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The convolution sums sum ∑m<n/8 σ1 (m) σ3 (n-8m) and ∑m<n/8 σ3 (m) σ1 (n-8m)

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In this paper we evaluate the convolution sum formulae of Σm<n/8 σ1(m)σ3(n - 8m) and Σm<n/8 σ3(m)σ1(n - 8m) for all n ∈ N.

Abstract

In this paper we evaluate the convolution sum formulae of Σm<n/8 σ1(m)σ3(n - 8m) and Σm<n/8 σ3(m)σ1(n - 8m) for all n ∈ N.

참고문헌 (10)

  1. B. C. Berndt, "Ramanujan's notebooks" (1989), Springer-Verlag
  2. B. C. Berndt, "Ramanujan's notebooks, Part III" (1991), Springer
  3. N. Cheng, "Evaluation of some convolution sums involving the sum of divisors functions", Yokohama Mathematical J, 52, pp.39~57 (2005)
  4. A. Kim, "Evaluation of certain convolution sums involving divisor functions and in nite product sums", British Journal of Mathematics & Computer Science, 5(6) (2015)[DOI]
  5. A. Kim, "Evaluation of some convolution sums and sums of triangular numbers", Asian Journal of Mathematics and Computer research, 9(2), pp.167~213 (2015)
  6. A. Kim, "The convolution sum Σm<n=8 σ3(m)σ3(n - 8m)", British Journal of Mathe-matics & Computer Science, 4(6), pp.774~789 (2014)
  7. A. Kim, "The convolution sums Σm<n4mσ3(m)σ5(n-4m) and Σm<n4mσ5(m)σ3(n-4m)", British Journal of Mathematics & Computer Science, 4(21), pp.3028~3053 (2014)
  8. D. B. Lahiri, "On Ramanujan's function  (n) and the divisor function σk(n), I, Bull", Calcutta Math. Soc, 38, pp.193~206 (1946)
  9. K. S. Williams, "Number Theory in the Spirit of Liouville, London Mathematical Soci-ety" (2011), Student Texts 76, Cambridge
  10. K. S. Williams, "The convolution sum Σm<n=8 σ(m)σ(n - 8m)", Paciffc Journal of Mathematics, 228(2), pp.387~396 (2006)

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