Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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CONVOLUTION SUMS INCLUDING RESTRICTED DIVISOR FUNCTIONS DERIVED FROM DIRICHLET CONVOLUTION

  • NOHYUN KIM (Department of Mathematics, Jeonbuk National University) ;
  • DAEYEOUL KIM (Department of Mathematics and Institute of Pure and Applied Mathematics, Jeonbuk National University)
  • Received : 2024.01.29
  • Accepted : 2024.12.28
  • Published : 2025.01.30
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Abstract

The study of the general convolution sums including divisor functions, ${{\sum}_{i=1}^{N-1}}{\sigma}(i){\sigma}(N-i)$ has been implemented by a lot of mathematicians. In this article, we define ${\bar{\mathcal{A}}}$ and ${\bar{\mathcal{K}}}$, which are slightly different from the general convolution sums. Moreover, those formulas are calculated using the Dirichlet convolution sum of arithmetical functions.

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References

  1. N. Cheng and K.S. Williams, Convolution sums involving the divisor function, Proceedings of the Edinburgh Mathematical Society 47 (2004), 561-572. https://doi.org/10.1017/S0013091503000956
  2. E. Cohen, Rings of arithmetic functions, Duke Math. J. 19 (1952), 115-129. https://doi.org/10.1215/S0012-7094-52-01913-3
  3. J.W.L. Glaisher, On certain sums of products of quantities depending upon the divisors of a number, Mess. Math. 15 (1885), 1-20.
  4. J.W.L. Glaisher, On the square of the series in which the coefficients are the sums of the divisors of the exponents, Mess. Math. 14 (1884), 156-163.
  5. A. Kim, D. Kim and Y. Li, Convolution sums arising from divisor functions, J. Korean Math. Soc. 50 (2013), 331-360. https://doi.org/10.4134/JKMS.2013.50.2.331
  6. S. Kong, Y. Li and D. Kim, Convolution sums of restricted divisor functions derived from Dirichlet convolution, Publ. Math. Debrecen 105 (2024), 281–303. https://doi.org/10.5486/PMD.2024.9530
  7. D.B. Lahiri, On Ramanujan's function τ(n) and the divisor function σ(n)−I, Bull. Calcutta Math. Soc. 38 (1946), 193-206.
  8. D.H. Lehmer, Some functions of Ramanujan, Math. Student 27 (1959), 105-116.
  9. P.J. McCarthy, Introduction to arithmetical functions, Springer Science & Business Media, 2012.
  10. S. Ramanujan, Collected papers, AMS Chelsea Publishing, Providence, RI, 2000.
  11. S. Ramanujan, On certain arithmetical functions, Trans. Cambridge Philos. Soc. 22 (1916), 159-184.
  12. N.-P. Skoruppa, A quick combinatorial proof of Eisenstein series identities, J. Number Theory 43 (1993), 68-73. https://doi.org/10.1006/jnth.1993.1007
  13. R. Vaidyanathaswamy, The theory of multiplicative arithmetic functions, Trans. Amer. Math. Soc. 33 (1931), 579-662. https://doi.org/10.1090/S0002-9947-1931-1501607-1