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Symbol Definition
Bn(x), Bn The Bernoulli polynomials and Bernoulli numbers Bn = Bn(0). These polynomi- als can be used via Faulhaber’s formula, among others, to generate the integral th k k k k k power sums d|n φk(d)(n/d) = 1 + 2 + · · · + n . n n n! k The binomial coefficients,P k = k!(n−k)! . ⌈ x⌉ The ceiling function ⌈x⌉ := x+ 1 − {x} where 0 ≤ {x} < 1 denotes the fractional part of x ∈ R. [qn]F (q) The coefficient of qn in the power series expansion of F (q) about zero.
cq(n) Ramanujan’s sum, cq(n) := dµ(q/d). d|(q,n) (j) th P Dz The higher-order j derivative operator with respect to z. k dk(n) The generalized k-fold divisor function dk(n) = 1∗k (n) whose DGF is ζ(s) . Note that the divisor function d(n) ≡ d1(n).
d(n) The ordinary divisor function, d(n) := d|n 1. ε(n) The multiplicative identity with respectP to Dirichlet convolution, ε(n)= δn,1. ∗; f ∗ g The Dirichlet convolution of f and g, f ∗ g(n) := f(d)g(n/d), for n ≥ 1. This d|n symbol for the discrete convolution of two arithmeticP functions is the only notion of convolution of functions we employ within the article. f −1(n) The Dirichlet inverse of f with respect to convolution defined recursively by −1 1 −1 f (n)= − f(1) f(d)f (n/d) provided that f(1) 6= 0. d|n d>P1 ⌊x⌋ The floor function ⌊x⌋ := x − {x} where 0 ≤ {x} < 1 denotes the fractional part of x ∈ R.
f∗j Sequence of nested j-convolutions of an arithmetic function f with itself for in-
tegers j ≥ 1. We define f∗0 (n)= δn,1, the multiplicative identity with respect to Dirichlet convolution.
γ(n) The squarefree kernel of n, γ(n) := p|n p. gcd(m,n), (m,n) The greatest common divisor of m andQ n. Both notations for the GCD are used interchangably within the article.
Gj Denotes the interleaved (or generalized) sequence of pentagonal numbers defined 1 j 3j+1 explictly by the formula Gj := 2 2 2 . The sequence begins as {Gj }j≥0 = {0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51l,...ml}. m k Idk(n) The power-scaled identity function, Idk(n) := n for n ≥ 1. The Dirichlet inverse −1 k of this function is given by Idk (n) := n · µ(n). 1 1 N S, χcond(x) We use the notation , χ : → {0, 1} to denote indicator, or characteristic functions. In paticular, 1S(n) = 1 if and only if n ∈ S, and χcond(n) = 1 if and only if n satisfies the condition cond.
[n = k]δ Synonym for δn,k which is one if and only if n = k, and zero otherwise. cond cond cond cond [ ]δ For a boolean-valued , [ ]δ evaluates to one precisely when is true, and zero otherwise.
ϑi(z,q), ϑi(q) For i = 1, 2, 3, 4, these are the classical Jacobi theta functions where ϑi(q) ≡ ϑi(0,q).
2 Symbol Definition k −t Jt(n) The Jordan totient function Jt(n)= n p|n(1 − p ) also satisfies d|n Jt(d)= nt. Q P Λ(n) The von Mangoldt lambda function Λ(n)= d|n log(d)µ(n/d). Ω(n) λ(n) The Liouville lambda function λ(n) = (−1) P . k λk(n) The arithmetic function defined by λk(n)= d|n d λ(d). Lf (q) The lambert series generating function of anP arithmetic function f, defined by f(n)qn L (q) := , |q| < 1. f 1 − qn nX≥1 lcm(m,n), [m,n] The least common multiple of m and n. log The natural logarithm function, log(n) ≡ ln(n). lsb(n) The least significant bit of n in the base-2 expansion of n. α1 αk µ(n) The Möbius function, defined for n ≥ 1 with n = p1 · · · pk its factorization into distinct prime powers with αi ≥ 1 for all 1 ≤ i ≤ k by
1, if n = 1; k µ(n)= (−1) , if αi = 1, ∀1 ≤ i ≤ k; 0, otherwise.
OGF Ordinary generating function. Given a sequence {fn}n≥0, its OGF (or sometimes called ordinary power series, OPS) enumerates the sequence by powers of a typ- n C ically formal variable z: F (z) := n≥0 fnz . For z ∈ within some radius or abcissa of convergence for the series, asymptotic properties can be extracted from P the closed-form representation of F , and/or the original sequence terms can be recovered by performing an inverse Z-transform on the OGF. th ωa A primitive a root of unity ωa = exp(2πı/a) for integers a ≥ 1. α1 αr ω(n), Ω(n) If n = p1 · · · pr is the prime factorization of n into distinct prime powers, then ω(n)= r and Ω(n)= α1 + · · · + αr. k φk(n) Generalized totient function, φk(n) := d . 1≤d≤n (d,nP)=1 φ(n) Euler’s classical totient function, φ(n) := 1. 1≤d≤n (d,nP)=1 th 2πık/n Φn(z) The n cyclotomic polynomial in z defined by Φn(z) := (z − e ). 1≤k≤n (k,nQ)=1 p(n) The partition function generated by p(n) = [qn] (1 − qn)−1. n≥1 π(x) The prime counting function denotes the numberQ of primes p ≤ x.
p, p≤x Unless otherwise specified by context, we use the index variable p to denote that the summation is to be taken only over prime values within the summation P P bounds. th k −k Ψk(n) The k Dedekind totient function, Ψk(n) := n p|n(1 + p ). k 2 ψk(n) The arithmetic function defined by ψk(n) := dQ|n d µ (n/d). P
3 Symbol Definition n−1 (a; q)n, (q)n The q-Pochhammer symbol defined as the product (a; q)∞ := (1−aq ). We n≥1 adopt the notation that (q)n ≡ (q; q)n and that (a; q)∞ denotesQ the limiting case for |q| < 1 as n →∞. r (a1, . . . , ar; q)n We use the common shorthand that (a1, . . . , ar; q)n = i=1(ai; q)n. rk(n) The sum of k squares function denotes the number ofQ integer solutions to n = 2 2 n k x1 + · · · + xk. A generating function is given by rk(n) = [q ]ϑ3(q) . α σα(n) The generalized sum-of-divisors function, σα(n) := d , for any n ≥ 1 and d|n α ∈ C. Note that the divisor function is also denotedP by d(n) ≡ σ0(n). n n k , k The Stirling numbers of the first and second kinds, respectively. Alternate nota- tion for these triangles is given by s(n, k) = (−1)n−k n and S(n, k)= n . k k n−1 m 24 τ(n) The function defined by τ(n) := [x ] m≥1(1 − x ) . −s ζ(s) The Riemann zeta function, defined by ζQ(s) := n≥1 n when ℜ(s) > 1, and by analytic continuation to the entire complex plane with the exception of a simple P pole at s = 1.
4 2 Background and identities for Lambert series generating functions
2.1 Introduction
In the most general setting, we define the generalized Lambert series expansion for integers 0 ≤ β < α and any fixed arithmetic function as f(n)qαn−β L (α, β; q) := ; |q| < 1, f 1 − qαn−β nX≥1 where the series coefficients of the Lambert series generating function are given by the divisor sums
n [q ]Lf (α, β; q)= f(d). αdX−β|n If we set (α, β) := (1, 0), the we recover the classical form of the Lambert series construction, which we will denote by the function Lf (q) ≡ Lf (1, 0; q). There is a natural correspondence between a sequence’s ordinary generating function (OGF), and its Lambert series m generating function. Namely, if F (q) := m≥1 f(m)q is the OGF of f, then
P αn−β e Lf (α, β; q)= F (q ). Xn≥1 e We also have a so-called Lambert transform defined by [1, §2]
∞ ta(t) F (x) := dt. a ext − 1 Z0 This transform satisfies an inversion relation of the form
k+1 k+1 (−1) k k (k) nk τa(τ) = lim µ(n)n Fa . k→∞ k! τ τ nX≥1 The interpretation of this invertible transform as a so-called Lambert transformation considers taking the mappings −x ta(t) ↔ an and e ↔ q.
2.2 Higher-order derivatives of Lambert series
Ramanujan discovered the following remarkable identities [1, §2]:
(−1)n−1qn qn = (2.1a) 1 − qn 1+ qn Xn≥1 Xn≥1 nqn qn = (2.1b) 1 − qn (1 − qn)2 nX≥1 Xn≥1 (−1)n−1nqn qn = (2.1c) 1 − qn (1 + qn)2 nX≥1 Xn≥1 qn qn = (2.1d) n(1 − qn) 1+ qn nX≥1 Xn≥1 (−1)n−1qn 1 = log (2.1e) n(1 − qn) 1 − qn Xn≥1 Xn≥1 αnqn = log (1 + qn) (2.1f) 1 − qn nX≥1 Xn≥1 5 n n2qn qn 1 = . (2.1g) 1 − qn (1 − qn)2 1 − qk nX≥1 Xn≥1 Xk=1
It follows that we can relate the partition function generating functions (q; ±q)∞ to the exponential of the two logarithmically termed series above (cf. the remarks in Section 7.5). More generally, higher-order jth derivatives for integers j ≥ 1 can be obtained by differentiating the Lambert series expansions termwise in the forms of1
j m qn j m (−1)j−kk!im qj · D(j) = , (2.2a) q 1 − qn m k (1 − qi)k+1 m=0 X Xk=0 j j m j m j − k (−1)j−k−rk!im = q(r+1)i. (2.2b) m k r (1 − qi)k+1 r=0 "m=0 # X X Xk=0 n m+1 n+m By the binomial generating functions given by [z ](1 − z) = m , we find that f(n)qmn n − m + k [qn] = d f(d), (1 − qn)k+1 k Xn≥t Xd|n t≤d≤ n ⌊ m ⌋ for positive integers m,t ≥ 1 and k ≥ 0.
2.3 Relation of the coefficients of the classical Lambert series to Ramanujan sums
We define the functions Φn(q) as the change of variable into the logarithmic derivatives of the cyclotomic polynomials as e ′ Φn(1/q) 1 d d Φn(q)= = µ(n/d) log(w − 1) q · Φn(1/q) q dw d|n w=1/q X e dµ(n/d) = . 1 − qd Xd|n As derived in [13], we can express the component series terms for n ≥ 1 in the form of 1 1 = Φ (q). 1 − qn n d Xd|n e Then we can express the Lambert series coefficients, (f ∗ 1)(n), for each positive natural number x ≥ 1 as
x x ⌊ d ⌋ f(n) f(nd) f(n) [qx] = c (x) = c (x). 1 − qn d nd n d nX≤x Xd=1 nX=1 nX≤x Xd|n 1 Here, since we can express the coefficients for all finite n ≥ 1 as
m n n f(m)q [q ]Lf (q) = [q ] , X 1 − qm m n ≤ by partial sums of the generating functions, we need not worry about uniform convergence of the Lambert series generating function in q.
6 2.4 Factorization theorems
2.4.1 Classical series cases
The first form of the factorization theorems considered in [14, 9] expands two variants of Lf (q) as
n f(n)q 1 n n = (so(n, k) ± se(n, k)) f(k)q , 1 ± q (∓q; q)∞ nX≥1 nX≥1 n qk where so(n, k) ± se(n, k) = [q ](∓q; q)∞ 1±qk is defined as the sum (difference) of the functions so(n, k) and se(n, k), which respectively denote the number of k’s in all partitions of n into and odd (even) number of distinct parts. If we define sn,k = so(n, k) − se(n, k), then this sequence is lower triangular and invertible. Its inverse matrix is defined by [15, A133732] n s−1 = p(d − k)µ . n,k d Xd|n We can define the form of another factorization of Lf (q) where |C(q)| < ∞ for all |q| < 1 is such that C(0) 6= 0 as
n n f(n)q 1 n n = sn,k(γ)f(k)(γ) q , 1 − q C(q) ! nX≥1 nX≥1 Xk=1 e for any prescribed non-zero arithmetic function γ(n) with
f(k)(γ)= f(d)γ(r). d|k r| k X Xd e In this case, we have that 1 n s−1 (γ)= [qd−k] γ . n,k C(q) d Xd|n This notion of factorization can be generalized to expanding the generalized Lambert series Lf (α, β; q) from the first subsection [10]. n In either case, the coefficients generated by Lf (q) as [q ]Lf (q) = (f ∗ 1)(n) and their summatory functions, x Σ (x) := (f ∗ 1)(n)= f(d) , f d nX≤x Xd≤x j k inherit partition-function-like recurrence relations from the structure of the factorizations we have constructed. In particular, for n,x ≥ 1 we have that [14]
√24n+1 b j 6 − k n+1 k(3k + b) (f ∗ 1)(n +1) = (−1)k+1(f ∗ 1) n + 1 − + s f(k), 2 n+1,k bX=±1 Xk=1 Xk=1 √24x+1 b j 6 − k x n+1 k(3k + b) Σ (x +1) = (−1)k+1Σ n + 1 − + s f(k). f f 2 n+1,k n=0 bX=±1 Xk=1 X Xk=1
2.4.2 Generalized Lambert series expansions
Most generally in [10], we define the generalized Lambert series factorizations by the series expansions
αn−β n f(n)q 1 ¯ n Lf (α, β; q) := αn−β = s¯n,k(α, β)f(k) q ; |q| < 1, (2.3) 1 − q C(q) ! nX≥1 nX≥1 Xk=1 7 for integers 0 ≤ β < α, C(q) any convergent OGF for |q| < 1 such that C(0) 6= 0, and f¯ some function of the f(n)’s. For |q| < 1, 0 ≤ β < α,
∞ ∞ n qαn−β 1 a = (s (n, k) − s (n, k))a qn, n 1 − qαn−β (qα−β; qα) o e k n=1 ∞ n=1 ! X X Xk=1 where so(n, k) and se(n, k) denote the number of (αk − β)’s in all partitions of n into an odd (respectively even) number of distinct parts of the form αk − β. Similarly, for |q| < 1, 0 ≤ β < α,
∞ αn−β ∞ n q α−β α n an αn−β = (q ; q )∞ s(n, k)ak q , 1 − q ! nX=1 nX=1 Xk=1 where s(n, k) denotes the number of (αk − β)’s in all partitions of n into parts of the form αk − β.
If we define the factorization pair (C(q), s¯n,k) in (2.3) 1 s−1 := [qd−k] · γ(n/d), n,k C(q) Xd|n for some fixed arithmetic functions γ(n) and γ(n) := d|n γ(d), then we have that the sequence of a¯n is given by the following formula for all n ≥ 1: P e a¯n = a d β γ(n/d). −α d|n d≡βXmod α e
8 3 Ordinary Lambert series identities
3.1 Listings of identities for arithmetic functions
We have the following well-known “classical” examples of Lambert series identities [12, §27.7] [8, §17.10] [3, §11]:
µ(n)qn = q, (3.1a) 1 − qn Xn≥1 φ(n)qn q = , (3.1b) 1 − qn (1 − q)2 nX≥1 nαqn = σ (n)qn, (3.1c) 1 − qn α n≥1 m≥1 X n X λ(n)q 2 = qm , (3.1d) 1 − qn nX≥1 mX≥1 Λ(n)qn = log(m)qm, (3.1e) 1 − qn nX≥1 mX≥1 |µ(n)|qn = 2ω(m)qm, (3.1f) 1 − qn nX≥1 mX≥1 J (n)qn t = mtqm, (3.1g) 1 − qn n≥1 m≥1 X n X µ(αn)q n = − qα , α ∈ P (3.1h) 1 − qn nX≥1 Xn≥0 qn ψ (1) + log(1 − q) = q , (3.1i) 1 − qn log(q) nX≥1 n 2 lsb(n)q ψ 2 (1/2) + log(1 − q ) = q . (3.1j) 1 − qn 2 log(q) nX≥1
3.2 Other identities
For any arithmetic function f and integers k ≥ 1, we have that
k f(n)qn = f(d) qm. (3.2) nk 1 − q k nX≥1 mX≥1 dX|n For example, we have that
n λ (n)q k k = qm (3.3a) 1 − qn nX≥1 mX≥1 n k+1 |µk(n)|q = q (3.3b) n≥1 X 2 µ(n)qn = |µ(m)|qm. (3.3c) 1 − qn2 nX≥1 mX≥1
9 4 Modified Lambert series identities
4.1 Definitions
For |q| < 1 and f any arithmetic function, let f(n)qn L (q) := . f 1+ qn nX≥1 b We can write Lf (q) ≡ Lh(q), e.g., as an ordinary Lambert series expansion, where
b h(n), if n is odd; h(n)= (h(n) − 2h(n/2), if n is even. Thus we have that 2 Lf (q)= Lf (q) − 2Lf (q ). (4.1)
4.2 Examples b
The two primary Lambert series expansions from the previous section that admit “nice”, algebraic closed-form expressions are translated below: µ(n)qn = q − 2q2 (4.2a) 1+ qn nX≥1 φ(n)qn q(q + q2) = . (4.2b) 1+ qn (1 − q2)2 Xn≥1 Note that Section 6.2 also provides a pair of related series in the context of GCD sums of an arithmetic function.
5 Generalized Lambert series identities
Listings of identities
From [8, §17.10], we obtain that
4 · (−1)n+1q2n+1 = r (m)qm, |q| < 1. (5.1) 1 − q2n+1 2 nX≥1 mX≥1 We also have a number of classical theta function related series of the following forms [12, §20]: qn 1 = ϑ2(q) − 1 (5.2a) 1+ q2n 4 3 n≥1 X q2n+1 1 ϑ2(q2) = ϑ2(q) − ϑ2(q) = 2 (5.2b) 1+ q4n+2 4 3 2 4 n≥1 X qn = L (q) − L (q2) (5.2c) 1 − q2n 1 1 Xn≥1 q2n+1 = L (q) − 2L (q2)+ L (q4) (5.2d) 1 − q4n+2 1 1 1 nX≥1 2n ′ 4 sin(2nz)q ϑ1(z,q) 2n = (5.2e) 1 − q ϑ1(z,q) nX≥1 10 n 2n ′ 4(−1) sin(2nz)q ϑ2(z,q) 2n = (5.2f) 1 − q ϑ2(z,q) nX≥1 n n ′ 4(−1) sin(2nz)q ϑ3(z,q) 2n = (5.2g) 1 − q ϑ3(z,q) nX≥1 n ′ 4 sin(2nz)q ϑ4(z,q) 2n = (5.2h) 1 − q ϑ4(z,q) nX≥1 ∞ 2 (−1)ne2ınzqn ϑ (0,q)ϑ (z,q)ϑ (z,q) = 2 3 4 . (5.2i) q−ne−ız + qneız ϑ (z,q) n=−∞ 2 X There are a number of Lambert, and Lambert-like series, for mock theta functions of order 6 given in [1, §8]. These a m−a m m series expansions are cited as follows where Ja,m := (q ,q ,q ; q )∞:
2 ∞ (−1)nqn (q; q2) 2 qr(3r+1)/2 φ (q)= n = (5.3a) mock (−q) J 1+ q3r 2n 1,3 r=−∞ nX≥0 X 2 ∞ (−1)nq(n+1) (q; q2) 2 qr(3r+1)/2 Ψ (q)= n = (5.3b) mock (−q) J 1+ q3r+1 2n+1 1,3 r=−∞ nX≥0 X ∞ qn(n+1)/2(−q) 1 (−1)rqr(3r+4) ρ (q)= n = (5.3c) mock (q; q2) J 1 − q6r+1 n+1 1,6 r=−∞ nX≥0 X ∞ qn(n+2)/2(−q) 1 (−1)rq(r+1)(3r+1) σ (q)= n = (5.3d) mock (q; q2) J 1 − q6r+3 n+1 1,6 r=−∞ nX≥0 X 2 ∞ qn (q) 1 (−1)rqr(3r+1)/2 γ (q)= n = . (5.3e) mock (q3; q3) (q; q) 1+ qr + q2r n ∞ r=−∞ nX≥0 X 6 Lambert series over Dirichlet convolutions and Apostol divisor sums
6.1 Dirchlet convolutions
6.1.1 Definitions
Given two prescribed arithmetic functions f and g we define their Dirichlet convolution, denoted by h = f ∗ g, to be the function
(f ∗ g)(n) := f(d)g(n/d), Xd|n for all natural numbers n ≥ 1 [3, §2.6]. The usual Möbius inversion result is stated in terms of convolutions as follows, where µ is the Möbius function: h = f ∗1 if and only if f = h∗µ. There is a natural connection between the coefficients of the Lambert series of an arithmetic function an and its corresponding Dirichlet generating function, s C DGF(an; s) := n≥1 an/n . Namely, we have that for any s ∈ such that ℜ(s) > 1 P a qn b = [qn] n if and only if DGF(b ; s) = DGF(a ; s)ζ(s), n 1 − qn n n nX≥1 where ζ(s) is the Riemann zeta function. Moreover, we can further connect the coefficients of the Lambert series over a convolution of arithmetic functions to its associated Dirichlet series by noting that DGF(f ∗g; s) = DGF(f; s)· DGF(g; s).
11 Notation 6.1 (Expanding Dirichlet inverse functions). The Dirichlet inverse function of f(n), denoted f −1(n), is −1 −1 an arithmetic function such that (f ∗ f )(n) = δn,1 for all n ≥ 1. The function f exists and is unique if and only if f(1) 6= 0. In these cases, we can expand the inverse function in terms of weighted terms in f recursively according to the formula 1 f(1) , n = 1; f −1(n)= 1 −1 n − f(1) f(d)f d , n ≥ 2. d|n d>P1 We have that [11] Ω(n) j (−1) · (f − f(1)ε)∗ (n) f −1(n)= j . f(1)j+1 Xj=1 Note that Section 7 contains a formula enumerating the Dirichlet inverse of any Dirichlet invertible arithmetic function f.
6.1.2 General identities
We can see that the Lambert series over the convolution (f ∗ g)(n) is given by the double sum
n Lf∗g(q)= f(n)Lg(q ), |q| < 1. nX≥1 Similarly, n 2n Lf∗g(q)= f(n) Lg(q ) − 2Lg(q ) . n≥1 X Clearly we have by Möbius inversionb that the ordinary generating function (OGF) of f is given by
n Lf∗µ(q)= f(n)q . nX≥1
If F (x) := n≤x f(n) is the summatory function of f, then we have that
P L (qn) F (n)qn = µ(n) f . 1 − q nX≥1 nX≥1 Proof. The last identity follows by writing
L (q) [qn] f = (f ∗ 1)(k) 1 − q kX≤n n n ⌊ k ⌋ = F (k) 1 k=1 n +1 X ⌊ k+1X⌋ n =⇒ [qn]L (q)= (F (m) − F (m − 1)). f m Xm|n −1 k Thus it follows that since Idk (n)= µ(n)Idk(n)= µ(n)n as in [3, cf. §2], we get that
n Lf∗Id 1 (q) L (q ) 1− = µ(n) f = F (n)qn. 1 − q 1 − q nX≥1 nX≥1
12 6.1.3 Listing of particular identities
We have the following convolution identities for Lambert series expansions of special functions [7, §7.4] [12, §24.4(iii)]:
k ψ (n)qn k qmj k = j! × 2ω(m) (6.1a) 1 − qn j (1 − qm)j+1 nX≥1 Xj=0 mX≥1 k k n 1 + j = j! × 2ω(d) j d · qn; m ∈ N, j j k j j=0 n≥1 n X X dX|j j k k (σ ∗ µ)(n)qn k j!qj k = ; m ∈ N; σ ∗ µ = Id , (6.1b) 1 − qn j (1 − q)j+1 k k nX≥1 Xj=0 σ (n)qn d(n)qn 1 = ; σ = φ ∗ σ , (6.1c) 1 − qn (1 − qn)2 1 0 nX≥1 Xn≥1 (φ ∗ Id )(n)qn 1 k k = (B (m + 1) − B (0)) qm. (6.1d) 1 − qn k + 1 k+1 k+1 nX≥1 mX≥1 We also have some unique generating function expressions for common summatory functions, including the following identity: L (qn) µ(n) µ∗ω = π(x)qx. (6.2a) 1 − q Xn≥1 Xx≥1
6.1.4 Characteristic functions
The argument used to arrive at the last identity shows that if A ⊆ Z+ and its indicator function is denoted by χA(n), then we have that n a µ(n)LχA (q )= q . (6.3a) nX≥1 aX∈A For example, if Nsqfree denotes the set of positive squarefree integers, then
n k µ(n)Lµ2 (q )= q . (6.3b) N nX≥1 k∈Xsqfree
Moroever, if χA(n) = (µ ∗ gA)(n), then g (n)f(n)qn A = L (qa). (6.3c) 1 − qn f nX≥1 aX∈A For example, in (7.3) we prove a prime summation identity for the Lambert series over the pointwise products of ω(n)f(n) and λ(n)f(n) for any arithmetic f.
6.1.5 Expressions for series generating Dirichlet inverse functions
We denote by f∗j (n) the j-fold convolution of f with itself, i.e., the sequence defined recursively by
δn,1, if j = 0; f∗j (n)= ( d|n f(d)f∗j 1 (n/d), if j ≥ 1. − Then as in [11], we have that P Ω(n) Ω(n) (−1)j f −1(n)= f (n). j f(1)j+1 ∗j Xj=1 13 Hence, we have that
f −1(n)qn f(n) Ω(n) L (qn) L (qn) = 1 − f − f . (6.4) 1 − qn f(1) f(1)f(n) f(1)f(n) nX≥1 nX≥1 nX≥1
6.2 GCD transform sums
6.2.1 General identities
We have that
qn µ(d) f(d) = f(k)qk (6.5a) 1 − qn 1 − qd n≥1 1≤d≤n k≥1 d|k X X X X (d,n)=1 qn µ(d) f(d) = f(k)qk (6.5b) 1 − qn 1 − qmd n≥1 1≤d≤n k≥1 d|k X X X X (d,k)=m n qn (f ∗ φ)(n)qn f(gcd(d, n)) n = n (6.5c) ! 1 − q 1 − q nX≥1 Xd=1 nX≥1 qn = f(n) (1 − qn)2 nX≥1 n = (f ∗ 1)(gcd(k,n))qn. nX≥1 Xk=1
6.2.2 Known particular cases
In [16], formulas for the discrete Fourier transform of a function evaluated at a gcd argument are derived. The reference also connects Lambert series expansions of Lioville for the divisor sum functions φa(n) (non-standard notation) that generalize the classical Euler totient function as
2a (a − |k − a|)d(gcd(a − |k − a|, a))qk qn d · φ(n/d) = k=1 (6.6a) 1 − qn P (1 − qa)2 nX≥1 d|X(a,n) qn p[a](q) d · φ(n/d) = , (6.6b) 1+ qn (1 − q2a)2 nX≥1 d|X(a,n) where
4a k p[a](q) := [(2a − |k − 2a|)d(gcd(2a − |k − 2a|, a)) − [k even]δ (a − |k/2 − a|) d(gcd(a − |k/2 − a|, a))] q . Xk=1 There are related LCM Dirichlet series, or DGF, identities that we can cite to find a Lambert series expansion for these functions from [6]:
n qn 1 m m 2 [k,n] = σ (m)+ dσ (d)µ qm, (6.7a) 1 − qn 2 1 2 dr dr n≥1 k=1 ! m≥1 d|m r| m X X X X Xd 14 n m+1 qn m + 1 B [k,n]m = σ (n)+ m+1−i 1 ∗ Id ∗ Id ∗ Id−1 (n) qm. (6.7b) 1 − qn m i m + 1 m m+i 2m n≥1 k=1 ! n≥1 i=1 ! X X X X
In particular, we learn from [6, D-19; D-71] that the first Lambert series above is generated as Lf1 (q) when f1 = 1 −1 2 Id1 ∗ Id2 ·(Id2 +Id3) , and the second (LCM powers sum) series is generated as Lf2 (q) with
m+1 m + 1 B f = Id + m+1−i Id ∗ Id ∗ Id−1 . 2 m i m + 1 m m+i 2m i=1 X 6.3 Anderson-Apostol divisor sums
We consider the Lambert series generating functions, denoted L1,m(f, g; q), over the sums m S (f, g; n) := fe(d)g . 1,m d d|X(m,n) We have that S (f, g; n)qn 1,m = (f ∗ g ∗ 1)(gcd(m,n))qn. (6.8) 1 − qn nX≥1 nX≥1 Proof of (6.8). We prove the following for integers n ≥ 1:
f(r)g(m/r)= f(r)g(s/r) d|n r|(m,d) s|m r|s X X sX|d|n X = (f ∗ g)(s). s|X(m,n) The key transition step in the above equations is in noting that (d, m) is a divisor of both d and m for any integers d, m ≥ 1.
The primary special case of interest with these types of sums is the Ramanujan sum, cq(x) ≡ S1,x(Id1,µ; q). However, these sums are periodic modulo m ≥ 1 and have a finite Fourier series expansion with known coefficients. Let d a (f, g; m)= g(d)f(k/d) · . k k d|X(m,k) Then we have that [12, §27.10] m 2πı·kn/m S1,m(f, g; n)= am(f, g; m) · e . Xk=1 6.4 Another summation variant
We next consider the Lambert series generating functions, denoted by L2,m(f, g; q), over the sums mn S (f, g; n) := f(d)g e . 2,m d2 d|X(m,n) As an example, we have that the Ramanujan tau function, τ(n), satisfies mn τ(m)τ(n)= d11τ . d2 d|X(m,n)
15 Also, for any α ∈ C and m,n ≥ 1, mn σ (m)σ (n)= dασ . α α α d2 d|X(m,n) We have that S (f, g; n)qn 2,a = f(d)g(ar) qm. (6.9) 1 − qn n≥1 m≥1 d|(a,m) r| m X X X Xd Proof of (6.9). We prove the following for integers n,m ≥ 1 by interchanging the order of divisor sum summation indices: mn mn f(r)g = f(r)g r2 rd d|n r|(d,m) r|(n,m) d| n X X X Xr = f(r)g(md). r|(n,m) d| n X Xr n Then since [q ]Lf (q) = (f ∗ 1)(n) for n ≥ 1, we are done.
Note that if g is completely multiplicative, then [3, §2, Ex. 31] mn f(m)f(n)= g(d)f . d2 d|X(m,n) 7 Other special identities
7.1 Pointwise products of convolutions with arithmetic functions (not necessarily multiplica- tive)
For any arithmetic functions f,g,h, we have that
d−1 h(n)f(n)qn (h ∗ µ)(d) = L (ωmq), (7.1a) 1 − qn d f d m=0 nX≥1 Xd≥1 X h(n)(f ∗ g)(n)qn f(d)h(dk)g(k)qdk = . (7.1b) 1 − qn 1 − qdk Xn≥1 Xd≥1 Xk≥1 As another example, let 1 ≤ b ≤ a be integers and f, g be any arithmetic functions. We define n n ha,b(f, g; n) := f a g b . a d d dX|n For integers a ≥ 1, let na f(n)q m Lf,a(q) := na = f(d) q . 1 − q a nX≥1 mX≥1 dX|m b Then we have that Lha,b;f,g (q) satisfies
h (f, g; n)qn d−1 L (ωmq) a,b = g(ra−b)µ(d/r) f,a d , (7.2a) 1 − qn d m=0 Xn≥1 Xd≥1 Xr|d X b
16 and by Möbius inversion, we obtain the OGF
n n ha,b(f, g; n)q = µ(n)Lha,b;f,g (q ). (7.2b) nX≥1 nX≥1 For example, the k-fold Möbius function satisfies the recurrence relation n n µ (n)= µ µ . k k−1 k k−1 d k d dX|n
7.2 Hadamard products with special arithmetic functions
We have by Mobius inversion and our previous identities on Dirichlet convolutions that: ω(n)f(n)qn = L (qp) (7.3a) 1 − qn f n≥1 p prime X X 2 λ(n)f(n)qn µ(n)f(nd2)qnd = . (7.3b) 1 − qn 1 − qnd2 nX≥1 Xd≥1 nX≥1 Proof. These two equations follow from (6.3) by noting that the characteristic function of the primes is given by χP = ω ∗ µ and that the characteristic function of the squares is given by χsq = λ ∗ µ.
7.3 Results on divisor sums involving products of ω(n) and µ(n)
Suppose that f is multiplicative such that f(p) 6= +1, −1, respectively, for all primes p. Then we have that [17] f(p) µ(d)ω(d)f(d)= (1 − f(p)) × (7.4a) f(p) − 1 Xd|n Yp|n Xp|n f(p) |µ(d)|ω(d)f(d)= (1 + f(p)) × (7.4b) 1+ f(p) Xd|n Yp|n Xp|n Under the same respective conditions, suppose that f is indeed completely multiplicative. Then similarly, we obtain that f(p) µ(d)ω(d)f(d)= µ(d)f(d) × (7.5a) f(p) − 1 Xd|n Xd|n Xp|n f(p) |µ(d)|ω(d)f(d)= |µ(d)|f(d) × (7.5b) 1+ f(p) Xd|n Xd|n Xp|n
7.4 Divisor sum convolution identities involving other prime-related arithmetic functions
We have the following prime sum related divisor sum identities in the form of f ∗ 1 generated by a Lambert series generating function over a multiplicative f [5]: µ(d) log d φ(n) log p = , (7.6a) d n 1 − p Xd|n Xp|n |µ(d)| log d Ψ (n) log p = k , (7.6b) dk nk pk + 1 Xd|n Xp|n |µ(d)| log d n log p = , (7.6c) φ(d) φ(n) p Xd|n Xp|n
17 µ(d) log d = −2ω(n) log γ(n), (7.6d) σ0(d) Xd|n k log γ(n) µ(d)ed (d) log d = (1 + (−1)ek)ω(n) × ; e ∈ {1, 2}, k ≥ 2, (7.6e) k k + (−1)e Xd|n log p µ(d)σ (d) log d = (−1)ω(n)γ(n) log γ(n)+ , (7.6f) 1 p Xd|n Xp|n f(p) logp µ(d)ef(d) log d = (1 + (−1)ef(p)) × ; e ∈ {1, 2}, (7.6g) f(p) + (−1)e Xd|n Yp|n Xp|n |µ(d)|kω(d) = (k + 1)ω(n). (7.6h) Xd|n
7.5 Relations of generalized Lambert series to q-series expansions
We do not focus on connections of other forms of generalized Lambert series expansions to q-series and partition generating functions, nor consider their representations in the context of modular forms. In this sense, we note that one can consider a class of generalized Lambert series defined by tn L(α; t,q) := , 1 − xqn nX≥1 and then connect variants of this function to q-series (see, for example, the identities given in (5.3)). For an overview of that vast material, we refer the reader to a subset of relevant references in [4, 2, 1].
18 References
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