On $N$-Unital Functions

关于某些对称积分的下界

On some lower bounds of some symmetry integrals

研究了某些算术函数$f:\N \rightarrow \R$的对称积分PDMATHBMN PDMATHBMN，\thinspace称$I_f$;我们从$I_f$的下界得到（对于一大类算术函数$f$）\lq的\lq Selberg积分的下界\lq的\rq \rq \thinspace，比如说$J_f$（这两个积分给出了几乎所有短间隔$[x-h,x+h]$中关于$f$的信息，当$N\le x\le 2N$时）。特别是，当\thinspace $f=d_k$，除数函数（具有狄利克雷级数\thinspace $ζ^k$，\thinspace $ζ$ \thinspace黎曼zeta函数），其中$k\ge 3$是整数，我们给塞尔伯格积分的下界，说\thinspace $J_k=J_{d_k}$，\thinspace $d_k$。我们应用初等方法（柯西不等式得到大筛型界），以给出$I_f$的下界。

Counting subrings of the ring $\Bbb{Z}_m \times \Bbb{Z}_n$

Let $m,n\in \Bbb{N}$. We represent the additive subgroups of the ring $\Bbb{Z}_m \times \Bbb{Z}_n$, which are also (unital) subrings, and deduce explicit formulas for $N^{(s)}(m,n)$ and $N^{(us)}(m,n)$, denoting the number of subrings of the ring $\Bbb{Z}_m \times \Bbb{Z}_n$ and its unital subrings, respectively. We show that the functions $(m,n)\mapsto N^{(s)}(m,n)$ and $(m,n)\mapsto N^{(us)}(m,n)$ are multiplicative, viewed as functions of two variables, and their Dirichlet series can be expressed in terms of the Riemann zeta function. We also establish an asymptotic formula for the sum $\su…