Generating Special Arithmetic Functions by Lambert Series Factorizations
aa r X i v : . [ m a t h . N T ] A ug GENERATING SPECIAL ARITHMETIC FUNCTIONS BYLAMBERT SERIES FACTORIZATIONS
MIRCEA MERCAACADEMY OF ROMANIAN SCIENTISTSSPLAIUL INDEPENDENTEI 54, BUCHAREST, 050094 [email protected] D. SCHMIDTSCHOOL OF MATHEMATICSGEORGIA INSTITUTE OF TECHNOLOGYATLANTA, GA 30332 [email protected]
Abstract.
We summarize the known useful and interesting results and formu-las we have discovered so far in this collaborative article summarizing resultsfrom two related articles by Merca and Schmidt arriving at related so-termedLambert series factorization theorems. We unify the matrix representationsthat underlie two of our separate papers, and which commonly arise in iden-tities involving partition functions and other functions generated by Lambertseries. We provide a number of properties and conjectures related to the in-verse matrix entries defined in Schmidt’s article and the Euler partition func-tion p ( n ) which we prove through our new results unifying the expansions ofthe Lambert series factorization theorems within this article. Introduction
Lambert series factorization theorems.
We consider recurrence relationsand matrix equations related to
Lambert series expansions of the form [4, § Date : Monday 7 th August, 2017.2010
Mathematics Subject Classification.
Key words and phrases.
Lambert series; factorization theorem; matrix factorization; partitionfunction. § X n ≥ a n q n − q n = X m ≥ b m q m , | q | < , (1)for prescribed arithmetic functions a : Z + → C and b : Z + → C where b m = P d | m a d . There are many well-known Lambert series for special arithmetic func-tions of the form in (1). Examples include the following series where µ ( n ) denotesthe M¨obius function , φ ( n ) denotes Euler’s phi function , σ α ( n ) denotes the general-ized sum of divisors function , λ ( n ) denotes Liouville’s function , Λ( n ) denotes vonMangoldt’s function , ω ( n ) defines the number of distinct primes dividing n , and J t ( n ) is Jordan’s totient function for a fixed t ∈ C [4, § § a : X n ≥ µ ( n ) q n − q n = q, ( a n , b n ) := ( µ ( n ) , [ n = 1] δ ) (2) X n ≥ φ ( n ) q n − q n = q (1 − q ) , ( a n , b n ) := ( φ ( n ) , n ) X n ≥ n α q n − q n = X m ≥ σ α ( n ) q n , ( a n , b n ) := ( n α , σ α ( n )) X n ≥ λ ( n ) q n − q n = X m ≥ q m , ( a n , b n ) := ( λ ( n ) , [ n is a positive square] δ ) X n ≥ Λ( n ) q n − q n = X m ≥ log( m ) q m , ( a n , b n ) := (Λ( n ) , log n ) X n ≥ | µ ( n ) | q n − q n = X m ≥ ω ( m ) q m , ( a n , b n ) := ( | µ ( n ) | , ω ( n ) ) X n ≥ J t ( n ) q n − q n = X m ≥ m t q m , ( a n , b n ) := ( J t ( n ) , n t ) . In this article, our new results and conjectures extend and unify the related Lambertseries factorization theorems considered in two separate contexts in the references a Notation : Iverson’s convention compactly specifies boolean-valued conditions and is equivalentto the
Kronecker delta function , δ i,j , as [ n = k ] δ ≡ δ n,k . Similarly, [ cond = True ] δ ≡ δ cond , True inthe remainder of the article.
RITHMETIC FUNCTIONS GENERATED BY LAMBERT SERIES FACTORIZATIONS 3 [3, 6]. In particular, in [2] Merca notes that X n ≥ q n ± q n = 1( ∓ q ; q ) ∞ X n ≥ ( s o ( n ) ± s e ( n )) q n , where s o ( n ) and s e ( n ) respectively denote the number of parts in all partitions of n into an odd (even) number of distinct parts. More generally, Merca [3] proves that X n ≥ a n q n ± q n = 1( ∓ q ; q ) ∞ X n ≥ n X k =1 ( s o ( n, k ) ± s e ( n, k )) a k ! q n , (3)where s o ( n, k ) and s e ( n, k ) are respectively the number of k ’s in all partitions of n into an odd (even) number of distinct parts. Table 1.
The bottom row sequences in the matrices, A − n , in the defi-nition of (6) on page 4 for ≤ n ≤ . n r n,n − , r n,n − , . . . , r n, ,
14 2 , ,
15 4 , , ,
16 5 , , , ,
17 10 , , , , ,
18 12 , , , , , ,
19 20 , , , , , , ,
110 25 , , , , , , , ,
111 41 , , , , , , , , ,
112 47 , , , , , , , , , ,
113 76 , , , , , , , , , , , ,
114 90 , , , , , , , , , , , , ,
115 129 , , , , , , , , , , , , , ,
116 161 , , , , , , , , , , , , , , ,
117 230 , , , , , , , , , , , , , , , ,
118 270 , , , , , , , , , , , , , , , , , Matrix equations for the arithmetic functions generated by Lambertseries.
We then define the invertible n × n square matrices, A n , as in Schmidt’sarticle according to the convention from Merca’s article as [6, cf. § A n := ( s e ( i, j ) − s o ( i, j )) ≤ i,j ≤ n , (4) MIRCEA MERCA AND MAXIE D. SCHMIDT where the entries, s i,j := s e ( i, j ) − s o ( i, j ), of these matrices are generated by [3,Cor. 4.3] s i,j = s e ( i, j ) − s o ( i, j ) = [ q i ] q j − q j ( q ; q ) ∞ . We then have formulas for the Lambert series arithmetic functions, a n , in (1) andin the special cases from (2) for all n ≥ a a ... a n = A − n b m +1 − X s = ± ⌊ √ m +1 − s ⌋ X k =1 ( − k +1 b m +1 − k (3 k + s ) / | {z } := B b,m ≤ m RITHMETIC FUNCTIONS GENERATED BY LAMBERT SERIES FACTORIZATIONS 5 Significance of our new results and conjectures. Questions involvingdivisors of an integer have been studied for millennia and they underlie the deepestunsolved problems in number theory and related fields. The study of partitions,i.e., the ways to write a positive integer as a sum of positive integers, is muchyounger, with Euler considered to be the founder of the subject. The history ofboth subjects is rich and interesting but in the interest of brevity we will not gointo it here.The two branches of number theory, additive and multiplicative, turn out to berelated in many interesting ways. Even though there are a number of importantresults connecting the theory of divisors with that of partitions, these are somewhatscattered in their approach. There seem to be many other connections, in particularin terms of different convolutions involving these functions, waiting to be discovered.We propose to continue the study of the relationship between divisors and partitionswith the goal of identifying common threads and hopefully unifying the underlyingtheory. Moreover, it appears that, on the multiplicative number theory side, theseconnections can be extended to other important number theoretic functions such asEuler’s totient function, Jordan’s totient function, Liouville’s function, the M¨obiusfunction, and von Mangold’s function, among others.Our goal is to establish a unified global approach to studying the relationshipbetween the additive and multiplicative sides of number theory. In particular, wehope to obtain a unified view of convolutions involving the partition function andnumber theoretic functions. To our knowledge, such an approach has not beenattempted yet. Convolutions have been used in nearly all areas of pure and appliedmathematics. In a sense, they measure the overlap between two functions. Theidea for a unified approach for a large class of number theoretical functions has itsorigin in Merca’s article [3] and in Schmidt’s article [6].Perhaps our most interesting and important result, which we discovered com-putationally with Mathematica and Maple starting from an example formula givenin the Online Encyclopedia of Integer Sequences for the first column of the inverse MIRCEA MERCA AND MAXIE D. SCHMIDT matrices defined by (4) is stated in Theorem 3.2. The theorem provides an ex-act divisor sum formula for the inverse matrix entries, s ( − n,k , involving a M¨obiustransformation of the shifted Euler partition function, p ( n − k ). This result isthen employed to formulate new exact finite (divisor) sum formulas for each of theLambert series functions, a n , from the special cases in (2). These formulas areimportant since there are rarely such simple and universal identities expressing for-mulas for an entire class of special arithmetic functions considered in the context ofso many applications in number theory and combinatorics. Generalizations, furtherapplications, and topics for future research based on our work in this article aresuggested in Section 4.2. Exact and recursive formulas for the inverse matrices Proposition 2.1 (Recursive Matrix-Product-Like Formulas) . We let s i,j := s e ( i, j ) − s o ( i, j ) denote the terms in the original matrices, A n , from Schmidt’s article andlet s ( − i,j denote the corresponding entries in the inverse matrices, A − n . Then wehave that s ( − n,j = − n − j X k =1 s ( − n,n +1 − k · s n +1 − k,j + δ n,j = − n − j X k =1 s n,n − k · s ( − n − k,j + δ n,j = − n X k =1 s n,k − · s ( − k − ,j + δ n,j . Proof. The proof follows from the fact that for any n × n invertible matrices, A n and A − n , with entries given in the notation above, we have the following inversionformula for all 1 ≤ k, p ≤ n : n X j =1 s p,j · s ( − j,k = n X j =1 s ( − p,j · s j,k = [ p = k ] δ . It is easy to see that the matrix, A n , is lower triangular with ones on its diagonalfor all n ≥ RITHMETIC FUNCTIONS GENERATED BY LAMBERT SERIES FACTORIZATIONS 7 this property implies that s ( − i,j ≡ j < i by the adjoint (or adjugate)cofactor expansion for the inverse of a matrix. (cid:3) We can use the second of the two formulas given in the proposition repeatedly toobtain the following recursive, and then exact sums for the inverse matrix entries: s ( − i,j = − i X k =1 k − X k =1 k − X k =1 s i,k − · s k − ,k − · s k − ,k − · s ( − k − ,j + i X k = j +2 s i,k − · s k − ,j − s i,j + δ i,j . By inductively extending the expansions in the previous equation and noticing thatthe product terms in the multiple nested sums resulting from this procedure areeventually zero, we obtain the result in the next corollary. Corollary 2.2 (An Exact Nested Formula for the Inverse Matrices) . Let the no-tation for the next multiple, nested sums be defined as Σ m ( i, j ) := i X k = j +2 k − X k = j +2 · · · k m − − X k m = j +2 | {z } m total sums s i,k − · s k − ,k − × · · · × s k m − ,j . Then we may write an exact expansion for the inverse matrix entries as s ( − i,j = δ i,j − s i,j + Σ ( i, j ) − Σ ( i, j ) + · · · + ( − i + j +1 Σ i − j ( i, j ) . Proof. The proof is easily obtained by induction on j and repeated applications ofthe third recurrence relation stated in Proposition 2.1. (cid:3) The terms in the multiple sums defined in the corollary are reminiscent of theformula for the multiplication of two or more matrices. We may thus potentiallyobtain statements of more productive exact results providing expansions of theseinverse matrix terms by considering the nested, multiple sum formulas in Corollary2.2 as partial matrix products, though for the most part we leave the observationof such results as a topic for future investigation on these forms. However, giventhe likeness of the nested sums in the previous equations and in Corollary 2.2 to MIRCEA MERCA AND MAXIE D. SCHMIDT sums over powers of the matrix A n , we have computationally obtained the followingrelated formula for the corresponding inverse matrices A − n : A − n = n − X i =1 (cid:18) n − i (cid:19) ( − i +1 A i − n , n ≥ . We do not provide the proof by induction used to formally prove this identityhere due to the complexity of the forms of the powers of the matrix A n whichsomewhat limit the utility of the formula at this point. We also notice that thecorollary expresses the complicated inverse entry functions as a sum over productsof sequences with known and comparatively simple generating functions stated inthe introduction [3, cf. Cor. 4.3]. The results in Section 3 provide a more exactrepresentation of the entries of these inverse matrices for all n obtained by a separatemethod of proof. 3. Some experimental conjectures Figure 1. The first rows of the function s ( − i,j where the valuesof Euler’s partition function p ( n ) are highlighted in blueand the remaining values of the partition function q ( n ) arehighlighted in purple (in both sequences) or pink. RITHMETIC FUNCTIONS GENERATED BY LAMBERT SERIES FACTORIZATIONS 9 Several figures and exact formulas. Based on our experimental analysisand some intuition with partition functions, we expect that the inverse matrixentries, s ( − i,j , are deeply tied to the values of the Euler partition function p ( n ).In fact, we are able to plot the first few rows and columns of the two-dimensionalsequence in Figure 1 to obtain a highlighted listing of the values of special partitionfunctions in the sequence of these matrix inverse entries. A quick search of the firstfew columns of the table in the figure turns up the following special entry in theonline sequences database [7]. Conjecture 3.1 (The First Column of the Inverse Matrices) . The first column ofthe inverse matrix is given by a convolution (dot product) of the partition function p ( n ) and the M¨obius function µ ( n ) [7, A133732] . That is to say that s ( − n, = X d | n p ( d − µ ( n/d ) , , , , , , , , , , , , . . . } , i.e., so that by M¨obius inversion we have that p ( n − 1) = X d | n s ( − d, , , , , , , , , , . . . } . We are then able to explore further with the results from this first conjectureto build tables of the following two formulas involving our sequence, s ( − n,k , and theshifted forms of the partition function, p ( n − k ), where we take p ( n ) ≡ n < a ′ n,k := X d | n s ( − d,k (i) a ′′ n,k := X d | n p ( d − k ) µ ( n/d ) . (ii)The results of plotting these sequences for the first few rows and columns of 1 ≤ n ≤ 18 and 1 ≤ k ≤ 12, respectively, are found in the somewhat surprising andlucky results given in Figure 2. From this experimental data, we arrive at thefollowing second conjecture providing exact divisor sum formulas for the inverse n \ k (i) The Divisor Sums a ′ n,k n \ k (ii) The Divisor Sums a ′′ n,k Figure 2. A comparison of the two experimental divisor sum variants, a ′ n,k and a ′′ n,k , defined on page 9. Theorem 3.2 summarizesthe results shown in these two sequence plots. matrix entries. The corollary immediately following this conjecture is implied by acorrect proof of these results and from the formulas established in [6, § Theorem 3.2 (Exact Formulas for the Inverse Matrices) . For all n, k ≥ with ≤ k ≤ n , we have the following formula connecting the inverse matrices and theEuler partition function: s ( − n,k := X d | n p ( d − k ) µ ( n/d ) . (7) RITHMETIC FUNCTIONS GENERATED BY LAMBERT SERIES FACTORIZATIONS 11 Proof. We see that the first equation in (7) which we seek to prove is equivalent to p ( n − k ) := X d | n s ( − d,k . We next consider the variant of the Lambert series factorization theorem in (3)applied to the Lambert series in (1) with a n := s ( − n,k for a fixed integer k ≥ 1. Inparticular, the identity in (3) implies that X d | n s ( − d,k = n X m =0 n − m X j =1 ( s o ( n − m, j ) − s e ( n − m, j )) s ( − j,k · p ( m )= n X m =0 δ n − k,m · p ( m )= p ( n − k ) , where we have by our matrix formulation in (4) that m X j =1 ( s o ( m, j ) − s e ( m, j )) s ( − j,k = δ m,j . Thus by M¨obius inversion, we have our key formula for the inverse matrix entriesgiven in (7). (cid:3) We notice that the last equation given in the conjecture implies that we have aLambert series generating function for the inverse matrix entries given by X n ≥ s ( − n,k q n − q n = q k ( q ; q ) ∞ . for fixed integers k ≥ 1. We also note that where Merca’s article [3] provides thepartition function representation for the sequence s n,k in the matrix interpretationestablished in [6], the result in the theorem above effectively provides us with anexact identity for the corresponding sequence of inverse matrix entries, s ( − n,k , em-ployed as in Schmidt’s article to obtain the new expressions for several key specialmultiplicative functions. One important and interesting consequence of the result in Theorem 3.2 is thatwe have now completely specified several new formulas which provide exact rep-resentations for a number of classical and special multiplicative functions cited asexamples in (2) of the introduction. These formulas, which are each expanded inthe next corollary, connect the expansions of several special multiplicative functionsto sums over divisors of n involving Euler’s partition function p ( n ). In particular,we can now state several specific identities for classical number theoretic functionswhich connect the seemingly disparate branches of multiplicative number theorywith the additive nature of the theory of partitions and special partition functions.The results in the next corollary are expanded in the following forms: Corollary 3.3 (Exact Formulas for Special Arithmetic Functions) . For naturalnumbers m ≥ , let the next component sequences defined in [6, § be defined bythe formulas B φ,m = m + 1 − − · ( − u − − − u + ( − u ) m + 2( − u u (3 u + 2) + ( − u (6 u + 8 u − ! B µ,m = [ m = 0] δ + X b = ± ⌊ √ m +25 − b ⌋ X k =1 ( − k [ m + 1 − k (3 k + b ) / δ B λ,m = (cid:2) √ m + 1 ∈ Z (cid:3) δ − X b = ± ⌊ √ m +1 − b ⌋ X k =1 ( − k +1 hp m + 1 − k (3 k + b ) / ∈ Z i δ B Λ ,m = log( m + 1) − X b = ± ⌊ √ m +1 − b ⌋ X k =1 ( − k +1 log( m + 1 − k (3 k + b ) / B | µ | ,m = 2 ω ( m +1) − X b = ± ⌊ √ m +1 − b ⌋ X k =1 ( − k +1 ω ( m +1 − k (3 k + b ) / B J t ,m = ( m + 1) t − X b = ± ⌊ √ m +1 − b ⌋ X k =1 ( − k +1 ( m + 1 − k (3 k + b ) / t , RITHMETIC FUNCTIONS GENERATED BY LAMBERT SERIES FACTORIZATIONS 13 where u ≡ u ( m ) := ⌊ ( √ m + 1 + 1) / ⌋ and u ≡ u ( m ) := ⌊ ( √ m + 1 − / ⌋ .Then we have that φ ( n ) = n − X m =0 X d | n p ( d − m − µ ( n/d ) B φ,m µ ( n ) = n − X m =0 X d | n p ( d − m − µ ( n/d ) B µ,m λ ( n ) = n − X m =0 X d | n p ( d − m − µ ( n/d ) B λ,m Λ( n ) = n − X m =0 X d | n p ( d − m − µ ( n/d ) B Λ ,m | µ ( n ) | = n − X m =0 X d | n p ( d − m − µ ( n/d ) B | µ | ,m J t ( n ) = n − X m =0 X d | n p ( d − m − µ ( n/d ) B J t ,m . The corresponding formulas for the average orders, Σ a,x , of these special arithmeticfunctions are obtained in an initial form by summing the right-hand-sides of theprevious equations over all n ≤ x . We can also compare the results of the recurrence relations in the previous corol-lary to two other identical statements of these results. In particular, if we definethe sequence { G j } j ≥ = { , , , , , , , , , , , , . . . } as in [3, § 1] by theformula G j = 12 (cid:24) j (cid:25) (cid:24) j + 12 (cid:25) , then by performing a divisor sum over n in the previous equations, we see that thesequence pairs in the form of (1) satisfy b n = n X k =1 k − X j =0 p ( n − k )( − ⌈ j/ ⌉ b ( k − G j ) . We immediately notice the similarity of the recurrence relation for b n given in thelast equation to the known result from [6, Thm. 1.4] which states that b n = n X j =0 ( − ⌈ j/ ⌉ b n − G j , and which was proved by a separate non-experimental approach in the reference. Remark 3.4 (An Experimental Conjecture) . Since we have a well-known recur-rence relation for the partition function given by p ( n ) = n X k =1 ( − k +1 ( p ( n − k (3 k − / 2) + p ( n − k (3 k + 1) / , we attempt to formulate an analogous formula for the s ( − i,j using (7), which leadsus to the sums a ′′′ n := n X k =1 ( − k +1 (cid:16) s ( − n,k (3 k − / + s ( − n,k (3 k +1) / (cid:17) , , , , , , , , , , , , . . . } . A search in the integer sequences database suggests that this sequence denotesthe number of partitions of n into relatively prime parts, or alternately, aperiodicpartitions of n [7, A000837]. We notice the additional, and somewhat obvious andless interesting, identity which follows from the recurrence relation for p ( n ) givenabove expanded in the form of n X k =0 ( − ⌈ k/ ⌉ s ( − n,G k = 0 . Other properties related to the partition function.Proposition 3.5 (Partition Function Subsequences) . Let n be a positive integer.For ⌈ n/ ⌉ < k ≤ n , s ( − n,k = p ( n − k ) . RITHMETIC FUNCTIONS GENERATED BY LAMBERT SERIES FACTORIZATIONS 15 The indices of the first few rows such that s ( − n,k = p ( n − k ) is true for all < k ≤ n are { , , , , , , , , , , . . . } .Proof. This result is immediate from the divisor sum in (7) where the only divisorof n in the range ⌈ n/ ⌉ < k ≤ n is n itself. (cid:3) Proposition 3.6 (Partition Function Subsequences for Prime n ) . For n prime and ≤ k ≤ n , s ( − n,k = p ( n − k ) − δ ,k , where δ i,j is the Kronecker delta function.Proof. This result is also immediate from the divisor sum in (7) where the onlydivisors of the prime n are 1 and n and p (1 − k ) = δ k, by convention. In particular,we have that s ( − n,k = µ ( p ) p (1 − k ) + µ (1) p ( n − k ) , for all 1 ≤ k ≤ n . (cid:3) The next two results which we initially obtained experimentally from tables of thematrix inverse entries follow along the same lines as the previous two propositions.Given the ease with which we proved the last formulas for prime n , we omit theone-line proofs of the next two results below. Note that by the formula in (7), wemay also strengthen these results to prime powers of the form n = p k for k ≥ p . Proposition 3.7. For n prime, s ( − n ,k = p ( n − k ) − p ( n − k ) , for ≤ k ≤ n , p ( n − k ) , for n < k ≤ n . Proposition 3.8. For n prime, s ( − n,k = p (2 n − k ) − p ( n − k ) − p (2 − k ) + δ ,k , for ≤ k ≤ , p (2 n − k ) − p ( n − k ) , for < k ≤ n , p (2 n − k ) , for n < k ≤ n . A similar argument to the above can be used to show that if q, r ∈ Z + arerelatively prime positive integers, then we have that s ( − qr,k = δ ,k − p ( q − k ) − p ( r − k ) + p ( qr − k ) , which as we observe is another example of an additive formula we have obtaineddefining an inherently multplicative structure in terms of additive functions. No-tably, we can use this observation to show that if the arithmetic function a n in(1) is multiplicative, then we have that a q · a r = b qr − b p − b q + b for all positiveintegers p, q such that ( p, q ) = 1. We can then form subsequent generalizations forproducts of pairwise relatively prime integers, q , q , . . . , q m , accordingly.4. Conclusions Summary. We have proved a unified form of the Lambert series factoriza-tion theorems from the references [3, 6] which allows us to exactly express matrixequations between the implicit arithmetic sequences, a n and b n , in (1) and in theclassical special cases in (2). More precisely, we have noticed that the invertiblematrices, A n , from Schmidt’s article are expressed through the factorization theo-rem in (3) proved by Merca. We then proved new divisor sum formulas involvingthe partition function p ( n ) for the corresponding inverse matrices which define thesequences, a n , in terms of only these matrix entries and the secondary sequence of b n as in the results from [6].The primary application of our new matrix formula results is stated in Corol-lary 3.3. The corollary provides new exact finite (divisor) sum formulas for the RITHMETIC FUNCTIONS GENERATED BY LAMBERT SERIES FACTORIZATIONS 17 special arithmetic functions, φ ( n ), µ ( n ), λ ( n ), Λ( n ), | µ ( n ) | , and J t ( n ), and the cor-responding partial sums defining the average orders of these functions. One relatedresult not explicitly stated in Schmidt’s article provides a discrete (i.e., non-divisor-sum) convolution for the average order of the sum-of-divisors function, denoted byΣ σ,x := P n ≤ x σ ( n ), in the form of [6, § σ,x +1 = X s = ± X ≤ n ≤ x j √ n +25 − s k X k =1 ( − k +1 k (3 k + s )2 · p ( x − n ) . Other related divisor sum results that can be stated in terms of our new inversematrix formulas implied by Theorem 3.2 are found, for example, in Merca’s article[3, § Generalizations. Merca showed another variant of the Lambert series fac-torization theorem stated in the form of [3, Cor. 6.1] X n ≥ a n q n − q n = 1( q ; q ) ∞ X n ≥ ⌊ n/ ⌋ X k =1 ( s o ( n − k, k ) − s e ( n − k, k )) a k · q n . If we consider the generalized Lambert series formed by taking derivatives of (1)from [5] in the context of finding new relations between the generalized sum-of-divisors functions, σ α ( n ), we can similarly formulate new, alternate forms of thefactorization theorems unified by this article. For example, suppose that k, m ≥ X n ≥ a n q ( m +1) n (1 − q n ) k +1 = 1( q ; q ) ∞ X n ≥ ⌊ n/ ( m +1) ⌋ X i =1 s n − m,i a i (1 − q i ) k · q n , so that we have the factorization theorem providing that the previous series areexpanded by X n ≥ a n q ( m +1) n (1 − q n ) k +1 = 1( q ; q ) ∞ X n ≥ ⌊ nm +1 ⌋ X i =1 ⌊ n − mi ⌋ X j =0 (cid:18) k − jk − (cid:19) s n − m − ji,i · a i · q n , and so that when m ≥ k the series coefficients of these modified Lambert seriesgenerating functions are given by X d | nd ≤ ⌊ nm +1 ⌋ ( nd − − m + kk ) a d = n X q =0 ⌊ n − qm +1 ⌋ X i =1 ⌊ n − q − mi ⌋ X j =0 (cid:18) k − jk − (cid:19) s n − q − m − ji,i · a i · p ( q ) . Thus, again, as in Merca’s article, the applications and results in Corollary 3.3 canbe repeated in the context of a slightly different motivation for considering thesefactorization theorems.4.3. Topics for future research. Topics for future research based on the unifiedfactorization theorem results we have proved within the article include investigatingthe properties of the generalizations defined in the last subsection, considering con-gruences for the partition function and the inverse matrix entries, s ( − n,k , and findinguseful new asymptotic formulas for the average orders of the special functions inCorollary 3.3.The last topic is of particular interest since we have given an explicit formula forthe M¨obius function, µ ( n ), which holds for all n ≥ 0. The problem of determiningwhether the average order, M ( x ) := P n ≤ x µ ( n ), of this particular special functionis bounded by M ( x ) = O ( x / ε ) for all sufficiently small ε > x ≥ Acknowledgments. The authors thank the referees for their helpful insights andcomments on preparing the manuscript. References 1. G. H. Hardy and E. M. Wright. An Introduction to the Theory of Numbers . Oxford UniversityPress, 2008.2. M. Merca, Combinatorial interpretations of a recent convolution for the number of divisors ofa positive integer, J. of Number Theor. , 160, pp. 60–75 (2016).3. M. Merca, The Lambert series factorization theorem, Ramanujan J. , pp. 1–19 (2017). RITHMETIC FUNCTIONS GENERATED BY LAMBERT SERIES FACTORIZATIONS 19 4. F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark. NIST Handbook of Mathe-matical Functions . Cambridge University Press, 2010.5. M. D. Schmidt, Combinatorial sums and identities involving generalized divisor functions withbounded divisors, 2017, https://arxiv.org/abs/1704.05595 .6. M. D. Schmidt, New recurrence relations and matrix equations for arithmetic functions gen-erated by Lambert series, 2017, https://arxiv.org/abs/1701.06257 . Tentatively accepted in Acta Arith. 7. N. J. A. Sloane, The Online Encyclopedia of Integer Sequences, 2017, https://oeis.org/https://oeis.org/