Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe Average Size of Ramanujan Sums over Cubic Number Fields
185
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Let K be a cubic number field. In this paper, we study the Ramanujan sums c_{J}(I), where I and J are integral ideals in O_{K}. The asymptotic behaviour of sums of c_{J}(I) over both I and J is investigated.
rate research
Read More
Let ${mathbb F}_q$ be the finite field with $q=p^k$ elements with $p$ being a prime and $k$ be a positive integer. For any $y, zinmathbb{F}_q$, let $N_s(z)$ and $T_s(y)$ denote the numbers of zeros of $x_1^{3}+cdots+x_s^3=z$ and $x_1^3+cdots+x_{s-1}^3+yx_s^3=0$, respectively. Gauss proved that if $q=p, pequiv1pmod3$ and $y$ is non-cubic, then $T_3(y)=p^2+frac{1}{2}(p-1)(-c+9d)$, where $c$ and $d$ are uniquely determined by $4p=c^2+27d^2,~cequiv 1 pmod 3$ except for the sign of $d$. In 1978, Chowla, Cowles and Cowles determined the sign of $d$ for the case of $2$ being a non-cubic element of ${mathbb F}_p$. But the sign problem is kept open for the remaining case of $2$ being cubic in ${mathbb F}_p$. In this paper, we solve this sign problem by determining the sign of $d$ when $2$ is cubic in ${mathbb F}_p$. Furthermore, we show that the generating functions $sum_{s=1}^{infty} N_{s}(z) x^{s}$ and $sum_{s=1}^{infty} T_{s}(y)x^{s}$ are rational functions for any $z, yinmathbb F_q^*:=mathbb F_qsetminus {0}$ with $y$ being non-cubic over ${mathbb F}_q$ and also give their explicit expressions. This extends the theorem of Myerson and that of Chowla, Cowles and Cowles.
We define a new kind of classical digamma function, and establish its some fundamental identities. Then we apply the formulas obtained, and extend tools developed by Flajolet and Salvy to study more general Euler type sums. The main results of Flajolet and Salvys paper cite{FS1998} are the immediate corollaries of main results in this paper. Furthermore, we provide some parameterized extensions of Ramanujan-type identities that involve hyperbolic series. Some interesting new consequences and illustrative examples are considered.
In this paper, we study simple cubic fields in the function field setting, and also generalize the notion of a set of exceptional units to cubic function fields, namely the notion of $k$-exceptional units. We give a simple proof that the Galois simple cubic function fields are the immediate analog of Shanks simplest cubic number fields. In addition to computing the invariants, including a formula for the regulator, we compute the class numbers of the Galois simple cubic function fields over $mathbb{F}_{5}$ and $mathbb{F}_{7}$ using truncated Euler products. Finally, as an additional application, we determine all Galois simple cubic function fields with class number one, subject to a mild restriction.
We present computational results on the divisor class number and the regulator of a cubic function field over a large base field. The underlying method is based on approximations of the Euler product representation of the zeta function of such a field. We give details on the implementation for purely cubic function fields of signatures $(3,1)$ and $(1, 1; 1, 2)$, operating in the ideal class group and infrastructure of the function field, respectively. Our implementation provides numerical evidence of the computational effectiveness of this algorithm. With the exception of special cases, such as purely cubic function fields defined by superelliptic curves, the examples provided are the largest divisor class numbers and regulators ever computed for a cubic function field over a large prime field. The ideas underlying the optimization of the class number algorithm can in turn be used to analyze the distribution of the zeros of the function fields zeta function. We provide a variety of data on a certain distribution of the divisor class number that verify heuristics by Katz and Sarnak on the distribution of the zeroes of the zeta function.
In this paper, we classify torsion groups of rational Mordell curves explicitly over cubic fields as well as over sextic fields. Also, we classify torsion groups of Mordell curves over cubic fields and for Mordell curves over sextic fields, we produce all possible torsion groups.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
