One proof is to note that φ( d) is also equal to the number of possible generators of the cyclic group C d specifically, if C d = ⟨ g⟩ with g d = 1, then g k is a generator for every k coprime to d. (See Arithmetical function for notational conventions.) This Demonstration uses the following formula to calculate : where is the complex zero of. Snapshot 3: for larger, we need more and more zeta zeros in order to closely match the step function. Snapshot 2: the graphs of and the formula using 30 pairs of zeta zeros. Where the sum is over all positive divisors d of n, can be proven in several ways. Snapshot 1: the graphs of the step function and the formula using no zeta zeros. It is written using the Greek letter phi as φ ( n ) In number theory, Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n. The points on the top line represent φ( p) when p is a prime number, which is p − 1.
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