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CONVOLUTION SUMS OF ODD AND EVEN DIVISOR FUNCTIONS
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  • Journal title : Honam Mathematical Journal
  • Volume 35, Issue 3,  2013, pp.445-506
  • Publisher : The Honam Mathematical Society
  • DOI : 10.5831/HMJ.2013.35.3.445
 Title & Authors
CONVOLUTION SUMS OF ODD AND EVEN DIVISOR FUNCTIONS
Kim, Daeyeoul;
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 Abstract
Let denote the sum of the s-th power of the positive divisors of N and with , > 0 and . In a celebrated paper [33], Ramanuja proved using elementary arguments. The coefficients' relation in this identity () motivated us to write this article. In this article, we found the convolution sums for odd and even divisor functions with , , and . If N is an odd positive integer, , , , and , then there exist satisfying with and () = 1(Theorem 1.1). We also give an elementary problem (O) and solve special cases of them in (O) (Corollary 3.27).
 Keywords
Divisor functions;Convolution sums;
 Language
English
 Cited by
1.
CONVOLUTION SUMS OF ODD AND EVEN DIVISOR FUNCTIONS II,;

호남수학학술지, 2015. vol.37. 2, pp.149-185 crossref(new window)
1.
CONVOLUTION SUMS OF ODD AND EVEN DIVISOR FUNCTIONS II, Honam Mathematical Journal, 2015, 37, 2, 149  crossref(new windwow)
 References
1.
A. Alaca, S. Alaca and K. S. Williams, The convolution sum ${\Sigma}_{m ${\sigma}(m){\sigma}(n-16m)$, Canad. Math. Bull. 51 (2008), 3-14. crossref(new window)

2.
A. Alaca, S. Alaca and K. S. Williams, The convolution sums ${\Sigma}_{l+24m=n}$ ${\sigma}(l){\sigma}(m)$ and ${\Sigma}_{3l+8m=n}$, M. J. Okayama Univ. 49 (2007), 93-111.

3.
A. Alaca, S. Alaca and K. S. Williams, Evaluation of the convolution sums ${\Sigma}_{l+12m=n}$ ${\sigma}(l){\sigma}(m)$ and ${\Sigma}_{3l+4m=n}$ ${\sigma}(l){\sigma}(m)$, Adv. Theor. and Appl. Math. 1 (2006), 27-48.

4.
A. Alaca, S. Alaca and K. S. Williams, Evaluation of the convolution sums ${\Sigma}_{l+18m=n}$ ${\sigma}(l){\sigma}(m)$ and ${\Sigma}_{2l+9m=n}$ ${\sigma}(l){\sigma}(m)$, Int. J. Math. Sci. 2 (2007), 45-68.

5.
A. Alaca, S. Alaca and K. S. Williams, Ottawa Evaluation of the sums ${\Sigma}_{m=1m{\equiv}a\;(mod\;4)}^{n-1}$ ${\sigma}(m){\sigma}(n-m)$, Czechoslovak Math. J. 134 (2009), 847-859.

6.
S. Alaca and K. S. Williams, Evaluation of the convolution sums ${\Sigma}_{l+6m=n}$ ${\sigma}(l){\sigma}(m)$ and ${\Sigma}_{2l+3m=n}$ ${\sigma}(l){\sigma}(m)$, J. Number Theory, 124 (2007), 491-510. crossref(new window)

7.
B. C. Berndt, Ramanujan's Notebooks, Part II. Springer-Verlag, New York, 1989.

8.
M. Besgue, Extrait d'une lettre de M. Besgue a M Liouville, J. Math. Pures Appl. 7 (1862), 256.

9.
H. H. Chan, and S. Cooper, Powers of theta functions, Pacific Journal of Mathematics, 235 (2008), 1-14. crossref(new window)

10.
N. Cheng and K. S. Williams, Convolution sums involving the divisor functions, Proc. Edinburgh Math. Soc. 47 (2004), 561-572. crossref(new window)

11.
B. Cho, D. Kim and J. K. Koo, Divisor functions arising from q-series, Publ. Math. Debrecen 76 (2010), 495-508.

12.
B. Cho, D. Kim and J. K. Koo, Modular forms arising from divisor functions, J. Math. Anal. Appl. 356 (2009), 537-547. crossref(new window)

13.
S. Cooper and P. C. Toh, Quintic and septic Eisenstein series, Ramanujan J. 19 (2009), 163-181. crossref(new window)

14.
L. E. Dickson, History of the Theory of Numbers, Vol.I, Chelsea Publ. Co., New York, 1952.

15.
L. E. Dickson, History of the Theory of Numbers, Vol.II, Chelsea Publ. Co., New York, 1952.

16.
N. J. Fine, Basic hypergeometric series and applications, American Mathematical Society, Providence, RI, 1988.

17.
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.

18.
J. W. L. Glaisher, On certain sums of products of quantities depending upon the divisors of a number, Mess. Math. 15 (1885), 1-20.

19.
J. W. L. Glaisher, Expressions for the five powers of the series in which the coefficients are the sums of the divisors of the exponents, Mess. Math. 15 (1885), 33-36.

20.
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).

21.
H. Hahn, Convolution sums of some functions on divisors, Rocky Mountain J. Math. 37 (2007), 1593-1622. crossref(new window)

22.
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.

23.
A. Kim, D. Kim, Yan Li, Convolution sums arising from divisor functions, J. Korean Math. Soc. 50(2) (2013), 331-360. crossref(new window)

24.
D. Kim and M. Kim, Divisor functions and Weierstrass functions arising from q series, Bull. of Korean Math. Soc. 49(4) (2012), 693-704. crossref(new window)

25.
D. B. Lahiri, On Ramanujan's function ${\tau}$ (n) and the divisor functions ${\sigma}$ (n), I, Bull. Calcutta Math. Soc. 38 (1946), 193-206.

26.
S. Lang, Elliptic functions, Addison-Wesly, 1973.

27.
D. H. Lehmer, Some functions of Ramanujan, Math. Student 27 (1959), 105-116.

28.
D. H. Lehmer, Selected papers, Vol. II, Charles Babbage Research Centre, St. Pierre, Manitoba, (1981).

29.
M. Lemire and K. S. Williams, Evaluation of two convolution sums involving the sum of divisor functions, Bull. Aust. Math. Soc. 73 (2005), 107-115.

30.
J. Liouville, Sur quelques formules generales qui peuvent etre utiles dans la theorie des nombres, Jour. de Math. (2) (1858-1865).

31.
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc. (2) 19 (1920), 75-113.

32.
G. Melfi, On some modular identities, de Gruyter, Berlin, 1998, 371-382.

33.
S. Ramanujan, On certain arithmetical functions, Trans. Cambridge Philos. Soc. 22 (1916), 159-184.

34.
S. Ramanujan, Collected papers, AMS Chelsea Publishing, Providence, RI, (2000).

35.
E. Royer, Evaluating convolution sums of the divisor function by quasimodular forms, Int. J. Number Theory 3 (2007), 231-261. crossref(new window)

36.
J. H. Silverman, Advanced topics in the arithmetic of elliptic curves, Springer- Verlag, 1994.

37.
N. P. Skoruppa, A quick combinatorial proof of Eisenstein series identities, J. Number Theory 43 (1993), 68-73. crossref(new window)

38.
K. S. Williams, Number Theory in the Spirit of Liouville, London Mathematical Society, Student Texts 76, Cambridge, (2011).

39.
K. S. Williams, The convolution sum ${\Sigma}_{m ${\sigma}(m){\sigma}(n-8m)$, Pacific J. Math. 228 (2006), 387-396. crossref(new window)

40.
K. S. Williams, The convolution sum ${\Sigma}_{m ${\sigma}(m){\sigma}(n-9m)$, Int. J. Number Theory 2 (2005), 193-205.