Acknowledgement
This study was financially supported by NRF 2021R1F1A1055200.
References
- A. Alaca, S. Alaca and K.S. Williams, Evaluation of the convolution sums ∑l+12m=nσ(l)σ(m) and ∑3l+4m=nσ(l)σ(m), Adv. Theor. Appl. Math. 1 (2006), 27-48.
- A. Alaca, S. Alaca and K.S. Williams, Evaluation of the convolution sums ∑l+18m=nσ(l)σ(m) and ∑2l+9m=nσ(l)σ(m), Int. Math. Forum 2 (2007), 45-68. https://doi.org/10.12988/imf.2007.07003
- A. Alaca, S. Alaca and K.S. Williams, Evaluation of the convolution sums ∑l+24m=nσ(l)σ(m) and ∑3l+8m=nσ(l)σ(m), Math. J. Okayama Univ. 49 (2007), 93-111.
- A. Alaca, S. Alaca and K.S. Williams, The convolution sum ∑m<n/16σ(m)σ(n - 16m), Canad. Math. Bull. 51 (2008), 3-14. https://doi.org/10.4153/CMB-2008-001-1
- A. Alaca, S. Alaca and E. Ntienjem, The convolution sum ∑al+bm=nσ(l)σ(m) for (a, b) = (1, 28), (4, 7), (1, 14), (2, 7), (1, 7), Kyungpook Math. J. 59 (2019), 377-389. https://doi.org/10.5666/KMJ.2019.59.3.377
- S. Alaca and Y. Kesicioglu, Evaluation of the convolution sums ∑l+27m=nσ(l)σ(m) and ∑l+32m=nσ(l)σ(m), Int. J. Number Theory 12 (2016), 1-13. https://doi.org/10.1142/S1793042116500019
- S. Alaca and K.S. Williams, Evaluation of the convolution sums ∑l+6m=nσ(l)σ(m) and ∑2l+3m=nσ(l)σ(m), J. Number Theory 124 (2007), 491-510. https://doi.org/10.1016/j.jnt.2006.10.004
- M. Besge, Extrait dune lettre de M. Besge a M. Liouville, J. Math. Pures Appl. 7 (1862), 256.
- H.H. Chan and S. Cooper, Powers of theta functions, Pacific J. Math. 235 (2008), 1-14. https://doi.org/10.2140/pjm.2008.235.1
- B. Cho, Convolution sums of a divisor function for prime levels, Int. J. Number Theory 16 (2020), 537-546. https://doi.org/10.1142/s179304212050027x
- 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
- S. Cooper and D. Ye, Evaluation of the convolution sums ∑l+20m=nσ(l)σ(m), ∑4l+5m=nσ(l)σ(m) and ∑2l+5m=nσ(l)σ(m), Int. J. Number Theory 10 (2014), 1385-1394. https://doi.org/10.1142/S1793042114500341
- 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 (1885), 156-163.
- J.G. Huard, Z.M. Ou, B.K. Spearman and K.S. Williams, Elementary evaluation of certain convolution sums involving divisor functions, in Number Theory for the Millennium, II, A.K. Peters, Natick, MA, 2002, 229-274.
- M. Kaneko and D. Zagier, A generalized Jacobi theta function and quasimodular forms, in: The Moduli Spaces of Curves, in: Progress Mathematics, vol. 129, Birkhauser, Boston, MA, 1995, 165-172.
- D.B. Lahiri, On Ramanujan's function τ(n) and the divisor function σ(n), I, Bull. Calcutta Math. Soc. 38 (1946), 193-206.
- J. Lee and Y.K. Park, Evaluation of the convolution sums ∑a1m1+a2m2+a3m3+a4m4=nσ(m1)σ(m2)σ(m3)σ(m4) with lcm(a1, a2, a3, a4) ≤ 4, Int. J. Number Theory 13 (2017), 2155-2173. https://doi.org/10.1142/S1793042117501160
- 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. https://doi.org/10.1017/S0004972700038661
- LMFDB, The L-functions and Modular Forms Database, website https://www.lmfdb.org/
- E. Ntienjem, Evaluation of the convolution sums ∑αl+βm=nσ(l)σ(m), where (α, β) is in {(1, 14), (2, 7), (1, 26), (2, 13), (1, 28), (4, 7), (1, 30), (2, 15), (3, 10), (5, 6)}, M.Sc. thesis Carleton University, Ottawa, Ontario, Canada, 2015.
- E. Ntienjem, Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52, Open Math. 15 (2017), 446-458. https://doi.org/10.1515/math-2017-0041
- E. Ntienjem, Elementary evaluation of convolution sums involving the divisor function for a class of levels, North-W. Eur. J. of Math. 5 (2019), 101-165.
- Y.K. Park, Evaluation of the convolution sums ∑ak+bl+cm=nσ(k)σ(l)σ(m) with lcm(a, b, c) ≤ 6, J. Number Theory 168 (2016), 257-275. https://doi.org/10.1016/j.jnt.2016.04.025
- Y.K. Park, Evaluation of the convolution sums ∑al+bm=nlσ(l)σ(m) with ab ≤ 9, Open Math. 15 (2017), 1389-1399. https://doi.org/10.1515/math-2017-0116
- Y.K. Park, Evaluation of the convolution sums ∑ak+bl+cm=nσ(k)σ(l)σ(m) with lcm(a, b, c) = 7, 8 or 9, Int. J. Number Theory 14 (2018), 1637-1650. https://doi.org/10.1142/s1793042118500999
- B. Ramakrishnan and B. Sahu, Evaluation of the convolution sums ∑l+15m=nσ(l)σ(m) and ∑3l+5m=nσ(l)σ(m) and an application, Int. J. Number Theory 9 (2013), 799-809. https://doi.org/10.1142/S179304211250162X
- S. Ramanujan, On certain arithmetical functions, Trans. Cambridge Philos. Soc. 22 (1916), 159-184.
- 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
- E. Royer, Quasimodular forms: An introduction, Ann. Math. Blaise Pascal 19 (2012), 297-306. https://doi.org/10.5802/ambp.315
- W. Stein, Modular Forms: a Computational Approach, Graduate Studies in Mathematics, vol. 79, American Mathematical Society, 2007.
- K.S. Williams, The convolution sum ∑m<n/9σ(m)σ(n - 9m), Int. J. Number Theory 1 (2005), 193-205. https://doi.org/10.1142/S1793042105000091
- K.S. Williams, The convolution sum ∑m<n/8σ(m)σ(n-8m), Pacific J. Math. 228 (2006), 387-396. https://doi.org/10.2140/pjm.2006.228.387
- E.X.W. Xia, X.L. Tian and O.X.M. Yao, Evaluation of the convolution sums ∑i+25j=nσ(i)σ(j), Int. J. Number Theory 10 (2014), 1421-1430. https://doi.org/10.1142/S1793042114500365
- D. Ye, Evaluation of the convolution sums ∑l+36m=nσ(l)σ(m) and ∑4l+9m=nσ(l)σ(m), Int. J. Number Theory 11 (2015), 171-183. https://doi.org/10.1142/S1793042115500104