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Let s ( n ) = ∑ d ∣ n , d > n d s(n)=\sum _{d\mid n,~d>n} d denote the sum of the proper divisors of n n . The second-named author proved that ω ( s ( n ) ) \omega (s(n)) has normal order log log n \log \log {n} , the analogue for s s -values of a classical result of Hardy and Ramanujan [ The normal number of prime factors of a number n [Quart. J. Math. 48 (1917), 76–92], AMS Chelsea Publ., Providence, RI, 2000, pp. 262–275]. We establish the corresponding Erdős–Kac theorem: ω ( s ( n ) ) \omega (s(n)) is asymptotically normally distributed with mean and variance log log n \log \log {n} . The same method applies with s ( n ) s(n) replaced by any of several other unconventional arithmetic functions, such as β ( n ) ≔ ∑ p ∣ n p \beta (n)≔\sum _{p\mid n} p , n − φ ( n ) n-\varphi (n) , and n + τ ( n ) n+\tau (n) ( τ \tau being the divisor function).
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This content will become publicly available on November 1, 2026
The irrationality of an infinite series involving ω(n) under a prime tuples conjecture
Let $$\omega(n)$$ denote the number of distinct prime factors of $$n$$. Assuming a suitably uniform version of the prime $$k$$-tuples conjecture, we show that the number \begin{align*} \sum_{n=1}^\infty \frac{\omega(n)}{2^n} \end{align*} is irrational. This settles (conditionally) a question of Erd\H{o}s.
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- Award ID(s):
- 2418328
- PAR ID:
- 10628576
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Journal of Number Theory
- Volume:
- 276
- Issue:
- C
- ISSN:
- 0022-314X
- Page Range / eLocation ID:
- 57 to 71
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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