A Probabilistic Look at Series Involving Euler’s Totient Function
نویسنده
چکیده
We use a probabilistic method to evaluate the limit of ∑n x=1 φ(x)x r−1 n−(r+1), where φ(x) is the Euler totient function and r is a nonnegative integer. We extend the probabilistic method to evaluate two other generalized types of series that involve Euler’s totient function. In addition to the probabilistic method, an analytic approach is presented to evaluate the series when the exponent parameter r is a positive real number.
منابع مشابه
On Perfect Totient Numbers
Let n > 2 be a positive integer and let φ denote Euler’s totient function. Define φ(n) = φ(n) and φ(n) = φ(φ(n)) for all integers k ≥ 2. Define the arithmetic function S by S(n) = φ(n) + φ(n) + · · ·+ φ(n) + 1, where φ(n) = 2. We say n is a perfect totient number if S(n) = n. We give a list of known perfect totient numbers, and we give sufficient conditions for the existence of further perfect ...
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