arXiv subjects
Publications and source records attributed to Ezra Waxman.
We study the variance of the number of lattice points in smoothed narrow sectors, where the average is taken over the position of the sectors. There are three regimes corresponding to specific ranges for the open angle of the sectors and in each we obtain an asymptotic formula for the variance.
We study the average analytic rank in the family of $L$-functions $L(s, E_d)$ associated with the elliptic curves $E_d : y^2=x^3-dx$, as $d$ varies over fourth-power-free odd integers. Since this is a family of curves with complex multiplication, we have $L(s, E_d)=L(s - \frac12, ξ_d)$, where $ξ_d$ is a Hecke character over $\mathbb{Z}[i]$. Assuming the Generalized Riemann Hypothesis, we compute the one-level density of the low-lying zeros of this family for test functions whose Fourier transform is supported in $(-\frac35, \frac35)$. As a consequence, we obtain the upper bound $\frac{13}{6}$ for the average analytic rank $r(E_d)$ over the family. Under the additional assumption of a conjecture on the distribution of quartic Gauss sums at prime elements (a quartic analogue of Patterson's conjecture for cubic Gauss sums), we extend the admissible support to $(-1, 1)$ and improve the upper bound for the average analytic rank to $\frac32$. Both results imply that a positive proportion of twists satisfy $r(E_d) =1$, while the second also yields a positive proportion of twists with $r(E_d)=0$.
We study the low-lying zeros of a family of $L$-functions attached to the CM elliptic curve $E_d \;:\; y^2 = x^3 - dx$, for each odd and square-free integer $d$. Specifically, upon writing the $L$-function of $E_d$ as $L(s-\frac12, ξ_d)$ for the appropriate Grössencharakter $ξ_d$ of conductor $\mathfrak{f}_d$, we consider the collection $\mathcal{F}_d$ of $L$-functions attached to $ξ_{d,k}$, $k \geq 1$, where for each integer $k$, $ξ_{d, k}$ denotes the primitive character inducing $ξ_d^k$. We observe that $25\%$ of the $L$-functions in $\mathcal{F}_d$ have negative root number. $\mathcal{F}_d$ is thus not one of the essentially homogeneous families of the Universality Conjecture of Sarnak, Shin and Templier, with unitary, symplectic or orthogonal (odd or even) symmetry type. By computing the one-level density in the family of $L$-functions in $\mathcal{F}_{d}$ with conductor at most $K^2 \mathrm N (\mathfrak{f}_d)$, we find that $\mathcal{F}_d$ naturally decomposes into subfamilies: more specifically, a collection of symplectic ($L(s, ξ_{d,k})$ for $k \equiv α\bmod 8$, $α$ even) and orthogonal ($L(s, ξ_{d,k})$ for $k \equiv α\bmod 8$, $α$ odd) subfamilies. For each such subfamily, we moreover compute explicit lower order terms in decreasing powers of $\log (K^2 \mathrm N(\mathfrak{f}_d))$.
On the circle of radius $R$ centred at the origin, consider a ``thin'' sector about the fixed line $y = αx$ with edges given by the lines $y = (α\pm ε) x$, where $ε= ε_R \rightarrow 0$ as $ R \to \infty $. We establish an asymptotic count for $S_α(ε,R)$, the number of integer lattice points lying in such a sector. Our results depend both on the decay rate of $ε$ and on the rationality/irrationality type of $α$. In particular, we demonstrate that if $α$ is Diophantine, then $S_α(ε,R)$ is asymptotic to the area of the sector, so long as $εR^{t} \rightarrow \infty$ for some $ t<2 $.
In this paper we completely solve a simple quartic family of Thue equations over $\mathbb{C}(T)$. Specifically, we apply the ABC-Theorem to find all solutions $(x,y) \in \mathbb{C}[T] \times \mathbb{C}[T]$ to the set of Thue equations $F_λ(X,Y) = ξ$, where $ξ\in \mathbb{C}^{\times}$ and \begin{equation*} F_λ(X,Y):=X^4 -λX^3Y -6 X^2Y^2 + λXY^3 +Y^4, \quad \quad λ\in \mathbb{C}[T]/\{\mathbb{C}\} \end{equation*} denotes a family of quartic simple forms.
In 1927, E. Artin proposed a conjecture for the natural density of primes $p$ for which $g$ generates $(\mathbb{Z}/p\mathbb{Z})^\times$. By carefully observing numerical deviations from Artin's originally predicted asymptotic, Derrick and Emma Lehmer (1957) identified the need for an additional correction factor; leading to a modified conjecture which was eventually proved to be correct by Hooley (1967) under the assumption of the generalised Riemann hypothesis. An appropriate analogue of Artin's primitive root conjecture may moreover be formulated for an algebraic function field $K$ of $r$ variables over $\mathbb{F}_{q}$. Relying on a soon to be established theorem of Weil (1948), Bilharz (1937) provided a proof in the particular case that $K$ is a global function field (i.e. $r=1$), which is correct under the assumption that $g \in K$ is a $\textit{geometric}$ element. Under these same assumptions, Pappalardi and Shparlinski (1995) established a quantitative version of Bilharz's result. In this paper we build upon these works by both generalizing to function fields in $r$ variables over $\mathbb{F}_{q}$ and removing the assumption that $g \in K$ is geometric; thereby completing a proof of Artin's primitive root conjecture for function fields over $\mathbb{F}_{q}$. In doing so, we moreover identify an interesting correction factor which emerges when $g$ is not geometric. A crucial feature of our work is an exponential sum estimate over varieties that we derive from Weil's Theorem.
We say that a prime number $p$ is an $\textit{Artin prime}$ for $g$ if $g$ mod $p$ generates the group $(\mathbb{Z}/p\mathbb{Z})^{\times}$. For appropriately chosen integers $d$ and $g$, we present a conjecture for the asymptotic number $π_{d,g}(x)$ of primes $p \leq x$ such that both $p$ and $p+d$ are Artin primes for $g$. In particular, we identify a class of pairs $(d,g)$ for which $π_{d,g}(x) =0$. Our results suggest that the distribution of Artin prime pairs, amongst the ordinary prime pairs, is largely governed by a Poisson binomial distribution.
We derive a refined conjecture for the variance of Gaussian primes across sectors, with a power saving error term, by applying the L-functions Ratios Conjecture. We observe a bifurcation point in the main term, consistent with the Random Matrix Theory (RMT) heuristic previously proposed by Rudnick and Waxman. Our model also identifies a second bifurcation point, undetected by the RMT model, that emerges upon taking into account lower order terms. For sufficiently small sectors, we moreover prove an unconditional result that is consistent with our conjecture down to lower order terms.
In this paper we apply the $L$-function Ratios Conjecture to compute the one-level density for a symplectic family of $L$-functions attached to Hecke characters of infinite order. When the support of the Fourier transform of the corresponding test function $f$ reaches $1$, we observe a transition in the main term, as well as in the lower order term. The transition in the lower order term is in line with behavior recently observed by D. Fiorilli, J. Parks, and A. Södergren in their study of a symplectic family of quadratic Dirichlet $L$-functions. We then directly calculate main and lower order terms for test functions $f$ such that supp($\widehat{f}) \subset [-α,α]$ for some $α<1$, and observe that this unconditional result is in agreement with the prediction provided by the Ratios Conjecture. As the analytic conductor of these L-functions grow twice as large (on a logarithmic scale) as the cardinality of the family in question, this is the optimal support that can be expected with current methods. Finally as a corollary we deduce that, under GRH, at least 75$\%$ of these $L$-functions do not vanish at the central point.
Fermat showed that every prime p = 1 mod 4 is a sum of two squares: $p = a^2 + b^2$. To any of the 8 possible representations (a,b) we associate an angle whose tangent is the ratio b/a. In 1919 Hecke showed that these angles are uniformly distributed as p varies, and in the 1950's Kubilius proved uniform distribution in somewhat short arcs. We study fine scale statistics of these angles, in particular the variance of the number of such angles in a short arc. We present a conjecture for this variance, motivated both by a random matrix model, and by a function field analogue of this problem, for which we prove an asymptotic form for the corresponding variance.