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Analogues of Alladi's formula
Journal of Number Theory ( IF 0.6 ) Pub Date : 2021-04-01 , DOI: 10.1016/j.jnt.2020.06.001
Biao Wang

Abstract In this note, we mainly show the analogue of one of Alladi's formulas over Q with respect to the Dirichlet convolutions involving the Mobius 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 , l ) = 1 , then − ∑ n ⩾ 2 p ( n ) ≡ l ( mod k ) μ ( n ) φ ( n ) = 1 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.

中文翻译:

阿拉迪公式的类似物

摘要 在这篇笔记中,我们主要展示了关于涉及莫比乌斯函数 μ ( n ) 的狄利克雷卷积的 Alladi 公式之一在 Q 上的类比,这与 Dawsey, Sweeting 的最近工作有关素数集的自然密度和 Woo 和 Kural 等人。这将为我们提供几个新的类似物。特别地,我们得到如果 ( k , l ) = 1 ,则 − ∑ n ⩾ 2 p ( n ) ≡ l ( mod k ) μ ( n ) φ ( n ) = 1 φ ( k ) ,其中 p ( n ) 是 n 的最小素数除数,φ ( n ) 是欧拉的整体函数。这改进了 1921 年哈代的一个公式。最后,我们给出了一些例子,将 φ ( n ) 替换为“接近 n”的函数,其中包括除数和函数。
更新日期:2021-04-01
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