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).
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Communicated by J. Schoißengeier.
Supported in part by the grant no. N N201 605940 from the National Science Centre (Poland).
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Kaczorowski, J. On a generalization of the Euler totient function. Monatsh Math 170, 27–48 (2013). https://doi.org/10.1007/s00605-012-0417-6
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DOI: https://doi.org/10.1007/s00605-012-0417-6
Keywords
- Euler totient function
- Square mean value
- Selberg class
- Polynomial Euler products
- Converse theorems
- Riemann zeta function