International Journal of Number Theory

For any $x$ in $(0,1]$, let $x=[d_1(x),d_2(x),\dots ]$ be the Lüroth expansion of $x$. In this paper, we study the relative convergence speed of its convergents ${\{\frac{P_n(x)}{Q_n(x)}\}}_{n\ge 1}$ to the rate of growth of digits in the Lüroth expansion of an irrational number. For any $z$ in $[0,1]$, the sets

Let $N$ be a fixed positive number. Define $\mathrm{\Psi }(x,y)$ as the number of positive integers $\le x$ having no prime factors $>y$, and define ${\mathrm{\Psi }}_N(x,y,z)$ as the number of positive integers $\le x$ having $N$ prime factors $\in (z,y]$, with all the other prime factors $\le z$. In this paper, we give an asymptotic estimate for the ratio $\frac{{\mathrm{\Psi }}_N(x,y,z)}{\mathrm{\Psi }(x,z)}$, provided that $z<y\le z^2$, $\frac{logy}{logx}=o(1)$, and $\frac{logx}{logy}=o(\frac{logx}{loglogx})$ as $x\to \infty$. Also, combining this estimate with conventional ones for $\mathrm{\Psi }(x,z)$, we provide sharp estimates for ${\mathrm{\Psi }}_N(x,y,z)$.

In this paper, we employ a random dynamical systems approach to study generalized Lüroth series expansions of numbers in the unit interval. We prove that for each $c=\frac{1}{\ell }$ with $\ell \in {\mathbb{N}}_{\ge 3}\cup \{\infty \}$ Lebesgue almost all numbers in $[c,1]$ have uncountably many universal generalized Lüroth series expansions with digits less than or equal to $\ell$, so expansions in which each possible block of digits occurs. In particular this means that Lebesgue almost all $x$ have uncountably many universal generalized Lüroth series expansions using finitely many digits only. For $c=0,$ we show that typically the speed of convergence to an irrational number $x$ of the corresponding sequence of Lüroth approximants is equal to that of the standard Lüroth approximants. For other rational values of $c$ we use stationary measures to study the typical speed of convergence of the approximants and the digit frequencies.

It is a difficult question to generalize Gauss sums to a ring of algebraic integers of an arbitrary algebraic number field. In this paper, we define and discuss Gauss sums over a Dedekind domain of finite norm. In particular, we give a Davenport–Hasse type formula for some special Gauss sums. As an application, we give some more precise formulas for Gauss sums over the algebraic integer ring of an algebraic number field (see Theorems 4.1 and 4.2).

Let $p>3$ be a prime. Let $D$ be the discriminant of an imaginary quadratic order. Assume that $\left|D\right|<4\sqrt{p/3}$ and $(\frac{D}{p})=-1$. We compute the number of ${𝔽}_p$-roots of the class polynomials $H_D(X)$. Suppose $\mathbb{Q}(\sqrt{D_1})\ne \mathbb{Q}(\sqrt{D_2})$, we prove that two class polynomials $H_{D_1}(X)$ and $H_{D_2}(X)$ have a common root in ${𝔽}_p$ if and only if $D_1{D}_2-4p$ is a perfect square. Furthermore, any three class polynomials do not have a common root in ${𝔽}_p$. As an application, we propose a deterministic algorithm for computing the class number of $\mathbb{Q}(\sqrt{-p})$.

Let $q\ge 3$ be an integer, and let $\chi$ denote a Dirichlet character modulo $q$. For any real number $a\ge 0$, we define the generalized Dirichlet $L$-function as

We study the continued fractions $I_1(\tau )$ and $I_2(\tau )$ of order sixteen by adopting the theory of modular functions. These functions are analogues of Rogers–Ramanujan continued fraction $r(\tau )$ with modularity and many interesting properties. Here we prove the modularities of $I_1(\tau )$ and $I_2(\tau )$ to find the relation with the generator of the field of modular functions on ${\mathrm{\Gamma }}_0(16)$. Moreover we prove that the values $2(I_1{(\tau )}^2+1/I_1{(\tau )}^2)$ and $2(I_2{(\tau )}^2+1/I_2{(\tau )}^2)$ are algebraic integers for certain imaginary quadratic quantity $\tau$.

In this paper, we study rational periodic points of polynomial $f_{d,c}(x)=x^d+c$ over the field of rational numbers, where $d$ is an integer greater than two. For period two, we describe periodic points for degrees $d=4,6$. We also demonstrate the nonexistence of rational periodic points of exact period two for $d=2k$ such that $3|2k-1$ and $k$ has a prime factor greater than three. Moreover, assuming the $abc$-conjecture, we prove that $f_{d,c}$ has no rational periodic point of exact period greater than one for sufficiently large integer $d$ and $c\ne -1$.

Recently, Merca and Yee proved some partition identities involving two new partition statistics. We refine these statistics and generalize the results of Merca and Yee. We also correct a small mistake in a result of Merca and Yee.

Let ${\overline{\rho }}_{f,\lambda }$ be the residual Galois representation attached to a newform $f$ and a prime ideal $\lambda$ in the integer ring of its coefficient field. In this paper, we prove explicit bounds for the residue characteristic of the prime ideals $\lambda$ such that ${\overline{\rho }}_{f,\lambda }$ is exceptional, that is reducible, of projective dihedral image, or of projective image isomorphic to ${𝔄}_4$, ${𝔖}_4$ or ${𝔄}_5$. We also develop explicit criteria to check the reducibility of ${\overline{\rho }}_{f,\lambda }$, leading to an algorithm that computes the exact set of such $\lambda$’s. We have implemented this algorithm in PARI/GP. Along the way, we construct lifts of Katz’ $𝜃$ operator in characteristic zero, and we prove a new Sturm bound theorem.