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- Title
A note on the partial sum of Apostol's Möbius function.
- Authors
Banerjee, D.; Fujisawa, Y.; Minamide, T. M.; Tanigawa, Y.
- Abstract
T. M. Apostol introduced a certain Möbius function μ k (·) of order k, where k ≥ 2 is a fixed integer. Let k=1, then μ 1 (·) coincides with the Möbius function μ (·) , in the usual sense. For any fixed k ≥ 2 , he proved the asymptotic formula ∑ n ≤ x μ k (n) = A k x + O k (x 1 / k log x) as x → ∞ , where A k is a positive constant. Later, under the Riemann Hypothesis, D. Suryanarayana showed the O-term is O k (x 4 k 4 k 2 + 1 exp (D log x log log x ) ) with some positive constant D. In this paper, without using any unproved hypothesis we shall prove that the O-term obtained by Apostol can be improved to O k (x 1 / k exp (- D k (log x) 3 / 5 (log log x) 1 / 5 ) ) with some positive constant D k .
- Subjects
MOBIUS function; RIEMANN hypothesis; PARTIAL sums (Series)
- Publication
Acta Mathematica Hungarica, 2023, Vol 170, Issue 2, p635
- ISSN
0236-5294
- Publication type
Article
- DOI
10.1007/s10474-023-01363-1