TY - JOUR
T1 - Evaluation of the convolution sums σa1m1+a2m2+a3m3+a4m4=nσ (m1) σ (m2) σ (m3) σ (m4) with lcm (α1,a2,a3,a4) ≤ 4
AU - Lee, Joohee
AU - Park, Yoon Kyung
N1 - Publisher Copyright:
© 2017 World Scientific Publishing Company.
PY - 2017/9/1
Y1 - 2017/9/1
N2 - The generating functions of divisor functions are quasimodular forms of weight 2 and the product of them is a quasimodular form of higher weight. In this work, we evaluate the convolution sumsa1m1+a2m2+a3m3+a4m4=nσ(m1)σ(m2)σ(m3)σ(m4) for the positive integers a1,a2,a3,a4, and n with lcm(a1,a2,a3,a4) ≤ 4. We reprove the known formulas for the number of representations of a positive integer n by each of the quadratic forms j=016x j2 and j=18(x 2j-12 + x 2j-1x2j + x2j2) as an application of the new identities proved in this paper.
AB - The generating functions of divisor functions are quasimodular forms of weight 2 and the product of them is a quasimodular form of higher weight. In this work, we evaluate the convolution sumsa1m1+a2m2+a3m3+a4m4=nσ(m1)σ(m2)σ(m3)σ(m4) for the positive integers a1,a2,a3,a4, and n with lcm(a1,a2,a3,a4) ≤ 4. We reprove the known formulas for the number of representations of a positive integer n by each of the quadratic forms j=016x j2 and j=18(x 2j-12 + x 2j-1x2j + x2j2) as an application of the new identities proved in this paper.
KW - Convolution sum
KW - quasimodular form
KW - sums of divisor functions
KW - the number of representation by quadratic forms
UR - http://www.scopus.com/inward/record.url?scp=85017028256&partnerID=8YFLogxK
U2 - 10.1142/S1793042117501160
DO - 10.1142/S1793042117501160
M3 - Article
AN - SCOPUS:85017028256
SN - 1793-0421
VL - 13
SP - 2155
EP - 2173
JO - International Journal of Number Theory
JF - International Journal of Number Theory
IS - 8
ER -