arXiv:1712.00608v1 [math.NT] 2 Dec 2017 ftefr [ form the of of expansions generalized and variants several 1.1. n fteform the of and ucin aaadproduct. Hadamard function; ATRZTO HOESFRHDMR PRODUCTS HADAMARD FOR THEOREMS FACTORIZATION N IHRODRDRVTVSO ABR SERIES LAMBERT OF DERIVATIVES HIGHER-ORDER AND 2010 Date e od n phrases. and words Key abr eisfcoiaintheorems. factorization series Lambert usa 5 Tuesday : ahmtc ujc Classification. Subject Mathematics dniyfrtenme fdsic rmsdividing primes distinct fun of zeta number Riemann the the for for identity series new several functions, ating ftececet fteinteger-order pr the results of coefficients new the our summation-b of new of functions, Applications explicit multiplicative involving the partitions. sums to of an functions theory expansi multiplicative article, the characteristic special the the over connect within ries they expansions that their in to requi significant to give corre enough we coefficients important are attention series cases The function and generating series. genertwo series series Lambert Lambert Lambert of of of derivatives products expansions Hadamard cha the for which of factorizations theorems cases factorization articl primary original this cific these Within to theorems. analogs factorization series Lambert Abstract. 3 , 5 , th 6 efis umrz on oko eea rlmnr canonic preliminary several on work joint summarize first We , eebr 2017. December, 9 n n X X ] ERI NTTT FTECHNOLOGY OF INSTITUTE GEORGIA ≥ ≥ EEAIGFUNCTIONS GENERATING 1 1 abr eis atrzto hoe;mti factorizat matrix theorem; factorization series; Lambert 1 1 a a ¯ − − n n [email protected] COLO MATHEMATICS OF SCHOOL q q 8 HRYSRE NW STREET CHERRY 686 q q n n [email protected] n n 1 KLSBUILDING SKILES 117 AI .SCHMIDT D. MAXIE TAT,G 30332 GA ATLANTA, 1. = = Introduction ( ( q q ; ; 1 1 q q 12;1P1 51;05A19. 05A17; 11P81; 11A25; USA ) ) ∞ ∞ j 1 th n n X X ≥ ≥ eiaie fLmetsre gener- series Lambert of derivatives 1 1 abr eisfcoiaintheorems factorization series Lambert k X k X =1 =1 n n nterfrne ehv proved have we references the In s s e n,k n,k n . e a a k k yadtv aueof nature additive ly · · sdinterpretations ased n fLmetse- Lambert of ons eetbihnew establish we e to,ada exact an and ction, q q pnigt these to sponding atrz w spe- two racterize eteseildue special the re n n vd e exotic new ovide oevr are moreover, d , , o higher-order for tn functions: ating o;partition ion; al 2 M. D. SCHMIDT for an anda ¯n depending on an arbitrary arithmetic function an and where the n k k lower-triangular sequence sn,k := [q ](q; q) q /(1 − q ), which we typically require ∞ to beb independent of the an, is the difference of two partition functions counting the number of k’s in their respective odd (even) distinct partitions. In the concluding remarks to [3] we gave several specific examples of other con- structions of related Lambert series factorization theorems. In the reference, we also proved a few new properties of the factorizations of Lambert series generating functions over the convolution of two arithmetic functions, f ∗ g expanded by n (f ∗ g)(n)qn 1 = s (g)f(k) · qn, 1 − qn (q; q) n,k n 1 ∞ n 1 k=1 X≥ X≥ X which we cite in this article. The Lambert series factorizationse of these two forms and their variations, two to three of which we consider in this article, also imply matrix factorizations of these expansions which are dictated by the corresponding typically invertible matrix of sn,k or sn,k(g). The matrix-based interpretation of these factorization theorems is perhaps the most intuitive way to explore how these expansions “factor” into distinct matricese applied to vectors of special sequences.
1.2. Significance of our results. Our new results are rare and important because they provide a mechanism that effectively translates the divisor sums of the coef- ficients of a Lambert series generating function into ordinary sums which similarly generate the same prescribed arithmetic function, say an. Moreover, these fac- torization theorems connect the special functions in multiplicative number theory which are typically tied to a particular Lambert series expansion with the additive theory of partitions and partition functions in unusual and unexpected new ways. We are one of the first authors to examine such relations between the additive and multiplicative in detail (see also [4, 3, 5]). We note that we are not the first to consider the derivatives of Lambert series generating functions [8], though our perspective on the connections afforded by these factorizations is distinctly new.
1.3. Focus within this article. Within this article we explore the expansions of factorization theorems for two primary additional special case variants which are distinctive and important enough in their applications to require special atten- tion here: Hadamard products of two Lambert series generating functions and the higher-order integer derivatives of Lambert series generating functions. Section 2 proves several new properties of the first case, where the results proved in Section 3 consider the second case in detail. The significance of these two particular fac- torizations is that they have a broad range of applications to expanding special and classical arithmetic functions from multiplicative number theory. In Section 4 we tie up loose ends by offering two other related variants of the Lambert series factorizations. Namely, we prove factorization theorems for generating function convolutions and provide a purely matrix-based proof of a new formula for the coefficients enumerated by a Lambert series generating function.
New results and characterizations. The Hadamard product generating function cases lead to several forms of new so-termed “exotic” sums for classical special functions as illustrated in the explicit corollaries given in Example 2.4 of the next section. For example, if we form the Hadamard product of generating functions of FACTORIZATION THEOREMS FOR HADAMARD PRODUCTS AND DERIVATIVES 3 the two Lambert series over Euler’s totient function, φ(n), we obtain the following more exotic-looking sum for our multiplicative function of interest:
√24k 23 b −6 − n p(d − k) j k j(3j + b) 2 φ(n)= µ(n/d) k2 + (−1)j k − . d 2 k=1 d n b= 1 j=1 X X| X± X The importance of obtaining new formulas and identities for the derivatives of a se- ries whose coefficients we are interested in studying should be obvious, though our factorization results also provide new alternate characterizations of these deriva- tives apart from typical ODEs which we can form involving the derivatives of any sequence ordinary generating functions. Our results for the factorization theorems for derivatives of Lambert series generating functions in Section 3 provides two particular factorizations which characterize these expansions.
2. Factorization Theorems for Hadamard Products 2.1. Hadamard products of generating functions. The Hadamard product of two ordinary generating functions F (q) and G(q), respectively enumerating the sequences of {fn}n 0 and {gn}n 0 is defined by ≥ ≥ n (F ◦ G)(q) := fngn · q , for |q| < σF σG, n 0 X≥ where σF and σG denote the radii of convergence of each respective generating function. Analytically, we have an integral formula and corresponding coefficient extraction formula for the Hadamard product of two generating functions when F (q) is expandable in a fractional series respectively given by [1, §1.12(V); Ex. 1.30, p. 85] [11, §6.3] π 2 1 ıt ıt (F ◦ G)(q )= F (qe )G(qe− )dt 2π 0 Z q (F ◦ G)(q) = [x0]F G(x). x In the context of the factorization theorems we consider in this article and in the ref- erences, we consider the Hadamard products of two Lambert series generating func- tions for special arithmetic functions fn and gn which we define coefficient-wise to enumerate the product of the divisor sums over each sequence corresponding to the coefficients of the individual Lambert series over the two functions. This subtlety is discussed shortly in Definition 2.1. As we prove below, it turns out that we can formulate analogous factorization theorems for the cases of these Hadamard prod- ucts as well. The next definition makes the expansion of the particular Hadamard product functions we consider in this section more precise. Definition 2.1 (Hadamard Products for Lambert Series Generating Functions). For any fixed arithmetic functions f and g, we define the Hadamard product of the two Lambert series over f and g to be the auxiliary Lambert series generating function over the composite function afg(n) whose coefficients are given by
a (m)qm a (d) = [qn] fg := f g , fg 1 − qm d d d n m 1 d n d n X| X≥ X| X| :=fg(n) | {z } 4 M. D. SCHMIDT so that by M¨obius inversion we have that
afg(n)= fg(d)µ(n/d) = (fg ∗µ)(n). d n X| We note that this definition of our Hadamard product generating functions is al- ready slightly different than the one given above in that we define the Hadamard product of two Lambert series generating functions by the expansion of a third composite Lambert series which corresponds to the particular expansion of the factorization in (1) below. 2.2. Main results and applications. The next theorems in this section define ( 1) the key matrix sequences, sn,k(f) and sn,k− (f), in terms of the next factorization of the Lambert series over afg(n) from the definition above in the form of n a (n)qn 1 fg = s (f)g · qn, (1) 1 − qn (q; q) n,k k n 1 ∞ n 1 k=1 X≥ X≥ X where sn,k(f) is independent of the function g, which is equivalent to defining the factorization expansion by the inverse matrix sequences as
n n ( 1) k afg(n)q gn = s − · [q ] (q; q) × . (2) n,k ∞ 1 − qn k=1 n 1 X X≥ We also define the following function to expand divisor sums over arithmetic func- tions as ordinary sums for any integers 1 ≤ k ≤ n: 1, if k|n; Tdiv(n, k) := (0, otherwise. Theorem 2.2. For all integers 1 ≤ k ≤ n, we have the following definition of the factorization matrix sequence defining the expansion on the right-hand-side of (1) where we adopt the notation f(n) := d n fd: | sn,k(f)= Tdiv(n, k)f(n) e P √24(n k)+1 b − − 6e j(3j + b) j(3j + b) + (−1)j T n − , k · f n − . div 2 2 b= 1 j=1 X± X e Proof. By the factorization in (1) and the definition of afg(n) given above, we have that for f(n)= d n fd | n e P s (f) = [g ] f × g T (n, d) n,k k d d div d n d=1 X| X n n = [q ](q; q) × Tdiv(n, k)f(n)q , ∞ n 1 X≥ which equals the stated expansion of the sequence by Euler’se pentagonal number theorem which provides that j j(3j 1)/2 j(3j+1)/2 (q; q) =1+ (−1) q − + q . ∞ j 1 X≥ FACTORIZATION THEOREMS FOR HADAMARD PRODUCTS AND DERIVATIVES 5
Theorem 2.3 (Inverse Sequences). For all integers 1 ≤ k ≤ n, we have the next definition of the inverse factorization matrix sequence which equivalently defines the expansion on the right-hand-side of (1).
( 1) p(d − k) sn,k− (f)= µ(n/d), d n f(d) X| Proof. We expand the right-hand-side of the factorization in (1) for the sequence e ( 1) 1 gn := sn,r− (f), i.e., the exact inverse sequence, for some fixed r ≥ 1 as follows : n j ( 1) ( 1) f(n) · sd,r− (f)= sj,ksk,r− · p(n − j) d n j=0 k=1 X| X X e n = [j = r]δ p(n − j)= p(n − r). j=0 X Then the last equation implies that
( 1) p(n − r) sd,r− (f)= , d n f(n) X| which by M¨obius inversion implies our stated result. e Example 2.4 (Applications of the Theorem). For the arithmetic function pairs t s t (f,g) := (n ,n ), (φ(n), Λ(n)), (n ,Jt(n)), respectively, and some constants s,t ∈ C where σα(n) denotes the generalized sum-of-divisors function, Λ(n) is von Mangoldt’s function, φ(n) is Euler’s totient function, and Jt(n) is the Jordan totient function, we employ the equivalent expan- sions of the factorization result in (2) to formulate the following “exotic” sums as consequences of the theorems above: n s p(d − k) n = µ(n/d) σt(k)σs(k) (3) σt(d) k=1 d n " X X| √24k+1 b 6 − j k j(3j + b) j(3j + b) + (−1)jσ k − σ k − t 2 s 2 b= 1 j=1 # X± X n p(d − k) Λ(n)= µ(n/d) k log(k) d k=1 d n " X X| √24k 23 b −6 − j k j(3j + b) j(3j + b) + (−1)j k − log k − 2 2 b= 1 j=1 # X± X √24k 23 b n −6 − p(d − k) j k j(3j + b) 2t J (n)= µ(n/d) k2t + (−1)j k − . t dt 2 k=1 d n b= 1 j=1 X X| X± X 1 Notation: Iverson’s convention compactly specifies boolean-valued conditions and is equiva- lent to the Kronecker delta function, δi,j ,as[n = k]δ ≡ δn,k. Similarly, [cond = True]δ ≡ δcond,True in the remainder of the article. 6 M. D. SCHMIDT
By forming a second divisor sum over the divisors of n on both sides of the first equation above, the first more exotic-looking sum for the sum-of-divisors functions leads to an expression for σs(n) as a sum over the paired product functions, σt(n) · σs(n). We do not know of another such identity relating the generalized sum-of- divisors functions existing in the literature or in the references which we cite in this article. However, the following relations between the multiplicative generalized sum-of-divisors functions and special additive partition functions are known where pk(n) denotes the number of partitions of n into at most k parts and pp(n) denotes the number of plane, or planar, partitions of n [10, A000219] [7, §26.9, §26.12]: n 1 − n · p(n)= p(k)σ1(n) k=0 Xn n · pk(n)= pk(n − t) j, k ≥ 1 t=1 j t X Xj | k ≤ n n · pp(n)= pp(n − j)σ2(j). j=1 X Corollary 2.5 (New Series for the Riemann Zeta Function). For fixed s,t ∈ C such that ℜ(s) > 1 we have the following infinite sum representations of the Riemann zeta function where we denote the sequence of interleaved pentagonal numbers, ω(±n), 1 j 3j+1 by Gj = 2 2 2 for j ≥ 0 [10, A001318]: n p(d − k) j/2 σt(k − Gj )σs(k − Gj ) ζ(s)= µ(n/d) × (−1)⌈ ⌉ s σt(d) (k − Gj ) n 1 k=1 d n j:Gj j/2 Gj (q; q) = (−1)⌈ ⌉q . ∞ j 0 X≥ The convergence of these infinite series is guaranteed by our hypothesis that ℜ(s) > 1. We compare the results in the previous theorem to the known Dirichlet gener- ating functions for the sum-of-divisors functions which are expanded by [2, Thm. 291] [7, §27.4] σ (n) ζ(s)ζ(s − α)= α , ℜ(s) > 1, α +1 ns n 1 X≥ ζ(s)ζ(s − α)ζ(s − β)ζ(s − α − β) σ (n)σ (n) = α β , ℜ(s) > 1, α +1,β +1. ζ(2s − α − β) ns n 1 X≥ FACTORIZATION THEOREMS FOR HADAMARD PRODUCTS AND DERIVATIVES 7 s In particular, we note that while the series n σs(n)/n , and similarly for the second series, are divergent, our sums given in Corollary 2.5 do indeed converge for ℜ(s) > 1. P 3. Factorization Theorems for Derivatives of Lambert Series 1000000000 00 1200000000 00 0 −230000000 00 −1 2 −34000000 00 −2 −4 −3 −4500000 00 −2 2 6 −4 −56000 0 00 −2 −4 −6 0 −5 −6700 0 00 −1 2 −3 8 0 −6 −780 0 00 0 −2 9 −4 0 0 −7 −89 0 0 0 1 2 −6 −8 15 0 0 −8 −9 10 0 0 2 0 −3 4 −10 6 0 0 −9 −10 11 0 3 2 12 12 −5 12 7 0 0 −10 −11 12 (i) s1,n,k 100000000000 − 1 1 2 2 0000000000 − 1 1 1 3 3 3 000000000 1 1 1 4 0 4 4 00000000 3 2 1 1 0 5 5 5 5 0000000 1 1 1 1 1 0 6 3 6 6 000000 4 5 3 2 1 1 7 1 7 7 7 7 7 0 0 0 0 0 9 7 5 3 3 1 1 1 8 8 8 8 8 4 8 8 0 0 0 0 16 4 8 7 5 1 2 1 1 9 3 9 9 9 3 9 9 9 0 0 0 5 11 11 9 1 1 3 1 1 1 2 10 10 10 2 2 10 5 10 10 0 0 31 30 15 7 5 3 2 1 1 11 11 2 11 1 11 11 11 11 11 11 0 13 8 7 5 13 3 7 5 1 1 1 1 4 3 4 4 12 4 12 12 4 6 12 12 (−1) (ii) s1,n,k Figure 3.1. Factorization matrix sequences for the first-order deriva- tives of an arbitrary Lambert series generating function when 1 ≤ n, k ≤ 12 3.1. Derivatives of Lambert series generating functions. We seek analogous factorization theorems for the higher-order tth derivatives for all integers t ≥ 1 of 8 M. D. SCHMIDT an arbitrary Lambert series in the form of n a qn 1 qt · D n = s a · qn. (4) t 1 − qn (q; q) t,n,k k n t ∞ n t k=t X≥ X≥ X We note that we consider these sums over n ≥ t to produce an invertible matrix of the factorization sequences st,n,k. At first computation, the corresponding matrix sequences for these higher-order derivatives do not suggest any immediate intuitions ( 1) to exact formulas for st,n,k and st,n,k− as in the factorizations expanded above. The listings given in Figure 3.1 provide the first several rows of these sequences in the first-order derivative case where t := 1 for comparison with our intuition. However, may still prove using the method invoked in the proof of Theorem 2.3 to show that in the first-order case we have that2 ( 1) p(d − k) s − = µ(n/d), 1,n,k d d n X| which, in light of the construction of Corollary 2.5 and its notation, leads to the following further convergent series infinite expansion of the Riemann zeta function for ℜ(s) > 1: n p(d − k) j/2 σs(k − Gj ) ζ(s)= µ(n/d) × (−1)⌈ ⌉ s 1 . d (k − Gj ) − n 1 k=1 d n j:Gj 3.2. Main results. We will next require the statements of the next two results proved in [8] to state and prove the main results in this section. We refer the reader to the proofs of these two lemmas given in the reference. Lemma 3.1 (Modified Lambert Series Coefficients). For any fixed arithmetic func- tion an and integers m,t ≥ 1, n, k ≥ 0, we have the following identity for the series 2 More generally, if we expand the next mixed series of jth derivatives and initial terms annihilated by the differential operator as n j−1 n j anq i 1 n q · Dj + (a ∗ 1)(i)q = sj,n,kak · q , X 1 − qn X (q; q) X X n≥1 i=1 ∞ n≥t k=t n k k for integers j ≥ 2, we can easily prove that sj,n,k = sn,k = [q ]q /(1 − q )(q; q)∞ for 1 ≤ n Then we have the next two Lambert series expansions for the function At(n) given by a qn (A ∗ µ)(n)qn A (n) = [qn]qt · D n = [qn] t . t t 1 − qn 1 − qn n t n 1 X≥ X≥ Proof. The first equation follows from (5) applied to Lemma 3.2 when s := t. To prove the second form of a Lambert series generating function over some sequence cn enumerating At(n), we require that cd = At(n), d n X| which is true if and only if cd = At(d)µ(n/d) = (At ∗ µ)(n), d n X| by M¨obius inversion. Theorem 3.4 (Higher-Order Derivatives of Lambert Series Generating Functions). Let the notation for the function At(n) be defined as in Proposition 3.3. Then we have the next formulas for At(n) given by n 1 A (n) = [qn] s (µ)A (k) · qn t (q; q) n,k t ∞ n 1 k=1 X≥ X e 3 Notation: The bracket notation of n ≡ (−1)n−k s(n,k) denotes the unsigned triangle of k Stirling numbers of the first kind and n ≡ S(n,k) denotes the triangle of Stirling numbers of k the second kind. 10 M. D. SCHMIDT √24k 23 p −6 − n j k ( 1) j j(3j + p) A (n)= s − (µ) A (k)+ (−1) A k − , t n,k t t 2 k=1 p= 1 j=1 X X± X e where n sn,k(µ)= sn,kj · µ(j), j=1 X for sn,k := so(n, k) − se(n, k)ewhen the sequences so(n, k) and se(n, k) respectively denote the number of k’s in all partitions of n into an odd (even) number of distinct parts, and where a n-fold convolution formula involving µ for the inverse matrix ( 1) sequence sn,k− (µ) is proved explicitly in the references [3, §4]. Moreover, for all n ≥ 1 we have that the full Lambert series tth derivative formula is given by e n t 1 ⌊ i ⌋ n! − ( 1) (ik)!ai · a = s − (µ) · + A (n). (n − t)! d n,ik (ik − t)! t d n i=1 k=1 X| X X e Proof. The first result follows from the factorizations of the Lambert series over the convolution of two arithmetic functions proved in the reference [3, §4] where our Lambert series expansion in question is provided by Proposition 3.3 above. Similarly, the second result is a consequence of the first whose explicit expansions, i.e., for the inverse sequence are again proved in the reference. The last equation in the theorem follows from the proposition and adding back in the subtracted Lambert series terms when the summation for the series considered for At(n) starts from n ≥ t instead of from one. The multiples of ik in the last formula reflect that the coefficients of qi/(1 − qi) and its qt-scaled derivatives are always zero unless the coefficient index is a multiple of i. 3.3. Another related factorization. Remark 3.5 (Another Factorization). The first factorization expansion we con- sidered in (4) of this section is obtained by applying Lemma 3.6 in the case where bn,i = [ai](At ∗ µ)(n) d t m t − k i − 1 − r + k t k r = (−1) − − k!× m k r k 0 k m t d n ≤0X≤r t≤ X| ≤ ≤ d × imT (d, i)µ(n/d) · t ≤ i ≤ . div r +1 δ In this case, we can obtain a similar expansion of the middle identity in Theorem 3.4 in the form of √24k 23 p −6 − n j k ( 1) j j(3j + p) a = s − (b) A (k)+ (−1) A k − . n t,n,k t t 2 k=1 p= 1 j=1 X X± X FACTORIZATION THEOREMS FOR HADAMARD PRODUCTS AND DERIVATIVES 11 Lemma 3.6 (A Related Factorization Result). If we expand the Lambert series factorization n n n bn,j aj · q 1 j=1 = s (b)a · qn, 1 − qn (q; q) n,k k n 1 P ∞ n 1 k=1 X≥ X≥ X then we have the formula n sn,k(b)= sn,j · bj,k, j=1 X n k k where sn,k := [q ](q; q) q /(1−q )= so(n, k)−se(n, k) when the sequences so(n, k) ∞ and se(n, k) respectively denote the number of k’s in all partitions of n into an odd (even) number of distinct parts. Proof. If we take the nested coefficients first of ak and then of bj,k for some j, k ≥ 1 on both sides of the factorization cited above, we obtain that j q n · (q; q) = [bj,k] sn,k(b) · q . 1 − qj ∞ n 1 X≥ Then if we take the coefficients of qn on each side of the previous equation we arrive at the identity sn,j = [bj,k]sn,k(b), for j =1, 2,...,n. Finally, we multiply through both sides of the last equation by bj,k and then sum over all j to conclude that the stated formula for sn,k(b) in the lemma is correct. Equivalently, since both sequences of bn,k and sn,k(b) are lower triangular, we could have deduced this identity by truncating the partial sums up to n and employing a matrix argument to justify the formula above. 4. Expansions of other special factorization theorems 4.1. A factorization theorem for convolutions of Lambert series. In what follows we adopt the next notation for the Lambert series over a prescribed arith- metic function h defined by h(n)qn H (q) := . L 1 − qn n 1 X≥ We seek a factorization theorem for the convolution of two Lambert series generat- ing functions, FL(q) and GL(q), in the following form: 1 1 n · F (q)G (q)= s (g)f · qn. (6) q L L (q; q) n,k k ∞ n 1 k=1 X≥ X Theorem 4.1 (Factorization Theorem for Convolutions). For the Lambert series factorization defined in (6), we have the following exact expansions of the two matrix sequences characterizing the factorization where the difference of partition functions n k k sn,k := [q ](q; q) q /(1 − q ): ∞ n+1 s (g)= s g(d) n,k j,k j=1 d n+1 j X | X− 12 M. D. SCHMIDT k+1 ( 1) d q sn,k− (g)= [q ] µ(n/d). (q; q) GL(q) d n ∞ X| Proof. We prove our second result along the same lines as the proof of Theorem ( 1) 2.3 above. Namely, we choose FL(q) to denote the Lambert series over sn,k− (g) for some fixed k ≥ 1 and expand the right-hand-side of (6) as k n q ( 1) [q ] = sd,k− (g), (q; q) · GL(q)/q ∞ d n X| which by M¨oebius inversion implies our stated result. Then to obtain a formula for the sequence sn,k(g) from the identity expanding the inverse sequence, we observe a property of the product of any two inverse matrices which is that n ( 1) [n = k]δ = sn,j− (g)sj,k(g). j=1 X We then perform a divisor sum over n in the previous equation to obtain that ( 1) [k|n]δ = sd,k− (g) d n X| n n j 1 1 = [q − − ] × s (g), (q; q) · G (q) j,k j=1 L X ∞ which by another generating function argument implies that n n GL(q) sn,k(g) = [q ] [k|n] q (q; q) · δ ∞ q n 1 X≥ qk G (q) = (q; q) · L 1 − qk ∞ q n+1 = s g(d) , j,k j=1 d n+1 j X | X− as claimed. We notice that the factorization in (6) together with the theorem imply that we have the two expansions of the following form: n k+1 n d q k [q ]FL(q)= [q ] × µ(n/d) · [q ](q; q) FL(q)GL(q) (q; q) GL(q) ∞ k=1 d n ∞ X X| n k+1 n d q k 2 [q ]GL(q)= [q ] × µ(n/d) · [q ](q; q) GL(q) . (q; q) GL(q) ∞ k=1 d n ∞ X X| The special case where FL(q) := GL(q) in the last expansion provides a curious new relation between any Lambert series generating function GL(q), its reciprocal, and its square. This observation can be iterated to obtain even further multiple sum identities involving powers of GL(q). FACTORIZATION THEOREMS FOR HADAMARD PRODUCTS AND DERIVATIVES 13 4.2. A matrix-based proof of a factorization for sequences generated by Lambert series. As a last application of special cases of the Lambert series fac- torization theorems we have extended in this article, we consider another method of matrix-based proof which provides new formulas for the sequences generated by a Lambert series over an: b(n) = (a ∗ 1)(n). This approach is unique because un- like the factorization theorem variants we have explored so far which provide new identities and expansions for the sequence an itself, the result in Theorem 4.2 pro- vides useful new inverse sequence expansions exclusively for the terms b(n) whose ordinary generating function is the Lambert series generating function at hand [9, cf. Thm. 1.4]. The first factorization theorem expanded in the introduction implicitly provides a matrix-multiplication-based representation of the coefficients b(n) stated in terms of the matrix, (Tdiv(i, j))n n ≡ (Tdiv)n, in the explicit forms of × a1 a1 a2 a2 (T ) = a and (T ) 1 = a · µ(n/d), div n . d div n− . d . d n . d n X| X| a a n n where the corresponding inverse operation above is M¨obius inversion. For example, when n := 6 these matrices are given by 100000 1 0 0000 110000 −1 1 0 000 101000 1 −1 0 1 000 (Tdiv)6 = and (Tdiv)6− = 110100 0 −1 0 100 100010 −1 0 0 010 111001 1 −1 −1 0 0 1 Operations with our new definitions of the matrices above allow us to prove the next result. Theorem 4.2. For all n ≥ 1 and a fixed arithmetic function an we have the identity n k b(n)= sn,kCk,j aj , j=1 Xk=1 X where the inner matrix entries are given by [10, A000041] d Cn,k = p(d − ik)µ(n/d). d n i=1 X| X Proof. We first note that the theorem is equivalent to showing that we have a desired expansion of the form n n ( 1) sn,k− b(k)= Cn,kak (i) kX=1 kX=1 14 M. D. SCHMIDT The right-hand-side of (i) is equivalent to the expansion of the last row in the matrix-vector product a1 a1 a2 j a2 1 i q 1 (Ci,j )n n . = (Tdiv)n− [q ] . , × . 1 − qj (q; q) . . ∞ n n . × a a n n and that the left-hand-side of (i) is expanded by a1 a1 a2 a2 ( 1) 1 (sn,k− )n n(Tdiv)n . = (Tdiv)n− (p(i − j))n n (Tdiv)n . . × . . . × . a a n n Then we have that the sequence b(n) is expanded by multiplying the left-hand-side 1 of (i) by the matrix (p(i − j))−n n(Tdiv)n where × a1 a1 j a2 k a2 1 i q 1 n q (p(i − j))n− n [q ] . = [q ] . × 1 − qj (q; q) . 1 − qk . ∞ n n . . × a a n n n qk 7−→ [qn] · a = b(n) 1 − qk k kX=1 Corollary 4.3 (An Exact Formula for a Prime Counting Function). For all n ≥ 1, we have the exact formula for the function ω(n) which counts the number of distinct primes dividing n given by [10, A001221] n k d ω(n) = log p(d − ji) s · |µ(j)| , 2 n,k k=1 j=1 d k i=1 X X X| X where sn,k = so(n, k) − se(n, k) denotes the difference of the number of k’s in all partitions of n into an odd (even) number of distinct parts as in Section 1. Proof. We select the special case of (a,b) := (|µ|, 2ω) to arrive at the statement in the corollary. Acknowledgments. The authors thank the referees for their helpful insights and comments on preparing the manuscript. The author also thanks Mircea Merca for suggestions on the article and for collaboration on the prior articles on Lambert series factorization theorems. References [1] L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expressions, Reidel Publishing Company, 1974. [2] G. H. Hardy and E. M. Wright. An Introduction to the Theory of Numbers. Oxford University Press, 2008. FACTORIZATION THEOREMS FOR HADAMARD PRODUCTS AND DERIVATIVES 15 [3] M. Merca and M. D. Schmidt, Factorization theorems for generalized Lambert series and applications, 2017, submitted, preprint draft available upon request. [4] M. Merca and M. D. Schmidt, A partition identity related to Stanley’s theorem, 2018, ac- cepted for publication in the American Mathematical Monthly, to appear. [5] M. Merca and M. D. Schmidt, New Factor Pairs for Factorizations of Lambert Series Gener- ating Functions, 2017, https://arxiv.org/abs/1706.02359, submitted. [6] M. Merca, The Lambert series factorization theorem, The Ramanujan Journal, pp. 1–19 (2017). [7] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark. NIST Handbook of Mathe- matical Functions. Cambridge University Press, 2010. [8] M. D. Schmidt, Combinatorial sums and identities involving generalized divisor functions with bounded divisors, 2017, https://arxiv.org/pdf/1704.05595, submitted. [9] M. D. Schmidt, New recurrence relations and matrix equations for arithmetic functions gen- erated by Lambert series, 2017, accepted for publication in Acta Arithmetica. [10] N. J. A. Sloane, The Online Encyclopedia of Integer Sequences, 2017, https://oeis.org/. [11] R. P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge University Press, 1999. E-mail address: [email protected], [email protected] Georgia Institute of Technology, School of Mathematics, 117 Skiles Building, 686 Cherry Street NW, Atlanta, GA 30332, USA