In order to study integers with few prime factors, the average of $\unicode[STIX]{x1D6EC}_{k}=\unicode[STIX]{x1D707}\ast \log ^{k}$ has been a central object of research. One of the more important cases, $k=2$ , was considered by Selberg [‘An elementary proof of the prime-number theorem’, Ann. of Math. (2)50 (1949), 305–313]. For $k\geq 2$ , it was studied by Bombieri [‘The asymptotic sieve’, Rend. Accad. Naz. XL (5)1(2) (1975/76), 243–269; (1977)] and later by Friedlander and Iwaniec [‘On Bombieri’s asymptotic sieve’, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4)5(4) (1978), 719–756], as an application of the asymptotic sieve. Let $\unicode[STIX]{x1D6EC}_{j,k}:=\unicode[STIX]{x1D707}_{j}\ast \log ^{k}$ , where $\unicode[STIX]{x1D707}_{j}$ denotes the Liouville function for $(j+1)$ -free integers, and $0$ otherwise. In this paper we evaluate the average value of $\unicode[STIX]{x1D6EC}_{j,k}$ in a residue class $n\equiv a\text{ mod }q$ , $(a,q)=1$ , uniformly on $q$ . When $j\geq 2$ , we find that the average value in a residue class differs by a constant factor from the expected value. Moreover, an explicit formula of Weil type for $\unicode[STIX]{x1D6EC}_{k}(n)$ involving the zeros of the Riemann zeta function is derived for an arbitrary compactly supported ${\mathcal{C}}^{2}$ function.

中文翻译：

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残差类别中广义 VON MANGOLDT 函数的意外平均值

为了研究具有少数素因数的整数，平均 $\unicode[STIX]{x1D6EC}_{k}=\unicode[STIX]{x1D707}\ast \log ^{k}$ 一直是研究的中心对象。比较重要的案例之一， $k=2$ , 被 Selberg ['质数定理的基本证明',安。数学。(2)50(1949), 305–313]。为了 $k\geq 2$ , 它是由 Bombieri ['渐近筛',撕裂。阿卡德。纳兹。加大码 (5)1(2) (1975/76), 243–269; (1977)] 和后来的 Friedlander 和 Iwaniec ['关于 Bombieri 的渐近筛'，安。Sc。规范。极好的。比萨 科学。(4)5(4) (1978), 719–756]，作为渐近筛的应用。让 $\unicode[STIX]{x1D6EC}_{j,k}:=\unicode[STIX]{x1D707}_{j}\ast \log ^{k}$ ， 在哪里 $\unicode[STIX]{x1D707}_{j}$ 表示刘维尔函数 $(j+1)$ -自由整数，和 $0$ 否则。在本文中，我们评估的平均值 $\unicode[STIX]{x1D6EC}_{j,k}$ 在残留类别中 $n\equiv a\text{ mod }q$ , $(a,q)=1$ , 一致在 $q$ . 什么时候 $j\geq 2$ ，我们发现残差类别中的平均值与期望值相差一个常数因子。此外，Weil 类型的显式公式为 $\unicode[STIX]{x1D6EC}_{k}(n)$ 涉及黎曼 zeta 函数的零点是针对任意紧支持的 ${\mathcal{C}}^{2}$ 功能。