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On a Restricted Divisor Problem


Affiliations
1 Department of Integrated Human Sciences (Mathematics), Hamamatsu University School of Medicine, Handayama 1-20-1, Hamamatsu, Shizuoka, 431-3192, Japan
2 Yamaguchi University, Yoshida 1677-1, Yamaguchi 753-8512, Japan
3 Graduate School of Mathematics, Nagoya University, Furo-Cho, Nagoya, 464-8602, Japan
     

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Let 0 < α < 1/2 and let dα(n) be the number of positive divisors k of n such that nα ≤ k ≤ n1-α, which we call a restricted divisor function. In the case α = 1/N (N ∈ N) we derive an asymptotic representation of Σn≤xdα(n). Furthermore we study the mean square of Pα(x) = Σl≤xαφ (x/l), which seems to be a natural object in the case of a restricted divisor problem.

Keywords

The Dirichlet Divisor Problem, Mean Square, Chowla and Walum's Expression.
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  • On a Restricted Divisor Problem

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Authors

Jun Furuya
Department of Integrated Human Sciences (Mathematics), Hamamatsu University School of Medicine, Handayama 1-20-1, Hamamatsu, Shizuoka, 431-3192, Japan
Makoto Minamide
Yamaguchi University, Yoshida 1677-1, Yamaguchi 753-8512, Japan
Yoshio Tanigawa
Graduate School of Mathematics, Nagoya University, Furo-Cho, Nagoya, 464-8602, Japan

Abstract


Let 0 < α < 1/2 and let dα(n) be the number of positive divisors k of n such that nα ≤ k ≤ n1-α, which we call a restricted divisor function. In the case α = 1/N (N ∈ N) we derive an asymptotic representation of Σn≤xdα(n). Furthermore we study the mean square of Pα(x) = Σl≤xαφ (x/l), which seems to be a natural object in the case of a restricted divisor problem.

Keywords


The Dirichlet Divisor Problem, Mean Square, Chowla and Walum's Expression.

References