Related papers: The ideal counting function in cubic fields

Let $q\geqslant 2$ be a fixed prime power. We prove an asymptotic formula for counting the number of monic polynomials that are of degree $n$ and have exactly $k$ irreducible factors over the finite field $\mathbb{F}_q$. We also compare our…

We have for positive integers $n$, $k$ and finite field $\mathbb{F}_q$, $c(n,k,q)$, as the number of simultaneous similarity classes of $k$-tuples of commuting $n\times n$ matrices over the $\mathbb{F}_q$. In this paper, it has been shown…

If $N = {q^k}{n^2}$ is an odd perfect number, where $q$ is the Euler prime, then we show that $n < q$ is sufficient for Sorli's conjecture that $k = \nu_{q}(N) = 1$ to hold. We also prove that $q^k < 2/3{n^2}$, and that $I(q^k) < I(n)$,…

We consider the simplest quartic number fields $\mathbb{K}_m$ defined by the irreducible quartic polynomials $$x^4-mx^3-6x^2+mx+1,$$ where $m$ runs over the positive rational integers such that the odd part of $m^2+16$ is squarefree. In…

In this paper we study the distribution of the real algebraic numbers. Given an interval $I$, a positive integer $n$ and $Q>1$, define the counting function $\Phi_n(Q;I)$ to be the number of algebraic numbers in $I$ of degree $n$ and height…

By considering the norm of elements in the ring of integers in $\mathbb{Q}(\sqrt{-a})$, we give an algebraic approach to count the number of integral solutions of diophantine equations of the form $x^2+ay^2=n$ where $a$ is a Heegner number…

We study the capitulation of $2$-ideal classes of an infinite family of imaginary biquadratic number fields consisting of fields $k =Q(\sqrt{pq_1q_2}, i)$, where $i=\sqrt{-1}$ and $q_1\equiv q_2\equiv-p\equiv-1 \pmod 4$ are different…

We examine the sum of modified Bessel functions with argument depending non-linearly on the summation index given by \[S_{\nu,p}(a)=\sum_{n\geq 1} (an^p/2)^{-\nu} K_\nu(an^p)\qquad (a>0,\ 0\leq\nu<1)\] as the parameter $a\to 0+$, where $p$…

Let $x$ be a real number satisfying $x \geq 2$. For any positive integer $n$, we define $s(n)$ as the smallest non-negative integer such that $n + s(n)$ is a perfect square. In this paper, we derive an asymptotic formula for the sum…

For any $k>1$, we find the asymptotics of the counting function of $k$-th power-free elements in an additive arithmetic semigroup with exponential growth of the abstract prime counting function. This paper continues the authors' earlier…

Let $L$ be a nilpotent algebra of class two over a compact discrete valuation ring $A$ of characteristic zero or of sufficiently large positive characteristic. Let $q$ be the residue cardinality of $A$. The ideal zeta function of $L$ is a…

We provide explicit bounds for the number of integral ideals of norms at most $X$ is $\mathbb{Q}[\sqrt{d}]$ when $d <0$ is a fundamendal discriminant with an error term of size $O(X^{1/3})$. In particular, we prove that, when $\chi$ is the…

Let $K$ be a number field. This paper considers arithmetic functions over $K$, that are, complex valued functions on the set of nonzero integral ideals in $K$. Firstly we generalize some basic results on arithmetic functions. Next we define…

Let $K$ be the number field determined by a monic irreducible polynomial $f(x)$ with integer coefficients. In previous papers we parameterized the prime ideals of $K$ in terms of certain invariants attached to Newton polygons of higher…

This extends a theorem of Davenport and Erd\"os on sequences of rational integers to sequences of integral ideals in arbitrary number fields $K$. More precisely, we introduce a logarithmic density for sets of integral ideals in $K$ and…

Let $K$ be a tamely ramified abelian cubic number field with discriminant $D_K$. We prove that the number of trace-one monic integral polynomials with root field $K$ and height $H$ is equal to the number of ideals in the quadratic field…

This work proposes a proof of the simplest cubic primes counting problem. It shows that the subset of primes {p = n^3 + 2 is prime : n => 1} is an infinite subset of primes. Further, the expected order of magnitude of the cubic primes…

In this thesis, we prove the existence of a secondary term for the count of cubic extensions of the function field $\mathbb{F}_q(t)$ of fixed absolute norm of discriminant. We show that the number of cubic extensions with absolute norm of…

Let $K$ be an imaginary quadratic field different from $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$. For a positive integer $N$, let $K_\mathfrak{n}$ be the ray class field of $K$ modulo $\mathfrak{n}=N\mathcal{O}_K$. By using the…

Let $K$ be an imaginary quadratic field with discriminant $d_K\leq-7$. We deal with problems of constructing normal bases between abelian extensions of $K$ by making use of singular values of Siegel functions. First, we show that a…