1. Introduction. Let p be a prime number and ${\mathbb{Z}}_p$ the ring of p-adic integers. Let k be a finite extension of the rational number field ℚ, $k_{\infty }$ a ${\mathbb{Z}}_p$-extension of k, $k_n$ the nth layer of $k_{\infty }/k$, and $A_n$ the p-Sylow subgroup of the ideal class group of $k_n$. Iwasawa proved the following well-known theorem about the order $A_n$ of $A_n$: Theorem A (Iwasawa). Let $k_{\infty }/k$ be a ${\mathbb{Z}}_p$-extension and $A_n$ the p-Sylow subgroup of the ideal class group of $k_n$, where $k_n$ is the $n$th layer of $k_{\infty }/k$. Then there exist integers $\lambda =\lambda (k_{\infty }/k)\ge 0$, $\mu =\mu (k_{\infty }/k)\ge 0$, $\nu =\nu (k_{\infty }/k)$, and n₀ ≥ 0 such that $A_n=p^{\lambda n+\mu p^n+\nu }$ for...

Let $b>a>0$. We prove the following asymptotic formula $$\sum _{n⩾0}|\{x/(n+a)\}-\{x/(n+b)\}|=\frac{2}{\pi }\zeta (3/2)\sqrt{cx}+O\left(c^{2/9}{x}^{4/9}\right),$$ with $c=b-a$, uniformly for $x⩾40c^{-5}{(1+b)}^{27/2}$.

Let $N$ be a sufficiently large integer. We prove that almost all sufficiently large even integers $n\in [N-6U,N+6U]$ can be represented as $$n=p_1^2+p_2^3+p_3^3+p_4^3+p_5^3+p_6^3,\left|p_1^2-{\displaystyle \frac{N}{6}}\right|\le U,\left|p_i^3-{\displaystyle \frac{N}{6}}\right|\le U,i=2,3,4,5,6,$$ where $U=N^{1-\delta +\epsilon }$ with $\delta \le 8/225$.