论文标题
Alladi公式的类似物
Analogues of Alladi's formula
论文作者
论文摘要
在本说明中，我们主要显示了$ \ mathbb {q} $的Alladi公式之一的类似物，就涉及Möbius函数$μ（N）$的Dirichlet卷积而言，这与Dawsey，Dawsey，Sweeting，Sweeting和Woo的最新作品以及Kural and Kural eT and Kural et al and and and Al a an and Al Al a ail a grime有关。这将为我们提供几个新的类似物。特别是，如果$（k，\ ell）= 1 $，则$$ - \ sum _ {\ begin {smallmatrix} n \ geq 2 \\ p（n）\ equiv \ equiv \ eeld（\ eperatorname {mod} k） \ frac1 {φ（k）}，$$，其中$ p（n）$是$ n $的最小prime除法，而$φ（n）$是欧拉的基本函数。这是1921年Hardy的一个公式之一。最后，我们为$φ（n）$替换为“接近$ n $”的$φ（n）$的示例，其中包括划分的功能。
In this note, we mainly show the analogue of one of Alladi's formulas over $\mathbb{Q}$ with respect to the Dirichlet convolutions involving the Möbius function $μ(n)$, which is related to the natural densities of sets of primes by recent work of Dawsey, Sweeting and Woo, and Kural et al. This would give us several new analogues. In particular, we get that if $(k, \ell)=1$, then $$-\sum_{\begin{smallmatrix}n\geq 2\\ p(n)\equiv \ell (\operatorname{mod} k) \end{smallmatrix}} \frac{μ(n)}{φ(n)} = \frac1{φ(k)},$$ where $p(n)$ is the smallest prime divisor of $n$, and $φ(n)$ is Euler's totient function. This refines one of Hardy's formulas in 1921. At the end, we give some examples for the $φ(n)$ replaced by functions "near $n$", which include the sum-of-divisors function.
