The Image of Carmichael’s Λ-function
نویسنده
چکیده
In this paper, we show that the counting function of the set of values of the Carmichael λ-function is x/(log x), where η = 1 − (1 + log log 2)/(log 2) = 0.08607 . . ..
منابع مشابه
On the range of Carmichael’s universal-exponent function
Let λ denote Carmichael’s function, so λ(n) is the universal exponent for the multiplicative group modulo n. It is closely related to Euler’s φ-function, but we show here that the image of λ is much denser than the image of φ. In particular the number of λ-values to x exceeds x/(log x).36 for all large x, while for φ it is equal to x/(log x)1+o(1), an old result of Erdős. We also improve on an ...
متن کاملTHE NUMBER OF SOLUTIONS OF λ(x) = n
We study the question of whether for each n there is an m 6= n with λ(m) = λ(n), where λ is Carmichael’s function. We give a “near” proof of the fact that this is the case unconditionally, and a complete conditional proof under the Extended Riemann Hypothesis. To Professor Carl Pomerance on his 65th birthday
متن کامل9 INTEGERS 11 A ( 2011 ) Proceedings of Integers Conference 2009
We study the question of whether for each n there is an m �= n with λ(m) = λ(n), where λ is Carmichael’s function. We give a “near” proof of the fact that this is the case unconditionally, and a complete conditional proof under the Extended Riemann Hypothesis. –To Professor Carl Pomerance on his 65th birthday
متن کاملValues of the Euler and Carmichael Functions Which Are Sums of Three Squares
Let φ denote Euler’s totient function. The frequency with which φ(n) is a perfect square has been investigated by Banks, Friedlander, Pomerance, and Shparlinski, while the frequency with which φ(n) is a sum of two squares has been studied by Banks, Luca, Saidak, and Shparlinski. Here we look at the corresponding threesquares question. We show that φ(n) is a sum of three squares precisely sevene...
متن کاملCorrigendum to “period of the Power Generator and Small Values of Carmichael’s Function”
We are indebted to Kelly Postelmans whose question drew our attention to a slip in the proof of Theorem 8 of [1]. In particular, we asserted that for a fixed number n, the number of pairs of primes p, l with gcd(p− 1, l− 1) < D and λ(λ(pl)) = n is at most Dτ(n), an assertion which now seems unjustified. (The notation is defined below.) In this note we give a corrected proof of Theorem 8. As in ...
متن کامل