Exact Formulas for the Generalized Sum-of-Divisors Functions
aa r X i v : . [ m a t h . N T ] A p r Exact Formulas for the GeneralizedSum-of-Divisors Functions
Maxie D. SchmidtSchool of MathematicsGeorgia Institute of TechnologyAtlanta, GA 30332USA [email protected]@gatech.edu
Abstract
We prove new exact formulas for the generalized sum-of-divisors functions, σ α ( x ) := P d | x d α . The formulas for σ α ( x ) when α ∈ C is fixed and x ≥ n ≤ x and terms involving the r -order harmonic numbersequences and the Ramanujan sums c d ( x ). The generalized harmonic number sequencescorrespond to the partial sums of the Riemann zeta function when r > r ≤ n ( q ), which completely factorize the Lambert series terms (1 − q n ) − intoirreducible polynomials in q . We focus on the computational aspects of these exact ex-pressions, including their interplay with experimental mathematics, and comparisons ofthe new formulas for σ α ( n ) and the summatory functions P n ≤ x σ α ( n ). Mathematics Subject Classification : Primary 30B50; Secondary 11N64, 11B83.
Keywords:
Divisor function; sum-of-divisors function; Lambert series; cyclotomic polynomial.
Revised:
April 23, 2019
We begin our search for interesting formulas for the generalized sum-of-divisors functions, σ α ( n )for α ∈ C , by expanding the partial sums of the Lambert series which generate these functionsin the form of [5, § § e L α ( q ) := X n ≥ n α q n − q n = X m ≥ σ α ( m ) q m , | q | < . (1)1n particular, we arrive at new expansions of the partial sums of Lambert series generatingfunctions in (1) which generate our special arithmetic sequences as σ α ( x ) = [ q x ] x X n =1 n α q n − q n ! = X d | x d α , α ∈ Z + . (2) The technique employed in this article using the Lambert series expansions in (1) is to expandby repeated use of the properties of the well-known sequence of cyclotomic polynomials , Φ n ( q ),defined by [3, §
3] [9, § n ( q ) := Y ≤ k ≤ n gcd( k,n )=1 (cid:16) q − e πı kn (cid:17) . (3)For each integer n ≥ q n − Y d | n Φ d ( q ) , (4)or equivalently that Φ n ( q ) = Y d | n ( q d − µ ( n/d ) , (5)where µ ( n ) denotes the M¨obius function . If n = p m r with p prime and gcd( p, r ) = 1, wehave the identity that Φ n ( q ) = Φ pr ( q p m − ). In later results stated in the article, we use theknown expansions of the cyclotomic polynomials which reduce the order n of the polynomialsby exponentiation of the indeterminate q when n contains a factor of a prime power. A shortlist summarizing these transformations is given as follows for p and odd prime, k ≥
1, andwhere p ∤ r :Φ p ( q ) = Φ p ( − q ) , Φ p k ( q ) = Φ p (cid:16) q p k − (cid:17) , Φ p k r ( q ) = Φ pr (cid:16) q p k − (cid:17) , Φ k ( q ) = q k − + 1 , (6)The next definitions to expand our Lambert series generating functions further by factoring itsterms by the cyclotomic polynomials . Special notation : Iverson’s convention compactly specifies boolean-valued conditions and is equivalent tothe
Kronecker delta function , δ i,j , as [ n = k ] δ ≡ δ n,k . Similarly, [ cond = True ] δ ≡ δ cond , True ∈ { , } , which is 1if and only if cond is true, in the remainder of the article. efinition 1.1 (Notation and logatithmic derivatives) . For n ≥ q , wedefine the following rational functions related to the logarithmic derivatives of the cyclotomicpolynomials: Π n ( q ) := n − X j =0 ( n − − j ) q j (1 − q )(1 − q n ) = ( n − − nq − q n (1 − q )(1 − q n ) (7) e Φ n ( q ) := 1 q · ddw [log Φ n ( w )] (cid:12)(cid:12)(cid:12) w → q . For any natural number n ≥ p , we use ν p ( n ) to denote the largest power of p dividing n . If p ∤ n , then ν p ( n ) = 0 and if n = p γ p γ · · · p γ k k is the prime factorization of n then ν p i ( n ) = γ i . That is, ν p ( n ) is the valuation function indicating the exact non-negative exponentof the prime p dividing any n ≥
2. In the notation that follows, we consider sums indexed by p to be summed over only the primes p by convention unless specified otherwise. Finally, wedefine the function e χ PP ( n ) to denote the indicator function of the positive natural numbers n which are not of the form n = p k , p k for any primes p and exponents k ≥
1. The conventionswhich make this definition accessible will become clear in the next subsections.
To provide some intuition to the factorizations of the terms in our Lambert series generatingfunctions defined above, the listings in Table 1 provide the first several expansions of theright-hand-sides of the next equations according to the optimal applications of (6) in our newformulas. The components highlighted by the examples in the table form the key terms ofour new exact formula expansions. Notably, we see that we may write the expansions of theindividual Lambert series terms as nq n − q n + n − − q = X d | nd> e Φ d ( q ) , where we can reduce the index orders of the cyclotomic polynomials, Φ n ( q ), and their logarith-mic derivatives, e Φ d ( q ), in lower-indexed cyclotomic polynomials with q transformed into powersof q to powers of primes according to the identities noted in (6) [3, cf. §
3] [9, cf. § n ≥ q n − q n = − n (1 − q ) + 1 n X d | nd> e Φ d ( q ) . (8)3 Lambert Series Expan-sions (cid:16) nq n − q n + n − − q (cid:17) FormulaExpansions Reduced-IndexFormula q e Φ ( q ) - - q q + q e Φ ( q ) - - q + q e Φ ( q )+ e Φ ( q ) e Φ ( q )+2 e Φ ( q )5 q +2 q + q q + q + q + q e Φ ( q ) - - q + − q − q + q + q q + q e Φ ( q )+ e Φ ( q )+ e Φ ( q ) - - q +4 q +3 q +2 q + q q + q + q + q + q + q e Φ ( q ) - - q + q + q e Φ ( q )+ e Φ ( q )+ e Φ ( q ) e Φ ( q )+2 e Φ ( q ) +4 e Φ ( q )9 q q + q + ( q ) q + q e Φ ( q )+ e Φ ( q ) e Φ ( q )+3 e Φ ( q )10 q + − q +2 q − q − q + q − q + q + q +2 q + q q + q + q + q e Φ ( q )+ e Φ ( q )+ e Φ ( q ) - - q +8 q +7 q +6 q +5 q +4 q +3 q +2 q + q q + q + q + q + q + q + q + q + q + q e Φ ( q ) - - q + q + − q − q + q + q q + q − ( − q ) − q + q e Φ ( q )+ e Φ ( q )+ e Φ ( q ) e Φ ( q )+2 e Φ ( q ) + e Φ ( q )+ e Φ ( q )+ e Φ ( q ) + e Φ ( q )+2 e Φ ( q ) q +10 q +9 q +8 q +7 q +6 q +5 q +4 q +3 q +2 q + q q + q + q + q + q + q + q + q + q + q + q + q e Φ ( q ) - - q + − q +4 q − q +2 q − q − q + q − q + q − q + q + q +4 q +3 q +2 q + q q + q + q + q + q + q e Φ ( q )+ e Φ ( q )+ e Φ ( q ) - - q q + q + q +2 q + q q + q + q + q + − q +5 q − q +3 q − q − q + q − q + q − q + q e Φ ( q )+ e Φ ( q )+ e Φ ( q ) - - q + q + q + q e Φ ( q )+ e Φ ( q )+ e Φ ( q )+ e Φ ( q ) e Φ ( q )+2 e Φ ( q ) +4 e Φ ( q ) +8 e Φ ( q ) Table 1: Expansions of Lambert Series Terms by Cyclotomic PolynomialPrimitives.
The double dashes (--) in the rightmost column of the table indicatethat the entry is the same as the previous column to distinguish between the caseswhere we apply our special reduction formulas.
Remark 1.2 (Experimental Intuition for These Formulas) . We begin by pointing our thatthe genesis of the formulas proved in Section 2 (stated precisely below) came about by ex-perimentally observing the exact polynomial expansions of the key Lambert series terms q n − q n which provide generating functions for σ α ( n ) with Mathematica . Namely, the computer algebraroutines employed by default in
Mathematica are able to produce the semi-factored outputreproduced in Table 1. Without this computationally driven means for motivating experi-mental mathematics, we would most likely never have noticed these subtle formulas for theoften-studied class of sum-of-divisors functions!The motivation for our definition of the next three divisor sum variants given in Definition1.3 is to effectively exploit the particularly desirable properties of the coefficients of thesepolynomial expansions when they are treated as generating functions for the generalized sum-4f-divisors functions in the main results stated in the next section. More to the point, whena natural number d is of the form d = p k , p k for some prime p and exponent k ∈ Z + , wehave the reduction formulas cited in (6) above to translate the implicit forms of the cyclotomicpolynomials Φ d ( q ) into polynomials in now powers of q p k indexed only by the sums over primes p . The third and fourth columns of Table 1 naturally suggest by computation the exact formsof the (logarithmic derivative) polynomial expansions we are looking for to expand our Lambertseries terms. In effect, the observation of these trends in the polynomial expansions of 1 − q n led to the intuition motivating our new results within this article. In particular, we introducethe notation in the next definition corresponding to component sums employed to express sumsover the previous identity in our key results stated in the next pages of the article. Definition 1.3 (Notation for component divisor sums) . For fixed q and any n ≥
1, we definethe component sums, e S i,n ( q ) for i = 0 , , e S ,n ( q ) = X d | nd> d = p k , p k e Φ d ( q ) e S ,n ( q ) = X p | n Π p νp ( n ) ( q ) e S ,n ( q ) = X p | np> Π p νp ( n ) ( − q ) . Recall that for any α ∈ C the generalized sum-of-divisors function is defined by the divisor sum σ α ( x ) = X d | x d α . We use the following notation for the generalized α -order harmonic number sequences: H ( α ) n := n X k =1 k − α . Proposition 1.4 (Series coefficients of the component sums) . For any fixed α ∈ C and integers x ≥ , we have the following components of the partial sums of the Lambert series generatingcited next in Theorem 1.5: b S ( α )0 ( x ) := [ q x ] x X n =1 e S ,n ( q ) n α − =: τ α ( x ) (i)5 S ( α )1 ( x ) := [ q x ] x X n =1 e S ,n ( q ) n α − = X p ≤ x ν p ( x )+1 X k =1 p αk − H (1 − α ) j xpk k (cid:18) p (cid:22) xp k (cid:23) − p (cid:22) xp k − p (cid:23) − (cid:19) (ii) b S ( α )2 ( x ) := [ q x ] x X n =1 e S ,n ( q ) n α − = X ≤ p ≤ x ν p ( x )+1 X k =1 p αk − − α H (1 − α ) j x pk k ( − j xpk − k (cid:18) p (cid:22) xp k (cid:23) − p (cid:22) xp k − p (cid:23) − (cid:19) . (iii)The precise form of the expansions in (i) of the previous proposition and its connectionsto the Ramanujan sums, c q ( n ), is explored in the results stated in Proposition 2.3 of the nextsection. Theorem 1.5 (Exact formulas for the generalized sum-of-divisors functions) . For any fixed α ∈ C and natural numbers x ≥ , we have the following generating function formula: σ α ( x ) = H (1 − α ) x + b S ( α )0 ( x ) + b S ( α )1 ( x ) + b S ( α )2 ( x ) . We first have a few remarks about symmetry in the identity from the theorem in the contextof negative-order divisor functions of the form σ − α ( x ) = X d | x (cid:16) xd (cid:17) − α = σ α ( x ) x α , α ≥ , and a brief overview of the applications we feature in Section 3. For integers α ∈ N , we can express the “negative-order” harmonic numbers, H ( − α ) n , in terms ofthe generalized Bernoulli numbers (polynomials) as Faulhaber’s formula n X m =1 m α = 1 α + 1 ( B α +1 ( n + 1) − B α +1 )= 1( α + 1) α X j =0 (cid:18) α + 1 j (cid:19) B j n α +1 − j . Then since the convolution formula above proves that σ − β ( n ) = σ β ( n ) /n β whenever β >
0, wemay also expand the right-hand-side of the theorem in the symmetric form of σ α ( x ) = x α (cid:16) H ( α +1) x + τ − α ( x ) + b S ( − α )1 ( x ) + b S ( − α )2 ( x ) (cid:17) , α > c d ( x ) denotes a Ramanujansum (see Proposition 2.3):
Theorem 1.6 (Symmetric Forms of the Exact Formulas) . For any fixed α ∈ C and integers x ≥ , we have the following formulas: b S ( − α )0 ( x ) = x X d =1 H ( α +1) ⌊ xd ⌋ · c d ( x ) d α +1 · χ PP ( d ) (i) b S ( − α )1 ( x ) = X p ≤ xp prime ν p ( x ) X k =1 ( p − p αk +1 H ( α +1) xpk − p α · ν p ( x )+ α +1 H ( α +1) (cid:22) xpνp ( x )+1 (cid:23) (ii) b S ( − α )2 ( x ) = ( − x α +1 X p ≤ xp prime ν p ( x ) X k =1 ( p − p αk +1 H ( α +1) j x pk k − p α · ν p ( x )+ α +1 H ( α +1) (cid:22) x pνp ( x )+1 (cid:23) . (iii) The generalized sum of divisors functions then have the following explicit expansions involvingthese formulas as σ α ( x ) = x α (cid:16) H ( α +1) x + b S ( − α )0 ( x ) + b S ( − α )1 ( x ) + b S ( − α )2 ( x ) (cid:17) . (9)We notice that this symmetry identity given in Theorem 1.6 provides a curious, and neces-sarily deep, relation between the Bernoulli numbers and the partial sums of the Riemann zetafunction involving nested sums over the primes. It also leads to a direct proof of the knownasymptotic results for the summatory functions [13, § X n ≤ x σ α ( n ) = ζ ( α + 1) α + 1 x α +1 + O (cid:0) x max(1 ,α ) (cid:1) , α > , α = 1 . We will explore this direct proof based on Theorem 1.6 in more detail as an application givenin Section 3.1.
Example 2.1.
We first revisit a computational example of the rational functions defined bythe logarithmic derivatives in Definition 1.1 from Table 1. We make use of the next variant ofthe identity in (5) in the proof below which is obtained by M¨oebius inversion:Φ n ( q ) = Y d | n ( q d − µ ( n/d ) . (10)7n the case of our modified rational cyclotomic polynomial functions, e Φ n ( q ), when n := 15, weuse this product to expand the definition of the function as e Φ ( q ) = 1 x · ddq (cid:20) log (cid:18) (1 − q )(1 − q )(1 − q )(1 − q ) (cid:19)(cid:21)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q → /q = 31 − q + 51 − q − − q − − q = 8 − q + 5 q − q + 3 q − q − q + q − q + q − q + q . The procedure for transforming the difficult-looking terms involving the cyclotomic polynomialswhen the Lambert series terms, q n / (1 − q n ), are expanded in partial fractions as in Table 1 isessentially the same as this example for the cases we will encounter here. In general, we havethe next simple lemma when n is a positive integer. Lemma 2.2 (Key characterizations of the tau divisor sums) . For integers n ≥ and anyindeterminate q , we have the following expansion of the functions in (7) : e Φ n ( q ) = X d | n d · µ ( n/d )(1 − q d ) . In particular, we have that e S ,n ( q ) = X d | n X r | d r · e χ PP ( d ) · µ ( d/r )(1 − q r ) . Proof.
The proof is essentially the same as the example given above. Since we can refer tothis illustrative example, we only need to sketch the details to the remainder of the proof. Inparticular, we notice that since we have the known identity for the cyclotomic polynomialsgiven by Φ n ( x ) = Y d | n (1 − q d ) µ ( n/d ) we can take logarithmic derivatives to obtain that1 x · ddq h log (cid:0) − q d (cid:1) ± i(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q → /q = ∓ dq d (cid:16) − q d (cid:17) = ± d − q d , which applied inductively leads us to our result.8 roposition 2.3 (Connections to Ramanujan sums) . Let the following notation denote a short-hand for the divisor sum terms in Theorem 1.5: τ α ( x ) := b S ( α )0 ( x ) = [ q x ] x X n =1 e S ,n ( q ) n α − . We have the following two characterizations of the functions τ α ( x ) expanded in terms of Ra-manujan’s sum, c q ( n ) , where µ ( n ) denotes the M¨obius function and ϕ ( n ) is Euler’s totientfunction: τ α +1 ( x ) = x X d =1 d = p k , p k H ( − α ) ⌊ xd ⌋ · d α · c d ( x )= x X d =1 d = p k , p k H ( − α ) ⌊ xd ⌋ · d α · µ (cid:18) d ( d, x ) (cid:19) ϕ ( d ) ϕ (cid:16) d ( d,x ) (cid:17) . Proof.
First, we observe that the contribution of the first (zero-indexed) sums in Theorem 1.5correspond to the coefficients τ α +1 ( x ) = [ q x ] x X k =1 X d | kd = p s , p s X r | d r · µ ( d/r )(1 − q r ) k α = x X k =1 X r | x X d | kd = p s , p s r · µ ( d/r ) · [ r | d ] δ · k α = x X k =1 X d | kd = p s , p s X r | ( d,x ) r · µ ( d/r ) · k α . Then since we can easily prove the identity that x X k =1 X d | k f ( d ) g ( k/d ) = x X d =1 f ( d ) ⌊ xd ⌋ X k =1 g ( k ) , for any prescribed arithmetic functions f and g , we can also expand the right-hand-side of theprevious equation as τ α +1 ( x ) = x X d =1 d = p k , p k X r | ( d,x ) rµ ( d/r ) H ( − α ) ⌊ xd ⌋ · d α . (11)9hus the identities stated in the proposition follow by expanding out Ramanujan’s sum in theform of [13, § § A.7] [5, cf. § c q ( n ) = X d | ( q,n ) d · µ ( q/d ) . Remark 2.4.
Ramanujan’s sum also satisfies the convenient bound that | c q ( n ) | ≤ ( n, q ) forall n, q ≥
1, which can be used to obtain asymptotic estimates in the form of upper boundsfor these sums when q is not prime or a prime power. Additionally, it is related to periodicexponential sums (modulo k ) of the more general forms s k ( n ) = X d | ( n,k ) f ( d ) g (cid:18) kd (cid:19) , where s k ( n ) has the finite Fourier series expansion s k ( n ) = k X m =1 a k ( m ) e πın/k , with coefficients given by the divisor sums [13, § a k ( m ) = X d | ( m,k ) g ( d ) f (cid:18) kd (cid:19) dk . It turns out that the terms in the formulas for σ α ( x ) represented by these sums, τ α ( x ) providedetailed insight into the error estimates for the summatory functions over the generalized sum-of-divisors functions. We computationally investigate the properties of the new expansions wecan obtain for these sums using the new formulas from the theorem as applications in Section3. Proof of Theorem 1.5.
We begin with a well-known divisor product formula involving the cy-clotomic polynomials when n ≥ q is fixed: q n − Y d | n Φ d ( q ) . Then by logarithmic differentiation we can see that q n − q n = − n (1 − q ) + 1 n X d | nd> e Φ d ( q ) (12)10 − n (1 − q ) + 1 n (cid:16) e S ,n ( q ) + e S ,n ( q ) + e S ,n ( q ) (cid:17) . The last equation is obtained from the first expansion in (12) above by identifying the next twosums as Π n ( q ) = X d | nd> e Φ n (1 /q ) = n − X j =0 ( n − − j ) q j (1 − q )1 − q n . Here we are implicitly using the known expansions of the cyclotomic polynomials which con-dense the order n of the polynomials by exponentiation of the indeterminate q when n containsa factor of a prime power given by (6) in the introduction. Finally, we complete the proof bysumming the right-hand-side of (12) over n ≤ x times the weight n α to obtain the x th partialsum of the Lambert series generating function for σ α ( x ) [5, § § q n is ( x + 1)-order accurate to the terms in theinfinite series. Proof of Proposition 1.4.
The identity in (i) follows from Lemma 2.2. Since Φ p ( q ) = Φ p ( − q )for any prime p , we are essentially in the same case with the two component sums in (ii) and(iii). We outline the proof of our expansion for the first sum, e S ,n ( q ), and note the smallchanges necessary along the way to adapt the proof to the second sum case. By the propertiesof the cyclotomic polynomials expanded in (6), we may factor the denominators of Π p εp ( n ) ( q )into smaller irreducible factors of the same polynomial, Φ p ( q ), with inputs varying as specialprime-power powers of q . More precisely, we may expand Q ( n ) p,k ( q ) := p − P j =0 ( p − − j ) q p k − jp − P i =0 q p k − i , and e S ,n ( q ) = X p ≤ n ν p ( n ) X k =1 Q ( n ) p,k ( q ) · p k − . In performing the sum P n ≤ x Q ( n ) p,k ( q ) p k − n α − , these terms of the Q ( n ) p,k ( q ) occur again, or havea repeat coefficient, every p k terms, so we form the coefficient sums for these terms as j xpk k X i = i (cid:0) ip k (cid:1) α − · p k − = p kα − · H (1 − α ) j xpk k . We can also compute the inner sums in the previous equations exactly for any fixed t as p − X j =0 ( p − − j ) t j = ( p − − pt − t p (1 − t ) , t + t + · · · + t p − = (1 − t p ) / (1 − t ). We now assemble the full sum over n ≤ x we are after in this proof as X n ≤ x e S ,n ( q ) · n α − = X p ≤ x ε p ( x ) X k =1 p kα − H (1 − α ) j xpk k ( p − − pq p k − + q p k (1 − q p k − )(1 − q p k ) . The corresponding result for the second sums is obtained similarly with the exception of signchanges on the coefficients of the powers of q in the last expansion.We compute the series coefficients of one of the three cases in the previous equation to showour method of obtaining the full formula. In particular, the right-most term in these expansionsleads to the double sum C ,x,p := [ q x ] q p k (1 ∓ q p k − )(1 ∓ q p k )= [ q x ] X n,j ≥ ( ± n + j q p k − ( n + p + jp ) . Thus we must have that p k − | x in order to have a non-zero coefficient and for n := x/p k − − jp − p with 0 ≤ j ≤ x/p k − C ,x,p := ( ± ⌊ x/p k − ⌋ × ⌊ x/p k − ⌋ X j =0 ± ⌊ x/p k − ⌋ (cid:22) xp k − (cid:23) + 1 = ( ± ⌊ x/p k − ⌋ (cid:22) xp k (cid:23) . With minimal simplifications we have arrived at our claimed result in the proposition. Theother two similar computations follow similarly.
Proof of Theorem 1.6.
We first note that since for non-negative α ≥
0, we have σ − α ( x ) = X d | x d − α = X d | x (cid:16) xd (cid:17) − α = σ α ( x ) x α , we can see that the formula in (9) follows immediately from Theorem 1.5. It remains to provethe subformulas in (i)–(iii) of the theorem. The first formula for S ( − α )0 ( x ) corresponds to theformulas we derived in Proposition 2.3 of the previous subsection for these cases of negative-order α . The second two formulas follow from Proposition 1.4 by expanding the cases of thefloor function inputs according to the inner index k in the ranges k ∈ [1 , ν p ( x )], i.e., where x/p k ∈ Z + , and then the remainder index case of k := ν p ( x ) + 1. We can use the new exact formula proved by Theorem 1.5 to asymptotically estimate partialsums, or average orders of the respective arithmetic functions, of the next form for integers12 ≥
1: Σ ( α,β ) ( x ) := X n ≤ x σ α ( n ) n β . (13)We similarly define Σ α ( x ) := Σ ( α, ( x ) in this notation. In the special cases where α := 0 ,
1, werestate a few more famous formulas providing well-known classically (and newer) establishedasymptotic bounds for sums of this form as follows where γ ≈ . d ( n ) ≡ σ ( n ) denotes the (Dirichlet) divisor function , and σ ( n ) ≡ σ ( n ) the (ordinary) sum-of-divisors function [8, 1, 7] [13, cf. § ( x ) := X n ≤ x d ( n ) = x log x + (2 γ − x + O (cid:16) x (cid:17) (14)Σ (0 , ( x ) := X n ≤ x d ( n ) n = 12 (log x ) + 2 γ log x + O (cid:0) x − / (cid:1) Σ ( x ) := X n ≤ x σ ( n ) = π x + O ( x log / x ) . For the most part, we suggest tackling potential improvements to these possible asymptoticformulas through our new results given in the theorem and in the symmetric identity (9) asa topic for exploration and subsequent numerical verification using the computational tools atour disposal.
Example 3.1 (Average order of the divisor function) . For comparison with the leading termsin the first of the previous expansions, we can prove the next formula using summation by partsfor integers r ≥ n X j =1 H ( r ) j = ( n + 1) H ( r ) n − H ( r − n Then using inexact approximations for the summation terms in the theorem, we are able toevaluate the leading non-error term in the following sum for large integers t ≥ H (1) n ∼ log n + γ : Σ (0 , t = − t + ( t + 1) H (1) t + O ( t · log ( t )) ∼ ( t + 1) log t + ( γ − t + γ + O ( t · log ( t )) . It is similarly not difficult to obtain a related estimate for the second famous divisor sum, Σ (0 , t ,using the symmetric identity in (9) of the introduction.There is an obvious finite sum identity which generates partial sums of the generalizedsum-of-divisors functions in the following forms [1, cf. §
7] [2]:Σ α ( x ) := X n ≤ x σ α ( n ) = X d ≤ x j xd k · d α (15)13 ⌊ log x log 2 ⌋ X m =0 ⌊ x m ⌋ − ⌊ x m +1 ⌋ X d =1 (cid:22) xd + ⌊ x − ( m +1) ⌋ (cid:23) (cid:16) d + j x m +1 k(cid:17) k = ⌊ log x log p ⌋ X m =0 ⌊ xpm ⌋ − j xpm +1 k X d =1 (cid:22) xd + ⌊ xp − ( m +1) ⌋ (cid:23) (cid:18) d + (cid:22) xp m +1 (cid:23)(cid:19) k , p ∈ Z , p ≥ . Since the sums P md =1 ( d + a ) k are readily expanded by the Bernoulli polynomials, we mayapproach summing the last finite sum identity by parts [13, § α := 0 are considered in the contextof the Dirichlet divisor problem in [2] as are the evaluations of several sums involving the floorfunction such as we have in the statement of Theorem 1.5. Similarly, we can arrive at additionalexact identities for the average order sum variants defined above:Σ ( α,β ) x = x X d =1 d α − β · H ( β ) ⌊ xd ⌋ . We can extend the known classical result for the sums Σ α ( x ) given byΣ α ( x ) := X n ≤ x σ α ( x ) = ζ ( α + 1) α + 1 x α +1 + O (cid:0) x max(1 ,α ) (cid:1) , α > , α = 1 , to the cases of these modified summatory functions using the new formulas proved in Theorem1.6. The next result provides the exact details of the limiting asymptotic relations for the sumsΣ ( α,β ) ( x ). Theorem 3.2 (Asymptotics for Summatory Functions) . For integers α > and ≤ β ≤ α ,we have that the summatory functions Σ ( α,β ) ( x ) defined in (13) above satisfy the followingasymptotic properties: Σ ( α,β ) ( x ) = ζ ( α + 1) x α +1 − β ( α + 1 − β ) (1 − C ( α ) + C , ( α ) + C ( α ) + C ( β ) + C , ( α ))+ α − β X j =1 (cid:18) α + 1 − βj (cid:19) B j x α +1 − β − j α + 1 − β (1 + C ,j ( α ) + C ,j ( α )) + α − β X j =0 C ,j ( α, β ) ζ ( α + 1) x j + α − β X j =0 (cid:18) α − βj (cid:19) C ( α, β )( − α − β − j E j α +2 − β + O (cid:18) x log x (cid:19) , where the absolute constants (depending only on α and m ) are defined by C ( α ) := X p ≥ p prime ( p − p ( p − p α +1 − (cid:20) ( p − p ( p α − − p α (cid:21) ,m ( α ) := X p ≥ p prime ( p − p α +2 − m ( p α − C ( α ) := X p ≥ p prime ( p − α +1 p ( p − p α +1 − (cid:20) ( p − p ( p α − − p α (cid:21) C ,m ( α, β ) := X p ≥ p prime α − β X k =0 (cid:18) α − βk (cid:19)(cid:18) α − β − km (cid:19) ( − k + m E k · ( p − α +2 − β − m p β +1+ m ( p α − C ( α, β ) := X p ≥ p prime ( p − p β +1 ( p α − C ( β ) := X p ≥ p prime ( p − p ( p − p β +1 − C ,m ( α ) := − X p ≥ p prime p α +1 − m . In the previous equations, B n is a Bernoulli numbers and E n denotes the Euler numbers. Due to the length of estimating some of the intricate nested sums from the formulas inTheorem 1.6, we delay a complete proof of Theorem 3.2 to the last appendix section on page26. Compared to the classical result related to these sums cited above, the error terms are ofnotably small order. This shows how accurate the new exact formulas can be in obtaining newasymptotic estimates. The expense of the small error terms is that the involved main termshave expanded in number and complexity.
We turn our attention to an immediate application of our new results which is perhaps one ofthe most famous unresolved problems in number theory: that of determining the form of theperfect numbers. A perfect number p is a positive integer such that σ ( p ) = 2 p . The first fewperfect numbers are given by the sequence { , , , , , . . . } . It currently is notknown whether there are infinitely-many perfect numbers, or whether there exist odd perfectnumbers. References to work on the distribution of the perfect number counting function, V ( x ) := { n perfect : n ≤ x } are found in [14, § σ ( n ), we briefly attempt to formulate conditions foran integer to be perfect within the scope of this article.It is well known that given a Mersenne prime of the form q = 2 p − p , thenwe have corresponding perfect number of the form P = 2 p − (2 p −
1) [14, § P has the form P = 2 p − (2 p −
1) for some (prime) integer p ≥
2, and15onsider the expansion of the sum-of-divisors function on this input to our new exact formulas.Suppose that R := 2 p − r γ r γ · · · r γ k k is the prime factorization of this factor R of P wheregcd(2 , r i ) = 1 for all 1 ≤ i ≤ k and that R s := R/s ν s ( R ) . Then by the formulas derived inProposition 1.4 we have by Theorem 1.5 that σ ( P ) = ( p + 1)2 P + PR (cid:22) R (cid:23) (cid:18) (cid:22) R (cid:23) − (cid:22) R − (cid:23) − (cid:19) + τ ( P )+ X ≤ s ≤ Ps prime
32 ( s − s P · ν s ( R )+ X ≤ s ≤ Ps prime s ν s ( R ) (cid:18)(cid:22) p − R s s (cid:23) + (cid:22) p − R s s (cid:23)(cid:19) ×× (cid:18) s (cid:22) p − R s s (cid:23) − s (cid:22) p − R s − s (cid:23) − (cid:19) . If we set σ ( P ) = 2 P , i.e., construct ourselves a perfect number P by assumption to work with,and then finally solve for the linear equation in P from the last equation, we obtain that P isperfect implies that the following condition holds: P = − τ ( P ) + P ≤ s ≤ Ps prime s ν s ( R ) (cid:16)j p − R s s k + j p − R s s k(cid:17) (cid:16) s j p − R s s k − s j p − R s − s k − (cid:17) ( p − + R (cid:4) R (cid:5) (cid:0) (cid:4) R (cid:5) − (cid:4) R − (cid:5) − (cid:1) + P ≤ s ≤ Ps prime 3( s − s · ν s ( R ) . (16) Corollary 3.3.
We have the following two new noteworthy identities for σ α ( x ) defined in termsof the famous Ramanujan sums and special multiplicative functions: σ α ( x ) = x X d =1 H (1 − α ) ⌊ xd ⌋ · d α − · c d ( x )= x X d =1 H (1 − α ) ⌊ xd ⌋ · d α − · µ (cid:18) d ( d, x ) (cid:19) ϕ ( d ) ϕ (cid:16) d ( d,x ) (cid:17) . Proof.
We can easily derive the following consequences from a modification of the result in (8)from the introduction when x ≥ α ∈ C : σ α ( x ) = [ q x ] x X n =1 X d | n q · ddw h log Φ d ( w ) i(cid:12)(cid:12)(cid:12) w =1 /q × n α −
16 [ q x ] x X n =1 X d | n X r | d rµ ( d/r )1 − q r × n α − = [ q x ] x X d =1 H (1 − α ) ⌊ xd ⌋ · d α − × X r | d rµ ( d/r )1 − q r = x X d =1 H (1 − α ) ⌊ xd ⌋ · d α − X r | ( d,x ) rµ ( d/r ) . Here, we are explicitly employing the well-known identity expanding the cyclotomic polynomi-als, Φ n ( q ), cited in (4) to obtain our second main characterization of the sum-of-divisor sums.Specifically, forming the logarithmic derivatives of the divisor product forms of Φ n ( q ) impliedby this identity allows us to relate each divisor function, σ α ( x ), to the sums over harmonicnumbers of generalized orders and to the Ramanujan sums, c d ( x ), defined in the previous re-mark given in this section. These steps then naturally lead us to the claimed identities statedabove. Remark 3.4 (Conjectures at related divisor sum identities) . Another identity for the gener-alized sum-of-divisors functions which we have obtained based on computational experimentsinvolving the last step in the previous derivation steps, and which we only conjecture here, isgiven by σ α +1 ( x ) = X d | x d α +1 ⌊ xd ⌋ X k =1 µ ( k ) k α H ( − α ) ⌊ x/dk ⌋ . (17)Since the forward differences of the harmonic numbers at ratios of floored arguments withrespect to k are easily shown to satisfy the relation∆ (cid:20) H ( − α ) ⌊ x +1 k ⌋ (cid:21) ( k ) − ∆ (cid:20) H ( − α ) ⌊ xk ⌋ (cid:21) ( k ) = − (cid:18) x + 1 k (cid:19) α [ k | x + 1] δ + (cid:18) x + 1 k + 1 (cid:19) α [ k + 1 | x + 1] δ , we are able to relate the α -weighted cases of the Mertens summatory functions M α ( x ) = X n ≤ x µ ( n ) n α , defined in the earlier remarks connecting Ramanujan sums to our primary new results in thisarticle by applying partial summation. We have some additional new consequences of the main theorems in this article via their relationto logarithmic derivatives of special polynomials. In particular, we can restate Theorem 1.517 PL α ( n )0 = 11 = 12 = 1 + 2 α α + 3 α α + · α (1 + 2 α )5 = 1 + 3 α + 5 α + 6 α + · α (1 + 2 α )6 = 1 + 5 α + 2 α +1 (2 α + 3 α ) + α (3 + 3 α ) + α (11 + 7 · α ) . Table 2: Generalized Planar Partitions.
The first few values of the generalized planarpartition functions for symbolic α ∈ Z + .in terms of logarithmic derivatives of products of the cyclotomic polynomials via (4), (5), andDefinition 1.1 in the following forms for x ≥ α ≥ σ α +1 ( x ) x = [ q x ] x X n =1 log (cid:18) q n − (cid:19) n α ! = [ q x ] log Q ≤ n ≤ x (1 − q n ) n α . (18)The expansions of the sum-of-divisors functions given in (18) above then follow from writing(8) in the form nq − q n = nq − q X d | n e Φ d (cid:0) q − (cid:1) , and then applying the cited identities to the definition of the modified logarithmic derivativevariants of the cyclotomic polynomials defined in the introduction.We enumerate the generalized planar partitions PL α ( n ) for non-negative integers α andall n ≥ α ) generating functions (cf. A000219,A000991, A001452, A002799, A023871–A023878, A144048 and A225196–A225199):PL α ( n ) := [ q n ] Y n ≥ − q n ) n α ! = [ q n ] exp X m ≥ σ α +1 ( m ) q m m ! . The details of the generating-function-based argument based on an extension of the logarithmicderivatives implicit to the expansions in (8) allow us to effectively generalize the first two casesof the convoluted recurrence relations for p ( n ) and PL( n ) in the next forms. n · PL α ( n ) = n X k =1 σ α +1 ( k ) PL α ( n − k ) , n ≥ . α . We immediately can see that when α := 0 ,
1, these partition functions correspond to the partition function p ( n ) and the (ordinary) sequence of planar partitions PL( n ), respectively(A000041, A000219). Thus without motivating this class of recursive identities relating gener-alized planar partitions and the sum-of-divisors functions by physical or geometric interpreta-tions to counting such partition numbers (for example, as are known in the case of the ordinaryplanar partitions PL( n )), we have managed to relate the special multiplicative divisor functionswe study in this article to a distinctly more additive flavor of number theoretic functions . There is an infinite series for the ordinary sums-of-divisors function, σ ( n ), due to Ramanujanin the form of [6, §
9, p. 141] σ ( n ) = nπ " − n + 2 cos (cid:0) nπ (cid:1) + 2 cos (cid:0) nπ (cid:1) + 2 cos (cid:0) nπ (cid:1) + · · · (19)In similar form, we have a corresponding infinite sum providing an exact formula for the divisorfunction expanded in terms of the functions c q ( n ) defined in [6, §
9] of the form (cf. Remark 2.3) d ( n ) = − X k ≥ c k ( n ) k log( k ) = − c ( n )2 log 2 − c ( n )3 log 3 − c ( n )4 log 4 − · · · . (20)Recurrence relations between the generalized sum-of-divisors functions are proved in the ref-erences [ ? , 15]. There are also a number of known convolution sum identities involving thesum-of-divisors functions which are derived from their relations to Lambert series and Eisen-stein series. Exact formulas for the divisor function, d ( n ), of a much different characteristic nature areexpanded in the results of [1]. First, we compare our finite sum results with the infinite sums We remark that such relations between multiplicative number theory and the more additive theories ofpartitions and special functions are decidedly rare in the supporting literature. See [11, 15] for further examplesand references to other works relating these two branches of number theory (i.e., the additive and multiplicativestructures of the respective functions involved).
19n the (weighted) Voronoi formulas for the partial sums over the divisor function expanded as x ν − Γ( ν ) X n ≤ x (cid:16) − nx (cid:17) ν − d ( n ) = x ν − ν ) + x ν (log x + γ − ψ (1 + ν ))Γ( ν + 1) − πx ν X n ≥ d ( n ) F ν (cid:0) π √ nx (cid:1)X n ≤ x d ( n ) = 14 + (log x + 2 γ − x − √ xπ X n ≥ d ( n ) √ n (cid:16) K (cid:0) π √ nx (cid:1) + π Y (cid:0) π √ nx (cid:1)(cid:17) , where F ν ( z ) is some linear combination of the Bessel functions , K ν ( z ) and Y ν ( z ), and ψ ( z ) isthe digamma function . A third identity for the partial sums, or average order, of the divisorfunction is expanded directly in terms of the Riemann zeta function, ζ ( s ), and its non-trivialzeros ρ in the next equation. X n ≤ x d ( n ) = − π
12 + (log x + 2 γ − x + π X ρ : ζ ( ρ )=0 ρ = − , − , − ,... ζ ( ρ/ ρζ ′ ( ρ ) x ρ/ − π X n ≥ ζ ( − (2 n + 1)) x − (2 n +1) (2 n + 1) ζ ′ ( − (2 n + 1))While our new exact sum formulas in Theorem 1.5 are deeply tied to the prime numbers2 ≤ p ≤ x for any x , we once again observe that the last three infinite sum expansions of thepartial sums over the divisor function are of a much more distinctive character than our newexact finite sum formulas proved by the theorem. p ( n )Rademacher’s famous exact formula for the partition function p ( n ) when n ≥ p ( n ) = 1 π √ X k ≥ A k ( n ) √ k ddn sinh (cid:16) πk q (cid:0) n − (cid:1)(cid:17)q n − , where A k ( n ) := X ≤ h 20s a Kloosterman-like sum and s ( h, k ) and ω ( h, k ) are the (exponential) Dedekind sums definedby s ( h, k ) := k − X r =1 rk (cid:18) hrk − (cid:22) hrk (cid:23) − (cid:19) (21) ω ( h, k ) := exp ( πı · s ( h, k )) . In comparison to other somewhat related formulas for special functions, we note that unlikeRademacher’s series for the partition function, p ( n ), expressed in terms of finite Kloosterman-like sums, our expansions require only a sum over finitely-many primes p ≤ x to evaluate thespecial function σ α ( x ) at x . For comparison with the previous section, we also note that thepartition function p ( n ) is related to the sums-of-divisor function σ ( n ) through the convolutionidentity np ( n ) = n X k =1 σ ( n − k ) p ( k ) . There are multiple recurrence relations that can be given for p ( n ) including the following ex-pansions: p ( n ) = p ( n − 1) + p ( n − − p ( n − − p ( n − 7) + p ( n − 12) + p ( n − − p ( n − − · · · = ⌊ ( √ n +1 − / ⌋ X k = ⌊ − ( √ n +1+1) / ⌋ k =0 ( − k +1 p (cid:18) n − k (3 k + 1)2 (cid:19) . (Overpartitions) . An overpartition of an integer n is defined to be a representa-tion of n as a sum of positive integers with non-increasing summands such that the last instanceof a given summand in the overpartition may or may not have an overline bar associated withit. The total number of overpartitions of n , ¯ p ( n ), is generated by X n ≥ ¯ p ( n ) q n = Y m ≥ q m − q m = 1 + 2 q + 4 q + 8 q + 14 q + 24 q + 40 q + · · · . A convergent Rademacher-type infinite series providing an exact formula for the partition func-tion ¯ p ( n ) is given by¯ p ( n ) = X k ≥ ∤ k X ≤ h Gauss circle problem asks for the count of the number of lattice points, denoted N ( R ),inside a circle of radius R centered at the origin. It turns out that we have exact formulas for N ( R ) expanded as both a sum over the sum-of-squares function and in the form of N ( R ) = 1 + 4 ⌊ R ⌋ + 4 ⌊ R ⌋ X i =1 j √ R − i k . The Gauss circle problem is closely-related to the Dirichlet divisor problem concerning sumsover the divisor function, d ( n ) [1, § r ( n ), satisfies the next Sierpi´nski formula which is similar and of Voronoi type to the first twoidentities for the divisor function given in the previous subsection. However, we note that theVoronoi-type formulas in the previous section are expanded in terms of certain Bessel functions,where our identity here requires the alternate special function J ν ( z ) = X n ≥ ( − n n ! · Γ( ν + n + 1) (cid:16) z (cid:17) ν +2 n , < | z | < ∞ and any selection of theparameter ν ∈ C . X n ≤ x r ( n ) = πx + √ x X n ≥ r ( n ) √ n J (cid:0) π √ nx (cid:1) As in the cases of the weighted Voronoi formulas involving the divisor function stated in thelast section, we notice that this known exact formula for the special function r ( n ) has a muchdifferent nature to its expansion than our new exact finite sum formulas for σ α ( n ). We do,however, have an analog to the infinite series for the classical divisor functions in (19) and (20)which is expanded in terms of Ramanujan’s sum functions c q ( n ) = X ℓm = qm | n µ ( ℓ ) m, as follows [6, § r ( n ) = π (cid:18) c ( n ) − c ( n )3 + c ( n )5 − · · · (cid:19) . In this article, we again began by considering the building blocks of the Lambert series generat-ing functions for the sum-of-divisors functions in (1). The new exact formulas for these specialarithmetic functions are obtained in this case by our observation of the expansions of the seriesterms, q n / (1 − q n ), by cyclotomic polynomials and their logarithmic derivatives. We note thatin general it is hard to evaluate the series coefficients of e Φ n ( q ) without forming the divisor sumemployed in the proof of part (i) of Proposition 1.4. We employed the established, or at leasteasy to derive, key formulas for the logarithmic derivatives of the cyclotomic polynomials alongwith known formulas for reducing cyclotomic polynomials of the form Φ p r m ( q ) when p ∤ m toestablish the Lambert series term expansions in (8). The expansions of our new exact formulasfor the generalized sum-of-divisors functions are deeply related to the prime numbers and thedistribution of the primes 2 ≤ p ≤ x for any x ≥ r -order harmonic numbers, H ( r ) n , when r > r ≤ σ α ( n ), σ ( n ), and d ( n ) to formulateasymptotic formulas for the partial sums of the divisor function, or its average order , whichmatch more famous known asymptotic formulas for these sums. The primary application ofthe new exact formulas for σ ( n ) provided us with a new necessary and sufficient conditioncharacterizing the perfectness of a positive integer n in Section 3.2. Finally, in Section 4 wecompared the new results proved within this article to other known exact formulas for thesum-of-divisors functions, partition functions, and other special arithmetic sequences.23 .2 Extensions of these techniques to other special Lambert seriesexpansions Lambert series are special in form because they encode as their series coefficients key numbertheoretic functions which are otherwise inaccessible by standard combinatorial generating func-tion techniques. More generally, for any well behaved arithmetic function f , we can form itsLambert series generating function as L f ( q ) := X n ≥ f ( n ) q n − q n = X m ≥ X d | m f ( d ) q m . Then it is easy to extend our proofs from Section 2 to show that X d | x f ( d ) = x X k =1 f ( k ) k + X p ≤ x ν p ( x )+1 X k =1 j xpk k X r =1 f ( p k · r ) r + X p ≤ x ν p ( x )+1 X k =1 j x pk k X r =1 ( − j ppk k f ((2 p ) k · r ) r + x X d =1 χ PP ( d ) d · n c d ( x ) × ⌊ xd ⌋ X n =1 f ( d · n ) . A Mathematica notebook containing definitions that can be used to computationally verify theformulas proved in this manuscript is available online at the following link:https://drive.google.com/open?id=13GYUxEn5RXes6xgEPtL-BvgJA76TF4Jh.The functional definitions provided in this notebook are also of use to readers for experimentalmathematics based on the contents of this article. In particular, this data is available to thosereaders who wish to extend the results presented here or for conjecturing new properties relatedto the generalized sum-of-divisors functions based on empirical results obtained with the codefreely available in this supplementary reference. References [1] N. A. Carella, An explicit formula for the divisor function, 2014, https://arxiv.org/abs/1405.4784 .[2] B. Cloitre, On the circle and divisor problems, 2012, https://oeis.org/A013936/a013936.pdf .243] H. Davenport, Multiplicative Number Theory , Springer, 2000.[4] P. Erd¨os, On the integers having exactly k prime factors, Annals of Math. , pp. 53–66(1948).[5] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers , Oxford UniversityPress, 2008.[6] G. H. Hardy, Ramanujan: Twelve lectures on subjects suggested by his life and work , AMSChelsea Publishing, 1999.[7] H. Hasse, A. Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie (Math-ematische Forschungsberichte, Band XV). 231 S. Berlin 1963. Deutscher Verlag der Wis-senschaften. Preis brosch. DM 36,-, ZAMM , (12), p. 607 (1964).[8] Huxley, M. N. Exponential Sums and Lattice Points III. Proc. London Math. Soc. A classical introduction to modern number theory , Springer, 1990.[10] D. Lustig, The algebraic independence of the sum-of-divisors functions, Journal of NumberTheory , , pp. 2628–2633 (2010).[11] M. Merca and M. D. Schmidt, A partition identity related to Stanley’s theorem , Amer.Math. Monthly , to appear (2018).[12] M. B. Nathanson, Additive number theory: the classical bases , Springer, 1996.[13] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Math-ematical Functions , Cambridge University Press, 2010.[14] P. Ribenboim, The new book of prime number records , Springer, 1996.[15] M. D. Schmidt, New recurrence relations and matrix equations for arithmetic functionsgenerated by Lambert series, Acta. Arith. (2017).[16] A. V. Sills, A Rademacher type formula for partitions and over partitions, InternationalJournal of Mathematics and Mathematical Sciences , , Article ID 630458, 21 pages(2010).[17] M. B. Villarino, Mertens’ proof of Mertens’ Theorem, arXiv:math/0504289 (2005).25 Appendix: The complete proof of Theorem 3.2 Lemma A.1. For any arithmetic functions f, g, h : N → C , and natural numbers x ≥ , wehave the following pair of divisor sum simplification identities: x X n =1 f ( n ) X d | n g ( d ) h (cid:16) nd (cid:17) = X d ≤ x g ( d ) ⌊ xd ⌋ X n =1 h ( n ) f ( dn ) x X d =1 f ( d ) X r | ( d,x ) g ( r ) h (cid:18) dr (cid:19) = X r | x g ( r ) X ≤ d ≤ x/r h ( d ) f ( rd ) . Since the proofs of these identities are not difficult, and indeed are fairly standard exercises,we will not prove the two formulas in Lemma A.1. Proof of Theorem 3.2. We break down the proof into four separate tasks of estimating thecomponent term sums from Theorem 1.6. The sum of the dominant and error terms resultingfrom each of these cases then constitutes the proof of the result at hand. In particular, we willdefine and asymptotically analyze the formulas for the following key sums:Σ ′ α,β ( x ) := X n ≤ x n α − β · S ( − α )1 ( n )Σ ′′ α,β ( x ) := X n ≤ x n α − β · S ( − α )2 ( n )Σ ′′′ α,β ( x ) := X n ≤ x τ − α ( n ) n α − β . The combined approximation we are after is given in terms of this notation byΣ ( α,β ) ( x ) = X n ≤ x n α − β · H ( α +1) n + Σ ′ α,β ( x ) + Σ ′′ α,β ( x ) + Σ ′′′ α,β ( x ) . (I) Leading Term Estimates: For α > x ≥ 1, we have that the ( α + 1)-order harmonicnumbers satisfy H ( α +1) x = ζ ( α + 1) + O (cid:0) x − ( α +1) (cid:1) , where ζ ( s ) is the Riemann zeta function . Notice that the leading terms in the formula for σ α ( n )from Theorem 1.6 then correspond to X n ≤ x n α − β · H ( α +1) n = X n ≤ x n α − β (cid:18) ζ ( α + 1) + O (cid:18) n α +1 (cid:19)(cid:19) = X n ≤ x (cid:18) ζ ( α + 1) n α − β + O (cid:18) n β (cid:19)(cid:19) ζ ( α + 1) " x α +1 − β α + 1 − β + x α − β α − β X j =2 (cid:18) α + 1 − βj (cid:19) B j x α +1 − β − j α + 1 − β + ( O (log x ) , if β = 0; O (1) , if β > , (22)where B n are the Bernoulli numbers . This establishes the leading dominant term in the asymp-totic expansion which matches with the known classical result cited above. (II) Second Terms Estimate: Next, we can asymptotically expand the first component sum inTheorem 1.6 as S ( − α )1 ( n ) = X p ≤ n ν p ( n ) X k =1 ( p − p (cid:18) ζ ( α + 1) p αk + O (cid:18) p k n α +1 (cid:19)(cid:19) − ζ ( α + 1) p αν p ( n )+ α +1 + O (cid:18) p ν p ( n ) n α +1 (cid:19) = X p ≤ n (cid:20) ζ ( α + 1)( p − p αν p ( n )+1 ( p α − − ζ ( α + 1)( p − p α +1 ( p α − (cid:21) [ p | n ] δ − X p ≤ n ζ ( α + 1) p αν p ( n )+ α +1 + O (cid:18) n α − · log n (cid:19) , where the last error term results by observing that ν p ( n ) ≤ log p ( n ). We can use Abel summa-tion together with a divergent asymptotic expansion for the exponential integral function todetermine that for real r > 0, the prime sums X p ≤ x p r +1 = C r + r + 1 r · x r log x + O (cid:18) x r log x (cid:19)X p ≤ x p r +1 log p = D r + ( r + 2)2 x r log x − ( r + 2) log x + 12 x r log x + O (cid:18) x r log x (cid:19) , where the terms C r , D r are absolute constants depending only on r . Now we will perform thesum over n and multiply by a factor of n α as in Theorem 1.6, and then interchange the indicesof summation to obtain thatΣ ′ α,β ( x ) := X n ≤ x n α − β · S ( − α )1 ( n )= X p ≤ x " x X n = p n α − β (cid:18) ζ ( α + 1)( p − p αν p ( n )+1 ( p α − − ζ ( α + 1)( p − p α +1 ( p α − (cid:19) [ p | n ] δ − X p ≤ x x X n = p n α − β ζ ( α + 1) p αν p ( n )+ α +1 + O (cid:18) n β (cid:19) . For β > 1, the error term in the last equation corresponds to X p ≤ x x X n = p O (cid:18) n β (cid:19) = X p ≤ x (cid:20) ζ ( β ) + O (cid:18) x β (cid:19)(cid:21) = O (cid:18) x log x + 1 x β − log x (cid:19) = O (cid:18) x log x (cid:19) , 27y applying the known asymptotic estimate for the prime counting function , π ( x ) = x/ log x + O ( x/ log x ). At this point we must evaluate sums of the next form for fixed primes p : T ,p ( x ) := x X n = p n α − β [ p | n ] δ p αν p ( n ) . We can use Abel summation and then form the approximations to the next summatory functionsin the following way to approximate T ,p ( x ) for large x : A ,p ( t ) = X i ≤ t [ p | i ] δ p αν p ( i ) = X i ≤ t/p p αν p ( p · i ) = log p ( t ) X k =1 p αk { i ≤ t/p : ν p ( i ) = k } = ∞ X k =0 p α ( k +1) { i ≤ t/p : p k +1 | i } − ∞ X s = k +2 { i ≤ t/p : p s | i } ! = ∞ X k =0 p α ( k +1) " tp k +2 − ∞ X i = k +2 tp i +1 = ( p − tp ( p − 1) ( p α +1 − . Then we have by Abel’s summation formula that T ,p ( x ) = x α − β · A ,p ( x ) − Z x A ,p ( t ) D t [ t α − β ] dt = ( p − x α +1 − β ( α + 1 − β ) p ( p − 1) ( p α +1 − . Similarly, we can estimate the asymptotic order of the sums T ,p ( x ) := x X n = p n α − β p αν p ( n )+ α +1 = ( p − x α +1 − β ( α + 1 − β )( p − p α +1 ( p α +1 − . Thus it follows that X p ≤ x (cid:20) ζ ( α + 1)( p − p ( p α − T ,p ( x ) − ζ ( α + 1) T ,p ( x ) (cid:21) ∼ C ( α ) ζ ( α + 1) x α +1 − β ( α + 1 − β ) . 28e can also estimate the summands X p ≤ x x X n = p n α − β · ζ ( α + 1)( p − p α +1 ( p α − 1) [ p | n ] δ = X p ≤ x x/p X n =1 ( pn ) α − β · ζ ( α + 1)( p − p α +1 ( p α − α − β X j =0 (cid:18) α + 1 − βj (cid:19) B j x α +1 − β − j ( α + 1 − β ) · ζ ( α + 1)( p − p α +1 − j ( p α − ∼ α − β X j =0 (cid:18) α + 1 − βj (cid:19) C ,j ( α ) B j x α +1 − β − j ( α + 1 − β ) . In summary, we obtain thatΣ ′ α,β ( x ) = C ( α ) ζ ( α + 1) x α +1 − β ( α + 1 − β ) − α − β X j =0 (cid:18) α + 1 − βj (cid:19) C ,j ( α ) B j x α +1 − β − j ( α + 1 − β ) + O (cid:18) x log x (cid:19) . (23) (III) Third Component Terms Estimate: Following from the estimates in the previous equations,we see that the second component sum from Theorem 1.6 is similar to the constructions ofthe sums we estimated last in (II), weighted by an additional term of ( − n × − ( α +1) whenperforming the last (outer) sum on n ≤ x . In particular, we have thatΣ ′′ α,β ( x ) = X n ≤ x n α − β S ( − α )2 ( n )= X p ≤ x " x X n = p ( − n · n α − β α +1 (cid:18) ζ ( α + 1)( p − p αν p ( n )+1 ( p α − − ζ ( α + 1)( p − p α +1 ( p α − (cid:19) [ p | n ] δ − X p ≤ x x X n = p ( − n · n α − β ζ ( α + 1)2 α +1 p αν p ( n )+ α +1 + O (cid:18) n β (cid:19) . Hence, our estimates in this case boil down to summing the following formulas by Abel sum-mation for large x : U ,p ( x ) := x X n = p ( − n · n α − β [ p | n ] δ p αν p ( n ) . To bound the main and error terms in this sum for large enough x , we proceed as before toform the summatory functions B , ( t ) = X i ≤ t ( − i · [2 | i ] δ αν ( i ) = X ≤ i ≤ t/ αν (2 i ) = log ( t ) X k =1 αk { i ≤ t/ ν ( i ) = k } ∞ X k =0 α ( k +1) { i ≤ t/ k +1 | i } − ∞ X s = k +2 { i ≤ t/ s | i } ! = ( p − t ( p − 1) ( p α +1 − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p =2 = 0 , and for p ≥ B ,p ( t ) = X i ≤ t ( − i · [ p | i ] δ p αν p ( i ) = − X i ≤ t/p p αν p ( p · i ) = − log p ( t ) X k =1 p αk { i ≤ t/p : ν p ( i ) = k } = − ∞ X k =0 p α ( k +1) { i ≤ t/p : p k +1 | i } − ∞ X s = k +2 { i ≤ t/p : p s | i } ! = − ∞ X k =0 p α ( k +1) " tp k +2 − ∞ X i = k +2 tp i +1 = − ( p − tp ( p − 1) ( p α +1 − . It follows from Abel’s summation formula that U ,p ( x ) = − ( p − x α +1 − β ( α + 1 − β ) p ( p − 1) ( p α +1 − 1) [ p ≥ δ . Then we can pick out the similar terms from these sums as we did in the previous derivationsfrom step (II) and estimate the resulting formula in the form of U ,p ( x ) = x X n = p ( − n · n α − β α +1 p αν p ( n )+ α +1 = − ( p − x α +1 − β ( α + 1 − β )( p − p α +1 ( p α +1 − 1) [ p ≥ δ . When we sum the corresponding terms in these two functions over all odd primes p ≤ x , weobtain that X ≤ p ≤ x (cid:20) ζ ( α + 1)( p − p ( p α − U ,p ( x ) − ζ ( α + 1) U ,p ( x ) (cid:21) ∼ − C ( α ) ζ ( α + 1) x α +1 − β ( α + 1 − β ) . 30o complete the estimate for the formula in this section (III), it remains to perform the sum-mation estimates [13, cf. § U ,p ( x ) := x X n = p ( − n · n α − β α +1 (cid:18) ζ ( α + 1)( p − p α +1 ( p α − (cid:19) [ p | n ] δ = x/p X n =1 ( − n · ( pn ) α − β α +1 (cid:18) ζ ( α + 1)( p − p α +1 ( p α − (cid:19) = " α − β X k =0 (cid:18) α − βk (cid:19) ( − α − β E k k + α +2 (cid:18) xp − (cid:19) α − β − k + ( − α − β α +2 − β α − β X k =0 (cid:18) α − βk (cid:19) ( − k E k ×× (cid:18) ζ ( α + 1)( p − p β +1 ( p α − (cid:19) Thus in total, for estimate (III) we have thatΣ ′′ α,β ( x ) ∼ − C ( α ) ζ ( α + 1) x α +1 − β ( α + 1 − β ) + α − β X r =0 C ,r ( α, β ) x r + ( − α − β α +2 − β α − β X k =0 (cid:18) α − βk (cid:19) ( − k E k × X p ≥ ζ ( α + 1)( p − p β +1 ( p α − 1) + o (1) (24) (IV) Last Remaining Estimates (Tau Divisor Sum Terms): Finally, we have only one componentin the sums from Theorem 1.6 left to estimate. Namely, we must bound the sumsΣ ′′′ α,β ( x ) := X n ≤ x τ − α ( n ) n α − β = X n ≤ x n X d =1 n α − β d α +1 H ( α +1) ⌊ nd ⌋ c d ( n ) χ PP ( d ) . Now by expanding the previous sums according to the divisor sum identities in Lemma A.1, weobtain that Σ ′′′ α,β ( x ) = X n ≤ x n X d =1 X r | ( d,n ) rµ ( d/r ) H ( α +1) ⌊ nd ⌋ χ PP ( d ) d α +1 n α − β = X n ≤ x n X d =1 X r | ( d,n ) rµ ( d/r ) H ( α +1) ⌊ nd ⌋ n α − β d α +1 − X n ≤ x X p ≤ n log p ( n ) X k =1 C p k ( n ) (cid:18) ζ ( α + 1) + O (cid:18) p ( α +1) k n α +1 (cid:19)(cid:19) n α − β p ( α +1) k = X n ≤ x X r | n X ≤ d ≤ r r α n β µ ( d ) d α +1 H ( α +1) ⌊ rd ⌋ X p ≤ x x/p X n =1 ν p ( n ) − X k =1 p k × (cid:18) ζ ( α + 1) ( p k n ) α − β p ( α +1) k + O (cid:18) n β +1 (cid:19)(cid:19) (25)In the transition from the second to last to the previous equation, we have used a known factabout the Ramanujan sums , c p k ( n ), at prime powers. Namely, c p k ( n ) = 0 whenever p k − ∤ n , andwhere the function is given by c p k ( n ) = − p k − if p k − | n , but p k ∤ n or c p k ( n ) = φ ( p k ) = p k − p k − if p k | n . For the first sum in (25), we can extend the divisor sum over r to all 1 ≤ r ≤ n , takethe absolute values of all M¨obius function terms over d , and observe that the resulting formulawe obtain by applying summation by parts corresponds to the identity that H ( m ) ⌊ r +1 d ⌋ − H ( m ) ⌊ rd ⌋ = (cid:16) rd (cid:17) − m [ d | r ] δ . Then we see that this first sum is bounded by X n ≤ x X r | n X ≤ d ≤ r r α n β µ ( d ) d α +1 H ( α +1) ⌊ rd ⌋ = O X n ≤ x n X r =1 (cid:0) r α +1 + O ( r α ) (cid:1) X d | r | µ ( d ) | d α +1 (cid:18) dr (cid:19) α +1 = O X n ≤ x n β n X r =1 X d | r | µ ( d ) | = O X n ≤ x n β n X r =1 ω ( r ) ! = O X n ≤ x n β X k ≥ k · { ≤ r ≤ n : ω ( r ) = k } ! . Now we can draw upon the result of Erd¨os in [4] to approximately sum the right-hand-side ofthe last equation as follows for integers β > O X n ≤ x o (1)) n − β log n ! = ( O (cid:0) log x (cid:1) , if β = 2; O (cid:16) β − 2) log x +1( β − x β − (cid:17) , if β ≥ V ,p ( x ) := x/p X n =1 ν p ( n ) − X k =1 p k × (cid:18) ζ ( α + 1) ( p k n ) α − β [ p | n ] δ p ( α +1) k + O (cid:18) n β +1 (cid:19)(cid:19) = x/p X n =1 ζ ( α + 1) n α − β p βν p ( n ) − p β α − β X j =0 (cid:18) α + 1 − βj (cid:19) ζ ( α + 1) B j ( α + 1 − β ) (cid:18) xp (cid:19) α +1 − β − j O (cid:18) p − n β − p ( p − n β +1 (cid:19) , where we have again used the upper bound ν p ( n ) ≤ log p ( n ) to bound the error term in theprevious equation. To evaluate the first component sum in the previous equation, we can useAbel summation much like we have in the prior estimates in this proof. Namely, we form thesummatory functions A ,p ( t ) = X i ≤ t [ p | i ] δ p βν p ( i ) = X i ≤ t/p p βν p ( pi ) = X k ≥ p βk { i ≤ t/p : ν p ( i ) = k } = X k ≥ p βk " tp k +1 − X i ≥ k +1 tp i +1 = ( p − tp ( p − p β +1 − . Then the complete first sum above is given by X p ≤ x x/p X n =1 ζ ( α + 1) n α − β p βν p ( n ) = X p ≤ x ζ ( α + 1)( p − x α +1 − β ( α + 1 − β ) p ( p − p β +1 − . When we assemble the second two sums over all primes p ≤ x , and add it to the first, we obtain X p ≤ x V ,p ( x ) ∼ C ( β ) ζ ( α + 1) x α +1 − β ( α + 1 − β ) + α − β X j =0 (cid:18) α + 1 − βj (cid:19) C ,j ( α ) ζ ( α + 1) B j x α +1 − β − j ( α + 1 − β )+ O (cid:18) x log x (cid:19) ..