호남수학학술지 (Honam Mathematical Journal)

  • 제35권2호
  • /
  • Pages.251-302
  • /
  • 2013
  • /
  • 1225-293X(pISSN)
  • /
  • 2288-6176(eISSN)
호남수학회 (The Honam Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

ON THE CONVOLUTION SUMS OF CERTAIN RESTRICTED DIVISOR FUNCTIONS

  • Kim, Daeyeoul (National Institute for Mathematical Sciences) ;
  • Kim, Aeran (Department of Mathematics and Institute of Pure and Applied Mathematics, Chonbuk National University) ;
  • Sankaranarayanan, Ayyadurai (School of Mathematics, Tata Institute of Fundamental Research)
  • 투고 : 2013.03.28
  • 심사 : 2013.05.04
  • 발행 : 2013.06.25
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

We study convolution sums of certain restricted divisor functions in detail and present explicit evaluations in terms of usual divisor functions for some specific situations.

키워드

참고문헌

  1. A. A. Aygunes, Y. Simsek, The action of Hecke operators to families of Weierstrass-type functions and Weber-type functions and their applications, Applied Mathematics and Computation 218 (2011), 678-682. https://doi.org/10.1016/j.amc.2011.03.090
  2. B. C. Berndt, Ramanujan's Notebooks, Part II. Springer-Verlag, New York, 1989.
  3. B. C. Berndt and R. J. Evans, Chapter 15 of Ramanujan's second notebook, Part II. Modular forms, Acta Arith 47 (1986), 123-142. https://doi.org/10.4064/aa-47-2-123-142
  4. C.-H. Chang, H. M. Srivastava, A note on Bernoulli identities associated with the Weierstrass ${\wp}$-function, Integral Transform. Spec. Funct. 18 (2007), 245-253. https://doi.org/10.1080/10652460701210276
  5. C.-H. Chang, H. M. Srivastava, T.-C. Wu, Some families of Weierstrass-type functions and their applications, Integral Transform. Spec. Funct. 19 (2008), 621-632. https://doi.org/10.1080/10652460802230546
  6. N. Cheng and K. S. Williams, Evaluation of some convolution sums involving the sum of divisors functions, Yokohama Mathematical J. 52 (2005), 39-57.
  7. B. Cho, D. Kim and J. K. Koo, Modular forms arising from divisor functions, J. Math. Anal. Appl. 356 (2009), 537-547. https://doi.org/10.1016/j.jmaa.2009.03.003
  8. B. Cho, D. Kim and J. K. Koo, Divisor functions arising from q-series, Publ. Math. Debrecen 76 (2010), 495-508.
  9. L. E. Dickson, History of the Theory of Numbers, Vol. I, Chelsea Publ. Co., New York, 1952.
  10. L. E. Dickson, History of the Theory of Numbers, Vol. II, Chelsea Publ. Co., New York, 1952.
  11. N. J. Fine, Basic hypergeometric series and applications, American Mathematical Society, Providence, RI, 1988.
  12. J. W. L. Glaisher, Expressions for the first five powers of the series in which the coefficients are sums of the divisors of the exponents, Mess. Math. 15 (1885), 33-36.
  13. J. G. Huard, Z. M. Ou, B. K. Spearman, and K. S. Williams, Elementary Evaluation of Certain Convolution Sums Involving Divisor Functions, Number theory for the millennium, II, 2002, 229-274.
  14. D. Kim, A. Kim, and Y. Li, Convolution sums arising from the divisor functions, J. Korean Math. Soc. 50 (2013), 331-360. https://doi.org/10.4134/JKMS.2013.50.2.331
  15. D. Kim and J. K. Koo, Algebraic integer as values of elliptic functions, Acta Arith. 100 (2001), 105-116. https://doi.org/10.4064/aa100-2-1
  16. D. B. Lahiri, On Ramanujan's function ${\tau}$(n) and the divisor function ${\sigma}_1$(n)-I, Bull. Calcutta Math. Soc. 38 (1946), 193-206.
  17. D. B. Lahiri, On Ramanujan's function ${\tau}$(n) and the divisor function ${\sigma}_k$(n)-II, Bull. Calcutta Math. Soc. 39 (1947), 33-52.
  18. J. Levitt, On a Problem of Ramanujan, M. Phil thesis, University of Nottingham, 1978.
  19. G. Melfi, On some modular identities, de Gruyter, Berlin, 1998, 371-382.
  20. K. Ono, S. Robins and D. T. Wahl, On the representation of integer as sum of triangular numbers, Aequationes Math. 50 (1995), 73-94. https://doi.org/10.1007/BF01831114
  21. V. Ramamani, On some identities conjectured by Srinivasa Ramanujan found in his lithographed notes connected with partition theory and elliptic modular functions, Ph. D Thesis, University of Mysore, 1970.
  22. S. Ramanujan, On certain arithmetical functions, Trans. Cambridge Philos. Soc. 22 (1916), 159-184.
  23. S. Ramanujan, Collected papers, AMS Chelsea Publishing, Providence, RI, 2000.
  24. J. H. Silverman, Advanced topics in the arithmetic of elliptic curves, Springer-Verlag, 1994.
  25. K. S. Williams, The convolution sum ${\Sigma}_{m<\frac{n}{8}}{\sigma}_1(m){\sigma}_1$(n-8m), Pacific J. Math. 228 (2006), 387-396. https://doi.org/10.2140/pjm.2006.228.387
  26. K. S. Williams, On Liouville's twelve squares theorem, Far East J. Math. Sci. 29 (2008), 239-242.
  27. K. S. Williams, Number Theory in the Spirit of Liouville, London Mathematical Society, Student Texts 76, Cambridge, 2011.