The Journal of the Indian Mathematical Society

Open Access Open Access  Restricted Access Subscription Access
Open Access Open Access Open Access  Restricted Access Restricted Access Subscription Access

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
     

   Subscribe/Renew Journal


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.
Subscription Login to verify subscription
User
Notifications
Font Size


  • M. Aoki and M. Minamide, A zero density estimate for the derivatives of the Riemann zeta function JANTA 2 (2012), 361-375.

  • X. Cao, Y. Tanigawa and W. Zhai, On a conjecture of Chowla and Walum, Sci. China Math. 53 (2010), 2755-2771.

  • H. Cramer, Uber zwei Satze des Herrn G. H. Hardy, Math. Z. 15 (1922), 201-210.

  • J. Furuya, M. Minamide and Y. Tanigawa, Representations and evaluations of the error term in a certain divisor problem, to appear in Math. Slovaca.

  • S. W. Graham and G. Kolesnik, Van der Corput's Method of Exponential Sums, Cambridge Univ. Press, 1991.

  • R. R. Hall, The behaviour of the Riemann zeta-function on the critical line, Mathematika 46 (1999), 281-313.

  • D. R. Heath-Brown, The Pjatecki-Sapiro prime number theorem, J. Number Theory 16 (1983), 242-266.

  • M. N. Huxley, Exponential sums and lattice points III Proc. London Math. Soc. 87 (2003), 591-609.

  • A. Ivic, The Riemann Zeta-Function, John Wiley & Sons, New York, 1985.

  • C.H. Jia and A. Sankaranarayanan, The mean square of divisor function, Acta Arith. 164 (2014), 181-208.

  • A. A. Karatsuba and S. M. Voronin, The Riemann Zeta-Function, Walter de Gruyter, New York, 1992.

  • Y.K. Lau and K.M. Tsang, On the mean square formula of the error term in the Dirichlet divisor problem, Math. Proc. Camb. Phil. Soc. 146 (2009), 277-287.

  • M. Minamide, The truncated Vorono formula for the derivative of the Riemann zeta function, Indian J. Math. 55 (2013), 325-352.

  • S. Ramanujan, Some formul in the analytic theory of numbers, Messenger of Math. 45 (1916), 81-84.

  • E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, 2-nd ed. rev. by D. R. Heath-Brown, Oxford Univ. Press, 1986.

  • K.C. Tong, On divisor problems (III), Acta Math. Sinica 6 (1956), 515-541.

  • B. M. Wilson, Proofs of some formul enunciated by Ramanujan, Proc. London Math. Soc. 21 (1922), 235-255.


Abstract Views: 420

PDF Views: 0




  • On a Restricted Divisor Problem

Abstract Views: 420  |  PDF Views: 0

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