On a generalization of the Euler totient function

Abstract

For a general polynomial Euler product F(s) we define the associated Euler totient function ϕ(n, F) and study its asymptotic properties. We prove that for F(s) belonging to certain subclass of the Selberg class of L-functions, the error term in the asymptotic formula for the sum of ϕ(n, F) over positive integers n ≤ x behaves typically as a linear function of x. We show also that for the Riemann zeta function the square mean value of the error term in question is minimal among all polynomial Euler products from the Selberg class, and that this property uniquely characterizes ζ(s).

Jordan totient quotients

“…We consider not only the balanced Jordan totient quotients, but also a more general class of totient functions (see Section 3 for the definitions). This class is similar to the one earlier studied by Kaczorowski [5] in the context of inverse theorems for the Selberg class. An analog of Theorem 1 for non-zero weight can be easily established on invoking Lemma 6, and partial summation, but has such a long winding formulation that we leave writing this down to the interested reader.…”

Iterating the Sum of Möbius Divisor Function and Euler Totient Function

“…It is well known [1,[4][5][6][7][8][9][10][11][12][13][14] that Euler ϕ-function have several interesting formula. For example, if (x, y) = 1 with two positive integers x and y, then ϕ(xy) = ϕ(x)ϕ(y).…”

Counting invertible sums of squares modulo $n$ and a new generalization of Euler's totient function

“…Recall that ϕ(n) is defined as the number of positive integers less than or equal to n that are coprime to n. Many generalizations and analogs of Euler's function are known. See, for instance [5,6,8,9,13,16] or the special chapter on this topic in [15]. Among the generalizations, the most significant is probably the Jordan's totient function J k given by J k (n) = n k p|n (1 − p −k ) (n ∈ N := {1, 2, .…”