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2. An auxiliary result We first recall some notations from the theory of modular forms. The modular group SL2(Z) is defined by a b SL2(Z) := : a,b,c,d ∈ Z, ad − bc =1 . c d 

If N > 1 is an integer, the congruence subgroup Γ0(N) is defined by a b Γ0(N)= ∈ SL2(Z): c ≡ 0 (modN) . c d 

We denote by Mk(Γ) the space of modular forms of weight k for Γ if Γ is a subgroup of SL2(Z) with finite index. In order to prove (1.2) we need an auxiliary result. Lemma 2.1. For Im τ > 0, we have E4(τ) − 9E4(2τ)+8E4(4τ) (2.1) = 10(2E2(4τ) − E2(2τ))(3E2(2τ) − E2(τ) − 2E2(4τ)), where E2(τ) and E4(τ) are two Eisenstein series given by 2πniτ E2(τ) := 1 − 24 σ1(n)e , nX≥1 2πniτ E4(τ) := 1 + 240 σ3(n)e , nX≥1 with α σα(n) := d . Xd|n Proof. It follows from [1, Exercise 1.2.8(e)] that

E2(τ) − NE2(Nτ) ∈ M2(Γ0(N)). In particular, E2(τ) − 2E2(2τ) ∈ M2(Γ0(2)). Then 2E2(4τ) − E2(2τ) ∈ M2(Γ0(4)) and

3E2(2τ) − E2(τ) − 2E2(4τ)=2E2(2τ) − E2(τ)+ E2(2τ) − 2E2(4τ) ∈ M2(Γ0(4)).

Therefore, the right side of (2.1) belongs to M4(Γ0(4)). Since E4(τ) ∈ M4(SL2(Z)), we see that E4(2τ) ∈ M4(Γ0(2)) and E4(4τ) ∈ M4(Γ0(4)). From [1, Theorem 3.5.1] we know that dim(M4(Γ0(4))) = 3. It is clear that {E4(τ), E4(2τ), E4(4τ)} is a basis of M4(Γ0(4)). Then there exist three complex numbers a,b,c such that

10(2E2(4τ) − E2(2τ))(3E2(2τ) − E2(τ) − 2E2(4τ)) (2.2) = aE4(τ)+ bE4(2τ)+ cE4(4τ). Set q = e2πiτ . Comparing the coefficients of q0, q1, q2 on both sides of (2.2) we get a =1,b = −9,c =8. Then (2.1) follows readily by substituting these constants back into (2.2).  TWO LAMBERT SERIES IDENTITIES 3

3. Proof of Theorem 1.1 We first prove (1.2). Since

q2n q2n 2q4n = − , 1+ q2n 1 − q2n 1 − q4n qn qn q2n = − . 1 − q2n 1 − qn 1 − q2n we see that nq2n nq2n nq4n (3.1) = − 2 , 1+ q2n 1 − q2n 1 − q4n nX≥1 nX≥1 nX≥1

nqn nqn nq2n = − 1 − q2n 1 − qn 1 − q2n nX≥1 nX≥1 nX≥1 and nq2n nq2n nq4n = − . 1 − q4n 1 − q2n 1 − q4n nX≥1 nX≥1 nX≥1 From the last two identities above we deduce that (2n − 1)q2n−1 nqn 2nq2n = − 1 − q4n−2 1 − q2n 1 − q4n nX≥1 nX≥1 nX≥1 (3.2) nqn nq2n nq4n = − 3 +2 . 1 − qn 1 − q2n 1 − q4n nX≥1 nX≥1 nX≥1 Similarly,

2n 1 B3(n) q − 3 1 − q4n−2 nX≥1 2n−1 1 2 q = (2n − 1)((2n − 1) − 1) 24 1 − q4n−2 nX≥1 1 qn 1 q2n = n(n2 − 1) − n(4n2 − 1) 24 1 − q2n 12 1 − q4n nX≥1 nX≥1 1 qn 1 q2n = n(n2 − 1) − n(n2 − 1) (3.3) 24 1 − qn 24 1 − q2n nX≥1 nX≥1 2n 4n 1 2 q 1 2 q − n(4n − 1) + n(4n − 1) 12 1 − q2n 12 1 − q4n nX≥1 nX≥1 1 qn 1 q2n = n(n2 − 1) − (3n3 − n) 24 1 − qn 8 1 − q2n nX≥1 nX≥1 1 q4n + n(4n2 − 1) . 12 1 − q4n nX≥1 4 BING HE

2πiτ Let q = e with Im τ > 0. We rewrite the Eisenstein series E2(τ) and E4(τ) as nqn E2(τ)=1 − 24 , 1 − qn nX≥1 n3qn E4(τ)=1+240 . 1 − qn nX≥1 Then nqn 1 = (1 − E2(τ)), 1 − qn 24 nX≥1 n3qn 1 = (E4(τ) − 1). 1 − qn 240 nX≥1 Substituting these identities into (3.1), (3.2) and (3.3) and then simplifying we get nq2n 1 1 (3.4) = (2E2(4τ) − E2(2τ)) − , 1+ q2n 24 24 nX≥1 (2n − 1)q2n−1 1 (3.5) = (3E2(2τ) − E2(τ) − 2E2(4τ)), 1 − q4n−2 24 nX≥1 2n 1 B3(n) q − 1 (3.6) = (E4(τ) − 9E4(2τ)+8E4(4τ)) 3 1 − q4n−2 5760 nX≥1 1 + (E2(τ) − 3E2(2τ)+2E2(4τ)). 242 1 Then (1.2) follows easily by adding 242 (E2(τ) − 3E2(2τ)+2E2(4τ)) into both sides of (2.1) and then substituting (3.4), (3.5) and (3.6) into the resulting identity. We now show (1.3). It follows easily from the formula 1 = xn, |x| < 1 1 − x nX≥0 that x (3.7) = nxn, |x| < 1. (1 − x)2 nX≥1 Differentiating (3.7) with respect to x and then multiplying the resulting identity by x we get 2 x(1 + x) n xn = , |x| < 1. (1 − x)3 nX≥1 Differentiating this equation with respect to x again and then multiplying the re- sulting identity by x we arrive at x3 +4x2 + x n3xn = , |x| < 1. (1 − x)4 nX≥1 Then q3m +4q2m + qm n3qn = n3qmn = (1 − qm)4 1 − qn mX≥1 m,nX≥1 nX≥1 TWO LAMBERT SERIES IDENTITIES 5 and so q2m q3m +4q2m + qm qm 6 = − (1 − qm)4 (1 − qm)4 (1 − qm)2 mX≥1 mX≥1 mX≥1 n3qn qm = − . 1 − qn (1 − qm)2 nX≥1 mX≥1 Similarly, q4m n3q2n q2m 6 = − . (1 − q2m)4 1 − q2n (1 − q2m)2 mX≥1 nX≥1 mX≥1 We combine these two identities to give q4n−2 q2n q4n 6 =6 − 6 (1 − q2n−1)4 (1 − qn)4 (1 − q2n)4 nX≥1 nX≥1 nX≥1 n3qn q2m−1 = − . 1 − q2n (1 − q2m−1)2 nX≥1 mX≥1 From this we deduce (1.3). This completes the proof of Theorem 1.1. Acknowledgement

This work was partially supported by the National Natural Science Foundation of China (Grant No. 11801451) and the Natural Science Foundation of Hunan Province (Grant No. 2020JJ5682). References

[1] F. Diamond and J. Shurman, A first course in modular forms, Graduate texts in mathematics 228. Springer-Verlag, New York, 2005. [2] M. El Bachraoui, On series identities of Gosper and integrals of Ramanujan theta function ψ(q), Proc. Amer. Math. Soc. 147(10)(2019), 4451–4464. [3] R.W. Gosper, Experiments and discoveries in q-trigonometry, in: F.G. Garvan, M.E.H. Ismail (Eds.), Symbolic Computation, Number Theory, Special Functions, Physics and Combina- torics, Kluwer, Dordrecht, Netherlands, 2001, pp.79–105.

School of Mathematics and Statistics, Central South University, Changsha 410083, Hunan, People’s Republic of China Email address: [email protected]; [email protected]