Searcharxiv⌕ Search

arXiv subjects

Amir Akbary

Publications and source records attributed to Amir Akbary.

At least 19 recordsLinked to original sources
arXiv2025

Number Fields With Large Pólya Groups

The Pólya group ${\rm Po}(K)$ of a number field $K$ is the subgroup of the ideal class group ${\rm Cl}(K)$ of $K$ generated by the classes of all the products of the prime ideals of $K$ with the same norm. Motivated by the classical "one class in each genus problem", we prove general finiteness theorems for the number fields $K$ with a fixed Pólya index $\left[{\rm Cl}(K):{\rm Po}(K)\right]$ in the families of Galois number fields, solvable CM-fields, and real quadratic fields of extended R-D type. We also give classification results for specific families. Most notably, we classify, unconditionally, all imaginary bi-quadratic and imaginary tri-quadratic fields with the Pólya index one. Furthermore, we classify all real quadratic fields of extended R-D type (with possibly only one more field) with the Pólya index one. Also, under GRH, we give the complete list of 161 imaginary quadratic fields with the Pólya index two. Finally, as a byproduct of our results, we extend, from narrow R-D types to the extended R-D types, Dohmae's classification of real quadratic fields of narrow R-D type whose narrow genus numbers equal their narrow class numbers.

math.NT↗
arXiv2025

Quadratic Euler-Kronecker constants in positive characteristic

In 2006, Ihara defined and systematically studied a generalization of the Euler-Mascheroni constant for all global fields, named the Euler-Kronecker constants. This paper examines their distribution across geometric quadratic extensions of a rational global function field, via the values of logarithmic derivatives of Dirichlet $L$-functions at 1. Using a probabilistic model, we show that the values converge to a limiting distribution with a smooth, positive density function, as the genii of quadratic fields approach infinity. We then prove a discrepancy theorem for the convergence of the frequency of these values, and obtain information about the proportion of the small values. Finally, we prove omega results on the extreme values. Our theorems imply new distribution results on the stable Taguchi heights and logarithmic Weil heights of rank 2 Drinfeld modules with CM.

math.NT↗
arXiv2025

Binary quadratic forms of odd class number

Let $-D$ be a fundamental discriminant. We express the number of representations of an integer by a positive definite binary quadratic form of discriminant $-D$ with an odd class number $h(-D)$ as a rational linear expression involving the Kronecker symbol $\left(\frac{-D}{.}\right)$ and the Fourier coefficients of certain cusp forms. We prove these cusp forms have eta quotient representations only if $D=23$. This provides, using theta functions, a generalization of a result of F. van der Blij from 1952 for binary quadratic forms of discriminant $-23$ to the case of forms of discriminant $-D$ with odd $h(-D)$. We also classify all the eta quotients of prime level $D$ which are half the difference of two theta functions of level $D$.

math.NT↗
arXiv2024

Constants for Artin like problems in Kummer and division fields

We apply the character sums method of Lenstra, Moree, and Stevenhagen, to explicitly compute the constants in the Titchmarsh divisor problem for Kummer fields and for division fields of Serre curves. We derive our results as special cases of a general result on the product expressions for the sums in the form $$\sum_{n=1}^{\infty}\frac{g(n)}{\#G(n)}$$ in which $g(n)$ is a multiplicative arithmetic function and $\{G(n)\}$ is a certain family of Galois groups. Our results extend the application of the character sums method to the evaluation of constants, such as the Titchmarsh divisor constants, that are not density constants.

math.NT↗
arXiv2023

Bounds for orders of zeros of a class of Eisenstein series and their applications on dual pairs of eta quotients

Let $k$ be an even positive integer, $p$ be a prime and $m$ be a nonnegative integer. We find an upper bound for orders of zeros (at cusps) of a linear combination of classical Eisenstein series of weight $k$ and level $p^m$. As an immediate consequence we find the set of all eta quotients that are linear combinations of these Eisenstein series and hence the set of all eta quotients of level $p^m$ whose derivatives are also eta quotients.

math.NT↗
arXiv2023

Value-Distribution of Logarithmic Derivatives of Quadratic Twists of Automorphic $L$-functions

Let $d\in\mathbb{N}$, and let $π$ be a fixed cuspidal automorphic representation of $\mathrm{GL}_{d}(\mathbb{A}_{\mathbb{Q}})$ with unitary central character. We determine the limiting distribution of the family of values $-\frac{L'}{L}(1+it,π\otimesχ_D)$ as $D$ varies over fundamental discriminants. Here, $t$ is a fixed real number and $χ_D$ is the real character associated with $D$. We establish an upper bound on the discrepancy in the convergence of this family to its limiting distribution. As an application of this result, we obtain an upper bound on the small values of $\left|\frac{L'}{L}(1,π\otimesχ_D)\right|$ when $π$ is self-dual.

math.NT↗
arXiv2021

Sums of triangular numbers and sums of squares

For non-negative integers $a,b,$ and $n$, let $N(a, b; n)$ be the number of representations of $n$ as a sum of squares with coefficients $1$ or $3$ ($a$ of ones and $b$ of threes). Let $N^*(a,b; n)$ be the number of representations of $n$ as a sum of odd squares with coefficients $1$ or $3$ ($a$ of ones and $b$ of threes). We have that $N^*(a,b;8n+a+3b)$ is the number of representations of $n$ as a sum of triangular numbers with coefficients $1$ or $3$ ($a$ of ones and $b$ of threes). It is known that for $a$ and $b$ satisfying $1\leq a+3b \leq 7$, we have $$ N^*(a,b;8n+a+3b)= \frac{2}{2+{a\choose4}+ab} N(a,b;8n+a+3b) $$ and for $a$ and $b$ satisfying $a+3b=8$, we have $$ N^*(a,b;8n+a+3b) = \frac{2}{2+{a\choose4}+ab} \left( N(a,b;8n+a+3b) - N(a,b; (8n+a+3b)/4) \right). %& t(8,0;{n}) = \frac{1}{36} \left( N(8,0;8n+8) - N(8,0;2n+2) \right). \label{eq31_5} $$ Such identities are not known for $a+3b>8$. In this paper, for general $a$ and $b$ with $a+b$ even, we prove asymptotic equivalence of formulas similar to the above, as $n\rightarrow\infty$. One of our main results extends a theorem of Bateman, Datskovsky, and Knopp where the case $b=0$ and general $a$ was considered. Our approach is different from Bateman-Datskovsky-Knopp's proof where the circle method and singular series were used. We achieve our results by explicitly computing the Eisenstein components of the generating functions of $N^*(a,b;8n+a+3b)$ and $N(a,b;8n+a+3b)$. The method we use is robust and can be adapted in studying the asymptotics of other representation numbers with general coefficients.

math.NT↗
arXiv2021

Two dimensional value-distribution of cubic Hecke $L$-functions

We establish the two-dimensional asymptotic distributions of the logarithm and logarithmic derivative of $L$-functions associated with a family of cubic Hecke characters. A crucial ingredient in the proof of our main result is an exponential decay estimate for the characteristic functions of the distributions.

math.NT↗
arXiv2020

On the moments of torsion points modulo primes and their applications

Let $\mathbb{A}[n]$ be the group of $n$-torsion points of a commutative algebraic group $\mathbb{A}$ defined over a number field $F$. For a prime ideal $\mathfrak{p}$, we let $N_{\mathfrak{p}}(\mathbb{A}[n])$ be the number of $\mathbb{F}_\mathfrak{p}$-solutions of the system of polynomial equations defining $\mathbb{A}[n]$ when reduced modulo $\mathfrak{p}$. Here, $\mathbb{F}_{\mathfrak{p}}$ is the residue field at $\mathfrak{p}$. Let $π_F(x)$ denote the number of primes $\mathfrak{p}$ of $F$ whose norm $N(\mathfrak{p})$ do not exceed $x$. We then, for algebraic groups of dimension one, compute the $k$-th moment limit $$M_k(\mathbb{A}/F, n)=\lim_{x\rightarrow \infty} \frac{1}{π_F(x)} \sum_{N(\mathfrak{p}) \leq x} N_{\mathfrak{p}}^k(\mathbb{A}[n])$$ by appealing to the prime number theorem for arithmetic progressions and more generally the Chebotarev density theorem. We further interpret this limit as the number of orbits of the action of the absolute Galois group of $F$on $k$ copies of $\mathbb{A}[n]$ by an application of Burnside's Lemma. These concrete examples suggest a possible approach for determining the number of orbits of a group acting on $k$ copies of a set. We also show that for an algebraic set $Y$ of dimension zero, the corresponding arithmetic function $N_\mathfrak{p}(Y)$, defined on primes $\mathfrak{p}$ of $F$, has an asymptotic limiting distribution.

math.NT↗
arXiv2019

Value-distribution of cubic Hecke $L$-functions

Let $k=\mathbb{Q}(\sqrt{-3})$, and let $c\in \mathfrak{O}_k$ be a square free algebraic integer such that $c\equiv 1~({\rm mod}~{\langle9\rangle})$. Let $ζ_{k(c^{1/3})}(s)$ be the Dedekind zeta function of the cubic field $k(c^{1/3})$ and $ζ_k(s)$ be the Dedekind zeta function of $k$. For fixed real $σ>1/2$, we obtain asymptotic distribution functions $F_σ$ for the values of the logarithm and the logarithmic derivative of the Artin $L$-functions \begin{equation*} L_c(σ)= \frac{ζ_{k(c^{1/3})}(σ)}{ζ_k(σ)}, \end{equation*} as $c$ varies. Moreover, we express the characteristic function of $F_σ$ explicitly as a product indexed by the prime ideals of $\mathfrak{O}_k$. As a corollary of our results, we establish the existence of an asymptotic distribution function for the error term of the Brauer-Siegel asymptotic formula for the family of number fields $\{k(c^{1/3})\}_{c}$. We also deduce a similar result for the Euler-Kronecker constants of this family.

math.NT↗
arXiv2018

Euler's Function on Products of Primes in Progressions

We study generalizations of some results of Jean-Louis Nicolas regarding the relation between small values of Euler's function $φ(n)$ and the Riemann Hypothesis. Among other things, we prove that for $1\leq q\leq 10$ and for $q=12, 14$, the generalized Riemann Hypothesis for the Dedekind zeta function of the cyclotomic field $\mathbb{Q}(e^{2πi/q})$ is true if and only if for all integers $k\geq 1$ we have \[\frac{\bar{N}_k}{φ(\bar{N}_k)(\log(φ(q)\log{\bar{N}_k}))^{\frac{1}{φ(q)}}} > \frac{1}{C(q,1)}.\] Here $\bar{N}_k$ is the product of the first $k$ primes in the arithmetic progression $p\equiv 1~({\rm mod}~{q})$ and $C(q, 1)$ is the constant appearing in the asymptotic formula \[\prod_{\substack{p \leq x \\ p \equiv 1~({\rm mod}~{q})}} \left(1 - \frac{1}{p}\right) \sim \frac{C(q, 1)}{(\log{x})^\frac{1}{φ(q)}},\] as $x\rightarrow\infty$. We also prove that, for $q\leq 400,000$ and integers $a$ coprime to $q$, the analogous inequality \[\frac{\bar{N}_k}{φ(\bar{N}_k)(\log(φ(q)\log{\bar{N}_k}))^{\frac{1}{φ(q)}}} > \frac{1}{C(q,a)}\] holds for infinitely many values of $k$. If in addition $a$ is a not a square modulo $q$, then there are infinitely many $k$ for which this inequality holds and also infinitely many $k$ for which this inequality fails.

math.NT↗
arXiv2017

On the Lang-Trotter conjecture for two elliptic curves

Following Lang and Trotter we describe a probabilistic model that predicts the distribution of primes $p$ with given Frobenius traces at $p$ for two fixed elliptic curves over $\mathbb{Q}$. In addition, we propose explicit Euler product representations for the constant in the predicted asymptotic formula and describe in detail the universal component of this constant. A new feature is that in some cases the $\ell$-adic limits determining the $\ell$-factors of the universal constant, unlike the Lang-Trotter conjecture for a single elliptic curve, do not stabilize. We also prove the conjecture on average over a family of elliptic curves following the work of David, Koukoulopoulos, and Smith.

math.NT↗
arXiv2017

The Signs in Elliptic Nets

We give a generalization of a theorem of Silverman and Stephens regarding the signs in an elliptic divisibility sequence to the case of an elliptic net. We also describe applications of this theorem in the study of the distribution of the signs in elliptic nets and generating elliptic nets using the denominators of the linear combination of points on elliptic curves.

math.NT↗
arXiv2015

On the greatest prime factor of some divisibility sequences

Let $P(m)$ denote the greatest prime factor of $m$. For integer $a>1$, M. Ram Murty and S. Wong proved that, under the assumption of the ABC conjecture, $$P(a^n-1)\gg_{ε, a} n^{2-ε}$$ for any $ε>0$. We study analogues results for the corresponding divisibility sequence over the function field $\mathbb{F}_q(t)$ and for some divisibility sequences associated to elliptic curves over the rational field $\mathbb{Q}$.

math.NT↗
arXiv2014

On Symmetries of Elliptic Nets and Valuations of Net Polynomials

Under certain conditions, we prove that the set of zeros of an elliptic net forms an Abelian group. We present two applications of this fact. Firstly we give a generalization of a theorem of Ayad on valuations of division polynomials in the context of net polynomials. Secondly we generalize a theorem of Ward on symmetry of elliptic divisibility sequences to the case of elliptic nets.

math.NT↗
arXiv2014

On invariants of elliptic curves on average

We prove several results regarding some invariants of elliptic curves on average over the family of all elliptic curves inside a box of sides $A$ and $B$. As an example, let $E$ be an elliptic curve defined over $\mathbb{Q}$ and $p$ be a prime of good reduction for $E$. Let $e_{E}(p)$ be the exponent of the group of rational points of the reduction modulo $p$ of $E$ over the finite field $\mathbb{F}_p$. Let $\mathcal{C}$ be the family of elliptic curves $$E_{a,b}:~y^2=x^3+ax+b,$$ where $|a|\leq A$ and $|b|\leq B$. We prove that, for any $c>1$ and $k\in \mathbb{N}$, $$\frac{1}{|\mathcal{C}|} \sum_{E\in \mathcal{C}} \sum_{p\leq x} e_E^k(p) = C_k {\rm li}(x^{k+1})+O\left(\frac{x^{k+1}}{(\log{x})^c} \right),$$ as $x\rightarrow \infty$, as long as $A, B>\exp\left(c_{1} (\log{x})^{1/2} \right)$ and $AB>x(\log{x})^{4+2c}$, where $c_1$ is a suitable positive constant. Here $C_k$ is an explicit constant given in the paper which depends only on $k$, and ${\rm li}(x)=\int_{2}^x dt/\log{t}$. We prove several similar results as corollaries to a general theorem. The method of the proof is capable of improving some of the known results with $A, B>x^ε$ and $AB>x(\log{x})^δ$ to $A, B>\exp\left(c_1 (\log{x})^{1/2} \right)$ and $AB>x(\log{x})^δ$.

math.NT↗
arXiv2014

A log-free zero-density estimate and small gaps in coefficients of $L$-functions

Let $L(s, π\timesπ^\prime)$ be the Rankin--Selberg $L$-function attached to automorphic representations $π$ and $π^\prime$. Let $\tildeπ$ and $\tildeπ^\prime$ denote the contragredient representations associated to $π$ and $π^\prime$. Under the assumption of certain upper bounds for coefficients of the logarithmic derivatives of $L(s, π\times\tildeπ)$ and $L(s, π^\prime\times\tildeπ^\prime)$, we prove a log-free zero-density estimate for $L(s, π\timesπ^\prime)$ which generalises a result due to Fogels in the context of Dirichlet $L$-functions. We then employ this log-free estimate in studying the distribution of the Fourier coefficients of an automorphic representation $π$. As an application we examine the non-lacunarity of the Fourier coefficients $b_f(p)$ of a modular newform $f(z)=\sum_{n=1}^{\infty} b_f(n) e^{2πi n z}$ of weight $k$, level $N$, and character $χ$. More precisely for $f(z)$ and a prime $p$, set $j_f(p):=\max_{x;~x> p} J_{f} (p, x)$, where $J_{f} (p, x):=\#\{{\rm prime}~q;~a_π(q)=0~{\rm for~all~}p<q\leq x\}.$ We prove that $j_f(p)\ll_{f, θ} p^θ$ for some $0<θ<1$.

math.NT↗