Journal of Number Theory最新文献

Let π be a cuspidal unitary representation od $GL(m,A)$ where $A$ denotes the ring of adèles of $Q$. Let $L(s,\pi )$ be its L-function. We introduce a universal lower bound for the integral ${\int }_{-\infty }^{+\infty }|\frac{L(\frac{1}{2}+it,\pi )}{\frac{1}{2}+it-s}{|}^{2}dt$ where s is equal to 0 or is a zero of $L\left(s\right)$ on the critical line. In the main text, the proof is given for $m\le 2$ and under a few assumptions on π. It relies on the Mellin transform; the proof involves an extension of a deep result of Friedlander-Iwaniec. An application is given to the abscissa of convergence of the Dirichlet series $L(s,\pi )$. In the Appendix, written with Peter Sarnak, the proof is made unconditional for general m.

Roughly speaking, the spectrum of multiplicative functions is the set of all possible mean values. In this paper, we are interested in the spectra of multiplicative functions supported over powerfull numbers. We prove that its real logarithmic spectrum takes values from $-\sqrt{2}/(4+\sqrt{2})=-0.26160...$ to 1 while it is known that the logarithmic spectrum of real multiplicative functions over all natural numbers takes values from 0 to 1. In the course of this study, we correct and complete the proof of Granville and Soundararajan on the spectrum of all multiplicative functions.

A new algorithm for computing Hecke operators for ${\mathrm{SL}}_n$ was introduced in [14]. The algorithm uses tempered perfect lattices, which are certain pairs of lattices together with a quadratic form. These generalize the perfect lattices of Voronoi [17]. The present paper is the first step in characterizing tempered perfect lattices. We obtain a complete classification in the plane, where the Hecke operators are for ${\mathrm{SL}}_2\left(Z\right)$ and its arithmetic subgroups. The results depend on the class field theory of orders in imaginary quadratic number fields.

Let $k,t$ be coprime integers, and let $1\le r\le t$. We let $D_k^{\times }(r,t;n)$ denote the total number of parts among all k-indivisible partitions (i.e., those partitions where no part is divisible by k) of n which are congruent to r modulo t. In previous work of the authors [3], an asymptotic estimate for $D_k^{\times }(r,t;n)$ was shown to exhibit unpredictable biases between congruence classes. In the present paper, we confirm our earlier conjecture in [3] that there are no “ties” (i.e., equalities) in this asymptotic for different congruence classes. To obtain this result, we reframe this question in terms of L-functions, and we then employ a nonvanishing result due to Baker, Birch, and Wirsing [1] to conclude that there is always a bias towards one congruence class or another modulo t among all parts in k-indivisible partitions of n as n becomes large.

We speculate on the distribution of primes in exponentially growing, linear recurrence sequences ${\left(u_n\right)}_{n\ge 0}$ in the integers. By tweaking a heuristic which is successfully used to predict the number of prime values of polynomials, we guess that either there are only finitely many primes $u_n$, or else there exists a constant $c_u>0$ (which we can give good approximations to) such that there are $\sim c_u\log N$ primes $u_n$ with $n\le N$, as $N\to \infty$. We compare our conjecture to the limited amount of data that we can compile. One new feature is that the primes in our Euler product are not taken in order of their size, but rather in order of the size of the period of the $u_n\left(\mathrm{mod}p\right)$.

We evaluate asymptotically the smoothed first moment of central values of families of primitive cubic, quartic and sextic Dirichlet L-functions, using the method of double Dirichlet series. Quantitative non-vanishing results for these L-values are also proved.

Let S be a sequence over a finite abelian group G and $v_g\left(S\right)$ be the times that $g\in G$ occurs in S. A sequence S over G is called weak-regular if $v_g\left(S\right)\le \mathrm{ord}\left(g\right)$ for every $g\in G$. Denote by $N\left(G\right)$ the smallest integer t such that every weak-regular sequence S over G of length $\left|S\right|\ge t$ has a nonempty zero-sum subsequence T of S satisfying $v_g\left(T\right)=v_g\left(S\right)$ for some $g|S$. $N\left(G\right)$ has been formulated by Gao et al. very recently to study zero-sum problems in a unify way and determined only for cyclic groups of prime-power order and some other very special groups. As for general cyclic groups $G=C_n$, they gave that$2n-\lceil 3\sqrt{n}\rceil +1\le N\left(G\right)\le 2n-\lceil 2\sqrt{n+1}\rceil +1.$

In this paper, we first study the max gap of the unit group of the residue class ring and give an upper bound of it. Then we prove that there is always an integer $a\in [n^{\frac{1}{2}},n^{\frac{1}{2}}+n^{\frac{1}{4}}]$ such that $\gcd (a,n)=1$ for $n\ge 2227$. Finally, we improve the result of Gao et al. by showing that$2n-\lceil 2\sqrt{n+1}\rceil \le N($