Journal of Number Theory最新文献

This paper is inspired by Richards' work on large gaps between sums of two squares [10]. It is shown in [10] that there exist arbitrarily large values of λ and μ, where $\mu \ge C\log \lambda$, such that intervals $[\lambda ,\lambda +\mu ]$ do not contain any sums of two squares. Geometrically, these gaps between sums of two squares correspond to annuli in $R^2$ that do not contain any integer lattice points. A major objective of this paper is to investigate the sparse distribution of integer lattice points within annular regions in $R^2$. Specifically, we establish the existence of annuli $\{x\in R^2:\lambda \le |x{|}^2\le \lambda +\kappa \}$ with arbitrarily large λ and $\kappa \ge C{\lambda }^s$ for $0<s<\frac{1}{4}$, satisfying that any two integer lattice points within any one of these annuli must be sufficiently far apart. This result is sharp, as such a property ceases to hold at and beyond the threshold $s=\frac{1}{4}$. Furthermore, we extend our analysis to include the sparse distribution of lattice points in spherical shells in $R^3$.

Let $D,k$ be integers with D square free and k even. Let N be a positive integer so that $(N,D)=1$ when D has residue one modulo four and $(N,4D)=1$ when D has residue two or three modulo four. In this paper the asymptotic behavior of a resonance sum $S_X(\alpha ,\beta ;\pi )$ attached to the quadratic base change lift of a holomorphic cusp form f of level N and weight k over the quadratic extension generated by $\sqrt{D}$ is computed. First a Voronoi summation formula is derived that expresses $S_X(\alpha ,\beta ;\pi )$ in terms of the Meier-G function. Then, using the known asymptotics of the Meier-G function the asymptotic behavior of $S_X(\alpha ,\beta ;\pi )$ as X approaches infinity is determined. It is then shown that using only finitely many Fourier coefficients of the form, one can recover the weight k and the level N, which is a special case of the multiplicity one theorem.

Let E be an elliptic curve over $Q$, p an odd prime number and n a positive integer. In this article, we investigate the ideal class group $\mathrm{Cl}\left(Q\right(E\left[p^n\right]\left)\right)$ of the $p^n$-division field $Q\left(E\right[p^n\left]\right)$ of E. We introduce a certain subgroup $E{\left(Q\right)}_{\mathrm{ur},p^n}$ of $E\left(Q\right)$ and study the p-adic valuation of the class number $\#\mathrm{Cl}\left(Q\right(E\left[p^n\right]\left)\right)$.

In addition, when $n=1$, we further study $\mathrm{Cl}\left(Q\right(E\left[p\right]\left)\right)$ as a $\mathrm{Gal}\left(Q\right(E\left[p\right])/Q)$-module. More precisely, we study the semi-simplification ${\left(\mathrm{Cl}\right(Q\left(E\right[p\left]\right))\otimes Z_p)}^{\mathrm{ss}}$ of $\mathrm{Cl}\left(Q\right(E\left[p\right]\left)\right)\otimes Z_p$ as a $Z_p\left[\mathrm{Gal}\right(Q\left(E\right[p\left]\right)/Q\left)\right]$-module. We obtain a lower bound of the multiplicity of the $E\left[p\right]$-component in the semi-simplification when $E\left[p\right]$ is an irreducible $\mathrm{Gal}\left(Q\right(E\left[p\right])/Q)$-module.

We investigate the number of squares in a very broad family of binary recurrence sequences with $u_0=1$. We show that there are at most two distinct squares in such sequences (the best possible result), except under very special conditions where we prove there are at most three such squares.

We show that any convergent (shuffle) arborified zeta value admits a series representation. This justifies the introduction of a new generalisation to rooted forests of multiple zeta values, and we study its algebraic properties. As a consequence of the series representation, we derive elementary proofs of some results of Bradley and Zhou for Mordell-Tornheim zeta values and give explicit formulas. The series representation for shuffle arborified zeta values also implies that they are conical zeta values. We characterise which conical zeta values are arborified zeta values and evaluate them as sums of multiple zeta values with rational coefficients.

In this paper we prove that if A and B are infinite subsets of positive integers such that every positive integer n can be written as $n=ab$, $a\in A$, $b\in B$, then $\underset{x\to \infty }{\lim }\frac{A\left(x\right)B\left(x\right)}{x}=\infty$. We present some tight density bounds in connection with multiplicative complements.

For a positive integer m, a (positive definite integral) quadratic form is called primitively m-universal if it primitively represents all quadratic forms of rank m. It was proved in [9] that there are exactly 107 equivalence classes of primitively 1-universal quaternary quadratic forms. In this article, we prove that the minimal rank of primitively 2-universal quadratic forms is six, and there are exactly 201 equivalence classes of primitively 2-universal senary quadratic forms.

We study the distribution of consecutive sums of two squares in arithmetic progressions. If ${\left\{E_n\right\}}_{n\in N}$ is the sequence of sums of two squares in increasing order, we show that for any modulus q and any congruence classes $a_1,a_2,a_3\mathrm{mod}q$ which are admissible in the sense that there are solutions to $x^2+y^2\equiv a_i\mathrm{mod}q$, there exist infinitely many n with $E_{n+i-1}\equiv a_i\mathrm{mod}q$, for $i=1,2,3$. We also show that for any $r_1,r_2\ge 1$, there exist infinitely many n with $E_{n+i-1}\equiv a_1\mathrm{mod}q$ for $1\le i\le r_1$ and $E_{n+i-1}\equiv a_2\mathrm{mod}q$ for $r_1+1\le i\le r_1+r_2$.

In this article, we study hook lengths of ordinary partitions and t-regular partitions. We establish hook length biases for the ordinary partitions and motivated by them we find a few interesting hook length biases in 2-regular partitions. For a positive integer k, let $p_{\left(k\right)}\left(n\right)$ denote the number of hooks of length k in all the partitions of n. We prove that $p_{\left(k\right)}\left(n\right)\ge p_{(k+1)}\left(n\right)$ for all $n\ge 0$ and $n\ne k+1$; and $p_{\left(k\right)}(k+1)-p_{(k+1)}(k+1)=-1$ for $k\ge 2$. For integers $t\ge 2$ and $k\ge 1$, let $b_{t,k}\left(n\right)$ denote the number of hooks of length k in all the t-regular partitions of n. We find generating functions of $b_{t,k}\left(n\right)$ for certain values of t and k. Exploring hook length biases for $b_{t,k}\left(n\right)$, we observe that in certain cases biases are opposite to the biases for ordinary partitions. We prove that $b_{2,2}\left(n\right)\ge b_{2,1}\left(n\right)$ for all $n>4$, whereas $b_{2,2}\left(n\right)\ge b_{2,3}\left(n\right)$ for all $n\ge 0$. We also propose some conjectures on biases among