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For an algebraic number field $K$, the P\'{o}lya group of $K$, denoted by $Po(K),$ is the subgroup of the ideal class group $Cl_{K}$ generated by the ideal classes of the products of prime ideals of same norm. The number field $K$ is said to be P\'{o}lya if $Po(K)$ is trivial. Motivated by several recent studies on the group $Po(K)$ when $K$ is a totally real bi-quadratic field, we investigate the same with five distinct odd primes ramifying in $K/\mathbb{Q}$. This extends the previous results on this problem, where the number of distinct ramified primes was at most four., Comment: 7 - 2epsilon pages, comments are welcome

The P\'{o}lya group of an algebraic number field is the subgroup generated by the ideal classes of the products of prime ideals of equal norm inside the ideal class group. Inspired by a recent work on consecutive quadratic fields with large class numbers by Cherubini et al., we extend the notion of {\it consecutiveness} of number fields to certain parametric families of cyclic cubic fields and bi-quadratic fields and address the question of the existence of infinitely many such consecutive fields with large P\'{o}lya groups. This extends a recent result of the second author and Saikia for totally real bi-quadratic fields., Comment: 7 - epsilon pages. Comments are welcome

The P\'{o}lya group of an algebraic number field is a particular subgroup of the ideal class group. This article provides an overview of recent results on P\'{o}lya groups of number fields, their connection with the ring of integer-valued polynomials and touches upon some results on number fields having large P\'{o}lya groups. For the sake of completeness, we have included the proof of Zantema's theorem which laid the foundation to determine the P\'{o}lya groups of many finite Galois extensions over $\mathbb{Q}$. Towards the end of the article, we provide an elementary proof of a weaker version of a recent result of Cherubini et al., Comment: 11 pages

For a real quadratic field $K= \mathbb{Q}(\sqrt{d})$ with discriminant $D_{K}$ having four distinct prime factors, we study the structure of the $2$-class group $A(K_{1})$ of the first layer $K_{1} = \mathbb{Q}(\sqrt{2},\sqrt{d})$ of the cyclotomic $\mathbb{Z}_{2}$-extension of $K$. With some suitably convenient assumptions on the rank and the order of $A(K_{1})$, we characterize $K$ for which the $2$-class group $A(K)$ is isomorphic to $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$. We infer that the $2$-ranks of the class groups in each layer stabilizes by virtue of a result of Fukuda. This also provides an alternate way to establish that the Iwasawa $\mu$-invariant of $K$ vanishes. In some cases, we also provide sufficient conditions on the constituent prime factors of $D_{K}$ that imply $A(K) \simeq \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$, $A(K_{1}) \simeq \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/4\mathbb{Z}$ and $A(K^{\prime}) \simeq \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$, where $K^{\prime} = \mathbb{Q}(\sqrt{2d})$. This extends some results obtained by Mizusawa., Comment: 13 pages, 3 Tables

In 2016, in the work related to Galois representations, Greenberg conjectured the existence of multi-quadratic $p$-rational number fields of degree $2^{t}$ for any odd prime number $p$ and any integer $t \geq 1$. Using the criteria provided by him to check $p$-rationality for abelian number fields, certain infinite families of quadratic, biquadratic and triquadratic $p$-rational fields have been shown to exist in recent years. In this article, for any integer $k \geq 1$, we build upon the existing work and prove the existence of infinitely many prime numbers $p$ for which the imaginary quadratic fields $\mathbb{Q}(\sqrt{-(p - 1)}),\ldots,\mathbb{Q}(\sqrt{-(p - k)})$ and $\mathbb{Q}(\sqrt{-p(p - 1)}),\ldots, \mathbb{Q}(\sqrt{-p(p - k)})$ are all $p$-rational. This can be construed as analogous results in the spirit of Iizuka's conjecture on the divisibility of class numbers of consecutive quadratic fields. We also address a similar question of $p$-rationality for two consecutive real quadratic fields by proving the existence of infinitely many $p$-rational fields of the form $\mathbb{Q}(\sqrt{p^{2} + 1})$ and $\mathbb{Q}(\sqrt{p^{2} + 2})$. The result for imaginary quadratic fields is accomplished by producing infinitely many primes for which the corresponding consecutive discriminants have large square divisors and the same for real quadratic fields is proven using a result of Heath-Brown on the density of square-free values of polynomials at prime arguments., Comment: 10 pages, 3 tables. Comments are welcome

Direction sets, recently introduced by Leonetti and Sanna, are generalization of ratio sets of subsets of positive integers. In this article, we generalize the notion of direction sets and define {\it $k$-generalized direction sets} and {\it distinct $k$-generalized direction sets} for subsets of positive integers. We prove a necessary condition for a subset of $\mathcal{S}^{k - 1} := \{\underline{x} \in [0,1]^{k} : ||\underline{x}|| = 1\}$ to be realized as the set of accumulation points of a distinct $k$-generalized direction set. We provide sufficient conditions for some particular subsets of positive integers so that the corresponding $k$-generalized direction sets are dense in $\mathcal{S}^{k - 1}$. We also consider the denseness properties of certain direction sets and give a partial answer to a question posed by Leonetti and Sanna. Finally we consider a similar question in the framework of an algebraic number field., Comment: 8 pages, comments are welcome

For a prime number $p \geq 5$, we explicitly construct a family of imaginary quadratic fields $K$ with ideal class groups $Cl_{K}$ having $p$-rank ${{\rm{rk}}_{p}(Cl_{K})}$ at least $2$. We also quantitatively prove, under the assumption of the $abc$-conjecture, that for sufficiently large positive real numbers $X$ and any real number $\varepsilon$ with $0 < \varepsilon < \frac{1}{p - 1}$, the number of imaginary quadratic fields $K$ with the absolute value of the discriminant $d_{K}$ $\leq X$ and ${{\rm{rk}}_{p}(Cl_{K})} \geq 2$ is $\gg X^{\frac{1}{p - 1} - \varepsilon}$. This improves the previously known lower bound of $X^{\frac{1}{p} - \varepsilon}$ due to Byeon and the recent bound $X^{\frac{1}{p}}/(\log X)^{2}$ due to Kulkarni and Levin., Comment: 10 pages. Comments are welcome

For an algebraic number field $K$ with ring of integers $\mathcal{O}_{K}$, an important subgroup of the ideal class group $Cl_{K}$ is the {\it P\'{o}lya group}, denoted by $Po(K)$, which measures the failure of the $\mathcal{O}_{K}$-module $Int(\mathcal{O}_{K})$ of integer-valued polynomials on $\mathcal{O}_{K}$ from admitting a regular basis. In this paper, we prove that for any integer $n \geq 2$, there are infinitely many totally real bi-quadratic fields $K$ with $|Po(K)| = 2^{n}$. In fact, we explicitly construct such an infinite family of number fields. This extends an infinite family of bi-quadratic fields with P\'{o}lya group $\mathbb{Z}/2\mathbb{Z}$ given by the authors in \cite{self-ja}. This also provides an infinite family of bi-quadratic fields with class numbers divisible by $2^{n}$., Comment: 7 pages

For an integral domain $R$, the {\it ring of integer-valued polynomials} over $R$ consists of all polynomials $f(X) \in R[X]$ such that $f(R) \subseteq R$. An interesting case to study is when $R$ is a Dedekind domain, in particular when $R$ is the ring of integers of an algebraic number field. An algebraic number field $K$ with ring of integers $\mathcal{O}_{K}$ is said to be a P\'{o}lya field if the $\mathcal{O}_{K}$-module of integer-valued polynomials on $K$ admits a regular basis. Associated to $K$ is a subgroup $Po(K)$ of the ideal class group $Cl_{K}$, known as the {\it P\'{o}lya group of $K$}, that measures the failure of $K$ from being a P\'{o}lya field. In this paper, we prove the existence of three pairwise distinct totally real bi-quadratic fields, each having P\'{o}lya group isomorphic to $\mathbb{Z}/2\mathbb{Z}$. This extends the previously known families of number fields considered by Heidaryan and Rajaei in \cite{rajaei-jnt} and \cite{rajaei}. Our results also establish that under mild assumptions, the possibly infinite families of bi-quadratic fields having a non-principal Euclidean ideal class, considered in \cite{self-jnt}, fail to be P\'{o}lya fields., Comment: 12 pages

For a square-free integer $t$, Byeon \cite{byeon} proved the existence of infinitely many pairs of quadratic fields $\mathbb{Q}(\sqrt{D})$ and $\mathbb{Q}(\sqrt{tD})$ with $D > 0$ such that the class numbers of all of them are indivisible by $3$. In the same spirit, we prove that for a given integer $t \geq 1$ with $t \equiv 0 \pmod {4}$, a positive proportion of fundamental discriminants $D > 0$ exist for which the class numbers of both the real quadratic fields $\mathbb{Q}(\sqrt{D})$ and $\mathbb{Q}(\sqrt{D + t})$ are indivisible by $3$. This also addresses the complement of a weak form of a conjecture of Iizuka in \cite{iizuka}. As an application of our main result, we obtain that for any integer $t \geq 1$ with $t \equiv 0 \pmod{12}$, there are infinitely many pairs of real quadratic fields $\mathbb{Q}(\sqrt{D})$ and $\mathbb{Q}(\sqrt{D + t})$ such that the Iwasawa $\lambda$-invariants associated with the basic $\mathbb{Z}_{3}$-extensions of both $\mathbb{Q}(\sqrt{D})$ and $\mathbb{Q}(\sqrt{D + t})$ are $0$. For $p = 3$, this supports Greenberg's conjecture which asserts that $\lambda_{p}(K) = 0$ for any prime number $p$ and any totally real number field $K$., Comment: Keywords and MSc classifications added. Minor modifications are made

For every positive integer $n$, Sita Ramaiah's identity states that \medskip \begin{equation*} \sum_{a_1, a_2, a_1+a_2 \in (\mathbb{Z}/n\mathbb{Z})^*} \gcd(a_1+a_2-1,n) = \phi_2(n)\sigma_0(n) \; \text{ where } \; \phi_2(n)= \sum_{a_1, a_2, a_1+a_2 \in (\mathbb{Z}/n\mathbb{Z})^*} 1, \end{equation*} \medskip where $(\mathbb{Z}/n\mathbb{Z})^*$ is the multiplicative group of units of the ring $\mathbb{Z}/n\mathbb{Z}$ and $\sigma_s(n) = \displaystyle\sum_{d\mid n}d^s$. \smallskip This identity can also be viewed as a generalization of Menon's identity. In this article, we generalize this identity to an algebraic number field $K$ involving a Dirichlet character $\chi$. Our result is a further generalization of a recent result in \cite{wj} and \cite{sury}., Comment: Keywords added. Minor modifications are made

Let $k \geq 1$ be a cube-free integer with $k \equiv 1 \pmod {9}$ and $\gcd(k, 7\cdot 571)=1$. In this paper, we prove the existence of infinitely many triples of imaginary quadratic fields $\mathbb{Q}(\sqrt{d})$, $\mathbb{Q}(\sqrt{d+1})$ and $\mathbb{Q}(\sqrt{d+k^2})$ with $d \in \mathbb{Z}$ such that the class number of each of them is divisible by $3$. This affirmatively answers a weaker version of a conjecture of Iizuka \cite{iizuka-jnt}., Comment: To appear in Acta Arithmetica

For an odd integer $d > 1$ and a finite Galois extension $K/\mathbb{Q}$ of degree $d$, G. L\"{u} and Z. Yang \cite{lu3} obtained an asymptotic formula for the mean values of the divisor function for $K$ over square integers. In this article, we obtain the same for finitely many number fields of odd degree and pairwise coprime discriminants, together with the moment of the error term arising here, following the method adapted by S. Shi in \cite{shi}. We also define the sum of divisor function over number fields and find the asymptotic behaviour of the summatory function of two number fields taken together., Comment: 16 pages. To appear in Res. Number Theory

Let $p$ be an odd prime number. In this article, we study the number of quadratic residues and non-residues modulo $p$ which are multiples of $2$ or $3$ or $4$ and lying in the interval $[1, p-1]$, by applying the Dirichlet's class number formula for the imaginary quadratic field $\mathbb{Q}(\sqrt{-p})$., Comment: To appear in the conference proceedings of ICCGNFRT-2017

H. W. Lenstra \cite{lenstra} introduced the notion of an Euclidean ideal class, which is a generalization of norm-Euclidean ideals in number fields. Later, families of number fields of small degree were obtained with an Euclidean ideal class (for instance, in \cite{hester1} and \cite{cathy}). In this paper, we construct certain new families of biquadratic number fields having a non-principal Euclidean ideal class and this extends the previously known families given by H. Graves \cite{hester1} and C. Hsu \cite{cathy}., Comment: It is a preliminary version. Comments and suggestions are welcome

Let $q\geq 1$ be any integer and let $ \epsilon \in [\frac{1}{11}, \frac{1}{2})$ be a given real number. In this short note, we prove that for all primes $p$ satisfying $$ p\equiv 1\pmod{q}, \quad \log\log p > \frac{\log 6.83}{\frac{1}{2}-\epsilon} \mbox{ and } \frac{\phi(p-1)}{p-1} \leq \frac{1}{2} - \epsilon, $$ there exists a quadratic non-residue $g$ which is not a primitive root modulo $p$ such that $gcd\left(g, \frac{p-1}{q}\right) = 1$., Comment: to appear in Bulletin of the Australian Mathematical Society

For any integer $l\geq 1$, let $p_1, p_2, \ldots, p_{l+2}$ be distinct prime numbers $\geq 5.$ For all real numbers $X>1,$ we let $N_{3,l}(X)$ denote the number of real quadratic fields $K$ whose absolute discriminant $d_K\leq X$ and $d_K$ is divisible by $(p_1\ldots p_{l+2})$ together with the class number $h_K$ of $K$ divisible by $2^{l}\cdot 3.$ Then, in this short note, by following the method in \cite{Byeonkoh}, we prove that $N_{3,l}(X) \gg X^\frac{7}{8}$ for all large enough $X$'s., Comment: To appear in Journal of The Ramanujan Mathematical Society

In this article, we prove some subsets of the set of natural numbers $\mathbb{N}$ and any non-zero ideals of an order of imaginary quadratic fields are fractionally dense in $\mathbb{R}_{>0}$ and $\mathbb{C}$ respectively., Comment: To appear in the Rocky Mountain Journal of Mathematics