We end with open questions in this area.

We connect several seemingly unrelated conjectures of Ankeny, Artin, Chowla and Mordell to a conjecture of Erdös on consecutive squarefull numbers. We then study the Erdös conjecture and relate it to the abc conjecture. We also derive by elementary methods several unconditional results pertaining to the Erdös conjecture.