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Minamide, Makoto
- On Zeros of the Derivative of the Modified Selberg Zeta Function for the Modular Group
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1 Graduate School of Mathematics, Nagoya University, Furocho, Chikusa-ku, Nagoya 464-8602, JP
1 Graduate School of Mathematics, Nagoya University, Furocho, Chikusa-ku, Nagoya 464-8602, JP
Source
The Journal of the Indian Mathematical Society, Vol 80, No 3-4 (2013), Pagination: 275-312Abstract
Asymptotic formulas for the number of zeros of the derivative of Selberg zeta functions were investigated in the celebrated paper due to Wenzhi Luo as an attempt to develop the study on the multiplicity of eigenvalues of hyperbolic Laplacians. The present paper is a succeeding study of the work, that is, it will deal with corresponding formulas for the modular group. In this case the Selberg zeta function has zeros which come from non-trivial zeros of the Riemann zeta function. But they are not our objects of this study. Therefore to remove these zeros, we shall define "the modified Selberg zeta function" and give the asymptotic formula for the vertical number of zeros of the derivative of the function. Moreover the paper deals with horizontal distribution of zeros.Keywords
The Derivative of the Selberg Zeta Function, the Distribution of Zeros of the Function for PSL(2, Z).
- On a Restricted Divisor Problem
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Authors
Affiliations
1 Department of Integrated Human Sciences (Mathematics), Hamamatsu University School of Medicine, Handayama 1-20-1, Hamamatsu, Shizuoka, 431-3192, JP
2 Yamaguchi University, Yoshida 1677-1, Yamaguchi 753-8512, JP
3 Graduate School of Mathematics, Nagoya University, Furo-Cho, Nagoya, 464-8602, JP
1 Department of Integrated Human Sciences (Mathematics), Hamamatsu University School of Medicine, Handayama 1-20-1, Hamamatsu, Shizuoka, 431-3192, JP
2 Yamaguchi University, Yoshida 1677-1, Yamaguchi 753-8512, JP
3 Graduate School of Mathematics, Nagoya University, Furo-Cho, Nagoya, 464-8602, JP
Source
The Journal of the Indian Mathematical Society, Vol 83, No 3-4 (2016), Pagination: 269-287Abstract
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
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