International Journal of Number Theory

Let $E/F$ be an elliptic curve defined over a number field $F$. Suppose that $E$ has complex multiplication over $\overline{F}$, i.e. ${\mathrm{\text{End}}}_{\overline{F}}(E)\otimes \mathbb{Q}$ is an imaginary quadratic field. With the aid of CM theory, we find elliptic curves whose quadratic twists have a constant root number.

Robin’s criterion states that the Riemann hypothesis is equivalent to $\sigma (n)<e^{\gamma }nloglogn$ for all integers $n\ge 5041$, where $\sigma (n)$ is the sum of divisors of $n$ and $\gamma$ is the Euler–Mascheroni constant. We prove that the Riemann hypothesis is equivalent to the statement that $\sigma (n)<\frac{e^{\gamma }}{2}nloglogn$ for all odd numbers $n\ge 3^4\cdot 5^3\cdot 7^2\cdot 11\cdots 67$. Lagarias’s criterion for the Riemann hypothesis states that the Riemann hypothesis is equivalent to $\sigma (n)<H_n+exp{H}_{n}log{H}_n$ for all integers $n\ge 1$, where $H_n$ is the $n$th harmonic number. We establish an analogue to Lagarias’s criterion for the Riemann hypothesis by creating a new harmonic series $H_n^{\prime }=2H_n-H_{2n}$ and demonstrating that the Riemann hypothesis is equivalent to $\sigma (n)\le \frac{3n}{logn}+exp{H}_n^{\prime }log{H}_n^{\prime }$ for all odd $n\ge 3$. We prove stronger analogues to Robin’s inequality for odd squarefree numbers. Furthermore, we find a general formula that studies the effect of the prime factorization of $n$ and its behavior in Robin’s inequality.

Let ${𝒮}$ be a finite cyclic semigroup written additively. An element $e$ of ${𝒮}$ is said to be idempotent if $e+e=e$. A sequence $T$ over ${𝒮}$ is called idempotent-sum-free provided that no idempotent of ${𝒮}$ can be represented as a sum of one or more terms from $T$. We prove that an idempotent-sum-free sequence over ${𝒮}$ of length over approximately a half of the size of ${𝒮}$ is well structured. This result generalizes the Savchev–Chen Structure Theorem for zero-sum-free sequences over finite cyclic groups.

For a positive definite ternary integral quadratic form $f$, let $r(n,f)$ be the number of representations of an integer $n$ by $f$. A ternary quadratic form $f$ is said to be a generalized Bell ternary quadratic form if $f$ is isometric to $x^2+2^{\alpha }{y}^2+2^{\beta }{z}^2$ for some nonnegative integers $\alpha ,\beta$. In this paper, we give a closed formula for $r(n,f)$ for a generalized Bell ternary quadratic form $f(x,y,z)=x^2+2^{\alpha }{y}^2+2^{\beta }{z}^2$ with $0\le \alpha \le \beta \le 6$ and class number greater than $1$ by using the Minkowski–Siegel formula and bases for spaces of cusp forms of weight $\frac{3}{2}$ and level $2^t$ with $t=6,7,8$ consisting of eta-quotients.

We consider the proportion of genus one curves over $\mathbb{Q}$ of the form $z^2=f(x,y)$ where $f(x,y)\in \mathbb{Z}[x,y]$ is a binary quartic form (or more generally of the form $z^2+h(x,y)z=f(x,y)$ where also $h(x,y)\in \mathbb{Z}[x,y]$ is a binary quadratic form) that have points everywhere locally. We show that the proportion of these curves that are locally soluble, computed as a product of local densities, is approximately 75.96%. We prove that the local density at a prime $p$ is given by a fixed degree-$9$ rational function of $p$ for all odd $p$ (and for the generalized equation, the same rational function gives the local density at every prime). An additional analysis is carried out to estimate rigorously the local density at the real place.

Let $q$ be a prime of the form $4k+3$. Then for any three mutually orthogonal great circles of the sphere $x^2+y^2+z^2=q$ lying on rational planes, at least one of these circles does not contain any rational points.

In this paper, we study the Iwasawa $\lambda$-invariant of the cyclotomic ${\mathbb{Z}}_2$-extension of $\mathbb{Q}(\sqrt{pq})$, where $p,q$ are distinct odd prime numbers satisfying certain arithmetic conditions. Moreover, we obtain an upper bound of the $2$-part of the class number of certain quartic number fields by calculating the Sinnott index explicitly.

For any $x\in [0,1)$, let $S_n(x)$ be the partial summation of the first $n$ digits in the binary expansion of $x$ and $R_n(x)$ be its run-length function. The classical Borel’s normal number theorem tells us that for almost all $x\in [0,1)$, the limit of $S_n(x)/n$ as $n$ goes to infinity is one half. On the other hand, the Erdös–Rényi limit theorem shows that $R_n(x)$ increases to infinity with the logarithmic speed ${log}_{2}n$ as $n\to \infty$ for almost every $x$ in $[0,1)$. In this paper, we are interested in the intersections of exceptional sets arising in the above two famous theorems. More precisely, for any $0\le {\alpha }_1\le {\alpha }_2\le 1$ and $0\le {\beta }_1\le {\beta }_2\le +\infty$, we completely determine the Hausdorff dimension of the following set:

For each algebraic number $\alpha$ and each positive real number $t$, the $t$-metric Mahler measure $m_t(\alpha )$ creates an extremal problem whose solution varies depending on the value of $t$. The second author studied the points $t$ at which the solution changes, called exceptional points for$\alpha$ . Although each algebraic number has only finitely many exceptional points, it is conjectured that, for every $N\in \mathbb{N}$, there exists a number having at least $N$ exceptional points. In this paper, we describe a polynomial time algorithm for establishing the existence of numbers with at least $N$ exceptional points. Our work constitutes an improvement over the best known existing algorithm which requires exponential time. We apply our main result to show that there exist numbers with at least $37$ exceptional points, another improvement over previous work which was only able to reach $11$ exceptional points.

$S\subset \{1,2,\dots ,n\}$ is called a Sidon set if $a+b$ are all distinct for any $a,b\in S$. Let $S_n$ be the largest cardinal number of such $S$. We are interested in the sum of elements in the Sidon set $S$. In this paper, we prove that for any $𝜖>0$,

In 2012, Chan posed six conjectures on congruences for Appell–Lerch sums and five of them were proved by mathematicians. In this paper, we confirm the remaining conjectural congruence of Chan by utilizing an identity due to Hickerson and Mortenson and the theory of modular forms.

This paper provides a mean value theorem for arithmetic functions $f$ defined by

Let $𝔸[n]$ be the group of $n$-torsion points of a commutative algebraic group $𝔸$ defined over a number field $F$. For a prime $𝔭$ of $F$, we let $N_{𝔭}(𝔸[n])$ be the number of ${𝔽}_{𝔭}$-solutions of the system of polynomial equations defining $𝔸[n]$ when reduced modulo $𝔭$. Here, ${𝔽}_{𝔭}$ is the residue field at $𝔭$. Let ${\pi }_F(x)$ denote the number of primes $𝔭$ of $F$ such that $N(𝔭)\le x$. We then, for algebraic groups of dimension one, compute the $k$th moment limit

Let $C$ be a smooth plane curve of degree $d\ge 4$ defined over a global field $k$ of characteristic $p=0$ or $p>(d-1)(d-2)/2$ (up to an extra condition on $\mathrm{\text{Jac}}(C)$). Unless the curve is bielliptic of degree four, we observe that it always admits finitely many quadratic points. We further show that there are only finitely many quadratic extensions $k(\sqrt{D})$ when $k$ is a number field, in which we may have more points of $C$ than these over $k$. In particular, we have this asymptotic phenomenon valid for Fermat’s and Klein’s equations. Second, we conjecture that there are two infinite sets ${ℰ}$ and ${𝒟}$ of isomorphism classes of smooth projective plane quartic curves over $k$ with a prescribed automorphism group, such that all members of ${ℰ}$ (respectively ${𝒟}$) are bielliptic and have finitely (respectively infinitely) many quadratic points over a number field $k$. We verify the conjecture over $k=\mathbb{Q}$ for $G=\mathbb{Z}/6\mathbb{Z}$ and $\mathrm{\text{GAP}}(16,13)$. The analog of the conjecture over global fields with $p>0$ is also considered.