References
-
A. Alaca, S. Alaca and K. S. Williams, The convolution sum
${\Sigma}_{m https://doi.org/10.4153/CMB-2008-001-1${\sigma}(m){\sigma}(n-16m)$ , Canad. Math. Bull. 51 (2008), 3-14. -
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. -
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. -
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. -
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. -
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. https://doi.org/10.1016/j.jnt.2006.10.004 - B. C. Berndt, Ramanujan's Notebooks, Part II. Springer-Verlag, New York, 1989.
- M. Besgue, Extrait d'une lettre de M. Besgue a M Liouville, J. Math. Pures Appl. 7 (1862), 256.
- H. H. Chan, and S. Cooper, Powers of theta functions, Pacific Journal of Mathematics, 235 (2008), 1-14. https://doi.org/10.2140/pjm.2008.235.1
- N. Cheng and K. S. Williams, Convolution sums involving the divisor functions, Proc. Edinburgh Math. Soc. 47 (2004), 561-572. https://doi.org/10.1017/S0013091503000956
- B. Cho, D. Kim and J. K. Koo, Divisor functions arising from q-series, Publ. Math. Debrecen 76 (2010), 495-508.
- 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
- S. Cooper and P. C. Toh, Quintic and septic Eisenstein series, Ramanujan J. 19 (2009), 163-181. https://doi.org/10.1007/s11139-008-9123-3
- L. E. Dickson, History of the Theory of Numbers, Vol.I, Chelsea Publ. Co., New York, 1952.
- L. E. Dickson, History of the Theory of Numbers, Vol.II, Chelsea Publ. Co., New York, 1952.
- N. J. Fine, Basic hypergeometric series and applications, American Mathematical Society, Providence, RI, 1988.
- 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.
- J. W. L. Glaisher, On certain sums of products of quantities depending upon the divisors of a number, Mess. Math. 15 (1885), 1-20.
- 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.
- 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).
- H. Hahn, Convolution sums of some functions on divisors, Rocky Mountain J. Math. 37 (2007), 1593-1622. https://doi.org/10.1216/rmjm/1194275937
- 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.
- A. Kim, D. Kim, Yan Li, Convolution sums arising from divisor functions, J. Korean Math. Soc. 50(2) (2013), 331-360. https://doi.org/10.4134/JKMS.2013.50.2.331
- D. Kim and M. Kim, Divisor functions and Weierstrass functions arising from q series, Bull. of Korean Math. Soc. 49(4) (2012), 693-704. https://doi.org/10.4134/BKMS.2012.49.4.693
-
D. B. Lahiri, On Ramanujan's function
${\tau}$ (n) and the divisor functions${\sigma}$ (n), I, Bull. Calcutta Math. Soc. 38 (1946), 193-206. - S. Lang, Elliptic functions, Addison-Wesly, 1973.
- D. H. Lehmer, Some functions of Ramanujan, Math. Student 27 (1959), 105-116.
- D. H. Lehmer, Selected papers, Vol. II, Charles Babbage Research Centre, St. Pierre, Manitoba, (1981).
- 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.
- J. Liouville, Sur quelques formules generales qui peuvent etre utiles dans la theorie des nombres, Jour. de Math. (2) (1858-1865).
- P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc. (2) 19 (1920), 75-113.
- G. Melfi, On some modular identities, de Gruyter, Berlin, 1998, 371-382.
- S. Ramanujan, On certain arithmetical functions, Trans. Cambridge Philos. Soc. 22 (1916), 159-184.
- S. Ramanujan, Collected papers, AMS Chelsea Publishing, Providence, RI, (2000).
- E. Royer, Evaluating convolution sums of the divisor function by quasimodular forms, Int. J. Number Theory 3 (2007), 231-261. https://doi.org/10.1142/S1793042107000924
- J. H. Silverman, Advanced topics in the arithmetic of elliptic curves, Springer- Verlag, 1994.
- 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
- K. S. Williams, Number Theory in the Spirit of Liouville, London Mathematical Society, Student Texts 76, Cambridge, (2011).
-
K. S. Williams, The convolution sum
${\Sigma}_{m https://doi.org/10.2140/pjm.2006.228.387${\sigma}(m){\sigma}(n-8m)$ , Pacific J. Math. 228 (2006), 387-396. -
K. S. Williams, The convolution sum
${\Sigma}_{m ${\sigma}(m){\sigma}(n-9m)$ , Int. J. Number Theory 2 (2005), 193-205.
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