ALGEBRAIC AND ANALYTIC PROPERTIES OF ARITHMETIC FUNCTIONS

Transcription

1 ALGEBRAIC AND ANALYTIC PROPERTIES OF ARITHMETIC FUNCTIONS MARK SCHACHNER Abstract. When consiere as an algebraic space, the set of arithmetic functions equippe with the operations of pointwise aition an Dirichlet convolution exhibits many nontrivial properties. In this paper, we elineate an prove several of these properties, an consier ways to quantify the behavior of certain functions. Contents 1. Introuction 1 2. The Space of Arithmetic Functions 2 3. A is an Integral Domain 3 4. Möbius Inversion an Relate Results 5 5. Polynomials in A 8 6. The Growth of Arithmetic Functions 13 References Introuction Consier a function ϱ : N N given by the recursive efinition ϱ(n = 1, if n = 1; ϱ(, if n > 1. n,<n This function, efine on the natural numbers, is an example of an arithmetic function. It exhibits erratic behavior in many ways: for example, for values of n with few prime factors it remains small, but it quickly grows arbitrarily large if n has many prime factors. However, using the techniques of analytic number theory, the behavior of this an relate functions can be quantitatively stuie. In this paper, we examine the properties of various important arithmetic functions an consier the space of such functions, equippe with the operations of aition, multiplication, an Dirichlet convolution. We also euce several facts about the existence an properties of roots of polynomials with arithmetic functions as coefficients. 1

2 2 MARK SCHACHNER 2. The Space of Arithmetic Functions In this section we introuce the concept of an arithmetic function, an efine a space of arithmetic functions with several operations. We then isolate a few specific functions which will be of use later. Definition 2.1. An arithmetic function is a complex-value function which is efine on the natural numbers, i.e., a function of the form f : N C. These functions are also known as number-theoretic functions, which is inicative of their interpretation as encoings of specific properties of the natural numbers. Definition 2.2. In this paper, we enote the set of arithmetic functions as A. We also equip A with the operation + : A A A given by, for f, g A, (f + g(n = f(n + g(n for all n N. Throughout this paper, we will enote arbitrary arithmetic functions by f, g, h, as well as A k for k N. Later, we will efine polynomial functions from A to itself, which will commonly be enote F, G, H. We also efine one other binary operation on A, which is important enough to merit its own efinition. Definition 2.3. Let f, g A. The Dirichlet convolution (or simply convolution of f, g, enote f g, is the function given by, for each n N, (f g(n = ( n f(g, n or, equivalently, (f g(n = f(ag(b. These two efinitions are equivalent via the substitution a =, b = n a = n. The following elements of A are important in the stuy of arithmetic functions: Definition 2.4. (i The ientity function ε A is given by, for each n N, { 1, if n = 1, ε(n = 0, otherwise. (ii The constant functions 0, 1 A are given by 1(n = 1 an 0(n = 0 for each n N. (iii The ivisor function σ i A, efine for each i C, is given by σ i (n = n i. for each n N. In this paper we enote σ 0 by τ an σ 1 by σ. Note that, while convolution will play a role similar to multiplication in this paper, the function 1 is not the ientity with respect to convolution.

3 ALGEBRAIC AND ANALYTIC PROPERTIES OF ARITHMETIC FUNCTIONS 3 3. A is an Integral Domain First, we evelop arithmetic on A an prove several statements regaring its algebraic structure. Lemma 3.1. The operation is commutative, associative, an istributive over aition. Proof. First, we show that convolution is commutative. To observe this, we use the secon efinition from Definition 2.3: (f g(n = f(ag(b = g(af(b = (g f(n. To show that convolution is associative, we again use the secon efinition: (f (g h(n = f(a (g h(b = ( f(a g(ch( c=b = ac=n = e=n ac=e f(ag(ch( f(ag(ch( = ((f g h(n. Lastly, we show that convolution istributes over aition: (f (g + h(n = f(a (g + h(b = f(ag(b + f(ah(b = f(ag(b + f(ah(b = (f g(n + (f h(n. We next prove an important result about the role of ε with respect to Dirichlet convolution: Lemma 3.2. The function ε A is the ientity element with respect to Dirichlet convolution, i.e., ε f = f ε = f for all f A. Proof. Fix some f A, an observe that, for all n N, (f ε(n = ( n f( ε n = f(nε(1 + ( n f( ε = f(n, n,<n

4 4 MARK SCHACHNER since ε ( n = 0 when < n. That ε f = f ε = f follows from Lemma 3.1. We can now make the following important statement about A: Theorem 3.3. The set A, along with the operations + an, forms a commutative ring. Proof. Pointwise aition is trivially commutative an associative, an has the ientity element 0 A, where 0(n = 0 for all n N. Furthermore, for any f A we efine f as the function such that ( f(n = (f(n for each n N; this represents the aitive inverse of the element f. From Lemma 3.1 an Lemma 3.2, we have that convolution is commutative, associative, an has an ientity element ε A. Lastly, from Lemma 3.1, we have that convolution istributes over aition. There, is, however, aitional algebraic structure on A with respect to convolution as emonstrate by the following lemma: Lemma 3.4. Suppose f A.Then there exists f 1 A such that f f 1 = ε if an only if f(1 0. Proof. To show the forwar irection, let f A an suppose that there exists f 1 A such that f f 1 = ε. Observe that f(1f 1 (1 = (f f 1 (1 = ε(1 = 1, so f 1 (1 = 1 f(1. Hence f(1 0. To show the backwar irection, fix f A such that f(1 0. We will show that g(n can be uniquely efine such that (f g(n = ε(n, proceeing by inuction on n : Base Case. Let n = 1 an g(1 = 1 f(1. Since f(1 0, we have g(1 is wellefine, an this efinition satisfies (f g(1 = f(1g(1 = 1 = ε(1. 1 Furthermore, any value of g(1 which satisfies the above must equal f(1, so this g(1 is efine uniquely. Inuctive Step. Fix n N an suppose that for each k < n we have that g(k is well-efine an unique. We now efine g(n as follows: g(n = 1 f(1 n,>1 f(g ( n. Observe that g(n is well-efine for each n N, since f(1 0 an n N, n < n, so g ( n is uniquely efine by the inuctive hypothesis for each n, > 1. Now, we show f g = ε. We have two cases: n = 1 an n > 1. If n = 1, then as before (f g(n = f(1g(1 = ε(n. If n > 1, then (f g(n = ( n f(g n = f(1g(n + ( n f(g n,>1 = f(1g(n f(1g(n = 0 = ε(n.

5 ALGEBRAIC AND ANALYTIC PROPERTIES OF ARITHMETIC FUNCTIONS 5 Thus g is the unique arithmetic function such that f g = ε, so for each f A where f(1 0, we can let f 1 = g so that f f 1 = ε. Again, that f 1 f = ε follows from the commutativity of Dirichlet convolution. Motivate by Lemma 3.4, we make the following efinition: Definition 3.5. A function f A is calle invertible if f(1 0. Using this property we can euce aitional algebraic structure on A : Theorem 3.6. The set A has no zero ivisors, i.e., if f, g A an f g = 0, then at least one of f = 0, g = 0 hols. Proof. We prove the contrapositive, that is, if f, g 0 then f g 0. Suppose f, g 0; we have that the sets M = {n N f(n 0} an N = {n N g(n 0} are nonempty. Let m, n be the least elements of M, N respectively, an consier the evaluation of f g at mn : (f g(mn = f(ag(b. ab=mn Now, each term on the right is of the form f(ag(b where ab = mn. For each of these terms, we have either a = m an b = n, in which case f(ag(b = f(mg(n 0, or one of a < m, b < n, in which case f(ag(b = 0 since we assume m, n were the least elements of M, N. Thus there is exactly one nonzero term on the right sie, so the right sie is nonzero; hence the left sie must also be nonzero, so f g 0. The combination of Theorems 3.3 an 3.6 gives the following result: Theorem 3.7. The set A, along with the operations of pointwise aition an Dirichlet convolution, forms an integral omain. Note that, since the conition that f(1 0 is weaker than the conition that f 0, Theorem 3.7 an Lemma 3.4 o not imply that A is a fiel. Inee, the existence of elements in A without inverses uner convolution will have consequences when we consier roots of polynomials in A. 4. Möbius Inversion an Relate Results In this section, we efine an arithmetic function introuce by Möbius in 1832, an prove several results regaring its relationship with other arithmetic functions. Definition 4.1. The Möbius function µ A is given by, for each n N, 1 if n is a prouct of an o number of istinct primes; µ(n = 1 if n is a prouct of an even number of istinct primes; 0 otherwise. Equivalently, if n is ivisible by the square of a prime number, then µ(n = 0; otherwise, µ(n = ( 1 ω(n, where ω(n enotes the number of prime factors of n. To iscuss the following results more concisely, we nee the following lemma, which soliifies the connection between ivisor sums an convolution: Lemma 4.2. For any f A, (f 1(n = n f(.

6 6 MARK SCHACHNER Proof. By efinition, (f 1(n = n f(1 ( n. Since 1(n = 1 for all n N, the lemma follows. Some results which follow from Lemma 4.2 are: (i 1 1 = τ (see Definition 2.4 (ii 1 } 1 {{ 1 } = τ n (see Definition 6.3 n times (iii 1 N = σ, where N(n = n for all n N Since µ is invertible, it is natural to ask what its inverse is uner Dirichlet convolution. The below theorem gives the surprising answer: Theorem 4.3. µ 1 = ε. The following lemma will prove useful to prove Theorem 4.3: Lemma 4.4. Let S be a finite nonempty set. Then the sets {S S : S is even} an {S S : S o} have the same carinality. Proof. We have two cases: S is o an S is even. If S is o, then we construct a bijection as follows: Fix some subset of S S. Trivially, S + S \ S = S, an, since S is o, this implies that S an S \ S have opposite parity. Thus each subset of S correspons to exactly one other subset of S such that every evensize subset correspons to an o-size subset, an vice versa. Hence we have a bijection between the set of even-size subsets of S an o-size subsets of S. If S is even, then we first fix some element p S. We construct a function f from the power set of S to itself as follows: for any subset S of S, p f(s if an only if p S, an for each element k S \ {p}, we have k f(s if an only if k S. Now, f is a bijection an has the property that if S has n elements, then f(s has either ( S + 1 n or ( S 1 n elements, epening on whether p S. However, since S is even, both S +1 an S 1 are o, so S + f(s is o for each S S. Similarly to the above, this implies that S an f(s have opposite parity; hence we again have a bijection between the even- an o-size subsets of S. Thus the lemma is prove. Now, we move on to the proof of Theorem 4.3. Proof of Theorem 4.3. If n = 1, then (µ 1(n = µ(1 = 1 = ε(n as require, so suppose n > 1. By Lemma 4.2, 1 µ = n µ(, so we nee to show that the sum of µ( across all of the ivisors of n is equal to ε(n. Since µ(n = 0 unless n is squarefree, we nee only consier the sum across all squarefree ivisors of n. Consier the set P n = {p 1, p 2,..., p m } of istinct prime factors of n; choosing a squarefree ivisor of n is equivalent to choosing some subset of P n. Observe that, for any subset A of P n, the value of µ at the squarefree ivisor which A represents is 1 if A is even an 1 if A is o. Since n > 1 we have P n is nonempty, so by Lemma 4.4 P n has an equal number of even- an o-size subsets. Thus each positive 1 in the sum is cancele by a -1 an the sum of µ across all subsets of P n is zero. The theorem follows. Since Theorem 4.3 states that the Möbius function is the inverse of the constant function 1 uner Dirichlet convolution, it is often calle Möbius inversion, although the name is also given to the following corollary:

7 ALGEBRAIC AND ANALYTIC PROPERTIES OF ARITHMETIC FUNCTIONS 7 Corollary 4.5 (Möbius Inversion. Let f, g A such that for any n N we have (4.6 f(n = n Then (4.7 g(n = n g(. ( n f(µ. Proof. By Lemma 4.2, (4.6 is equivalent to f = g 1. Convolving on both sies by µ, we obtain f µ = g 1 µ = g by Theorem 4.3. By efinition this is equivalent to eq. (4.7. As an application of Dirichlet convolution an Möbius inversion, we return to our function ϱ from Section 1. We have, for all n, ϱ(n = ε(n + ϱ(. So n,<n 2ϱ(n = ε(n + n ϱ( Thus we have = ε(n + (ϱ 1(n. (4.8 2ϱ = ε + ϱ 1 (2ϱ µ = ε µ + ϱ 1 µ 2(ϱ µ = µ + ϱ ϱ(n = µ(n + ( 2ϱ(n + ϱ(n = µ(n + n,>1 n,>1 ( 2µ(ϱ ( n ( 2µ(ϱ ( n. By evaluating eq. (4.8 at n an substituting for ϱ ( n, we obtain ϱ(n = µ(n + 1 n, 1>1 + ( n ( 2µ( 1 µ 1 n, 1>1 2 n, 1 2> m=n 1, 2,..., m 1>1 1 ( n ( 2 2 µ( 1 µ( 2 ϱ 1 2 Repeating this process of substitution, we arrive at ϱ(n = µ(n + ( 2 ( m n µ( i µ m which is a non-recursive efinition for ϱ. In general, for each k C we can efine a function ϱ k A which satisfies the functional equation i m kϱ k = (k 1ε + ϱ k 1,.,

8 8 MARK SCHACHNER where the factor (k 1 is inclue so ϱ k (1 = 1 for all k C. By a similar argument to the above, the solution to this functional equation is given by ( m k ( n ϱ k (n = µ(n + µ( i µ, 1 k m m=n 1, 2,..., m 1>1 a fact which has as a corollary ϱ 0 = µ. Thus Möbius inversion can be useful in etermining solutions to functional equations involving Dirichlet convolution. i m 5. Polynomials in A In this section, we evelop the theory of polynomials in A, an euce several results about the existence an properties of their roots. We begin with several efinitions. Definition 5.1. Let f A. We enote the repeate convolution f f f by }{{} n times f n, an set f 0 = ε. Definition 5.2. (i A polynomial in A is an expression of the form n A i X i = A 0 + A 1 X + A 2 X A n X n, i=0 where A 0, A 1,..., A n A such that A n 0 an X is an ineterminate. (ii The natural number n is calle the egree of the polynomial. (iii The elements A 0, A 1,..., A n are calle the coefficients or coefficient functions of the polynomial, with A n specifically calle the leaing coefficient. (iv If F is a polynomial in A an F(g = 0 for some g A, we call g a root of F. We enote the set of polynomials in A by A[X]. Since A is a ring, A[X] inherits the operations of aition, pointwise multiplication an Dirichlet convolution efine over A, so for any polynomials F, G A[X] we let (F + G(X = F(X + G(X an (F G(X = F(X G(X. Theorem 5.3. Let A 0, A 1 A such that A 1 is invertible. Then the polynomial F(X = A 0 + A 1 X has a single unique root. Proof. Since A 1 is invertible, there exists a unique arithmetic function A 1 1 such that A 1 A 1 1 = ε. Let g = A 0 A 1 1. Now F(g = 0. Also, this root is unique because any root g of F must satisfy A 0 A 1 1 = g, an the inverse of A 1 is unique by Lemma 3.4. There is also an analog to the quaratic formula for polynomials in A with all coefficients invertible. To prove this, we first nee the following lemma: Lemma 5.4. Let f A be invertible. g 1, g 2 A such that g 1 g 1 = g 2 g 2 = f. Then there exist exactly two functions Proof. For each natural number n, let L(n enote the sum of the exponents in the prime factorization of n. (For instance, since 288 = , we have L(288 = 5+2 = 7. We will show that we can always uniquely efine g 1 (n, g 2 (n by inuction on L(n.

9 ALGEBRAIC AND ANALYTIC PROPERTIES OF ARITHMETIC FUNCTIONS 9 Base Case. Let L(n = 0, so n = 1. If we efine g 1 (1 as the principal square root of f(1 an g 2 (1 as the negative square root of f(1, then we have that (g 1 g 1 (1 = f(1 = (g 2 g 2 (1. Moreover, any function h which satisfies (h h(1 = f(1 must satisfy h(1 2 = f(1, so g 1, g 2 are both well-efine an the only functions which satisfy the require property if L(n = 0. Inuctive Step. Suppose that for some n N we have that if L(s n then g 1 (s, g 2 (s are uniquely well-efine. Fix some m N such that L(m = n + 1 an let (5.5 (5.6 g 1 (m = 1 2g 1 (1 g 2 (m = 1 2g 2 (1 f(m m 1<<m f(m m 1<<m ( m g 1 (g 1, ( m g 2 (g 2 We first show that the above efinitions are well-efine. We have efine g 1 (1, g 2 (1 to be the principal an negative square roots of f(1; since f(1 0 by hypothesis, we thus have that 2g 1 (1, 2g 2 (1 0. Next, observe that for all m such that 1 < < m we have L(, L( m < L(m = n + 1, so by our inuctive hypothesis the above expressions are well-efine. Furthermore, since a rearrangement of the above yiels f(m = ( m g i (g i = (g i g i (m, m for i = 1, 2, any function h A which satisfies h h = f must satisfy one of (5.5,(5.6. Thus g 1 an g 2 are uniquely well-efine for any k N such that (k = n + 1. Hence the inuctive proof is complete. As the above lemma shows, for an invertible function f A, the principal Dirichlet square root (written g 1 above is well-efine; going forwar we enote this by f 1/2. This square root has many of the same properties as the usual square root; in particular, f 1/2 g 1/2 is a square root of f g for all invertible f, g A. It is also important to note that, while Lemma 5.4 states that all invertible functions have square roots uner convolution, there exist non-invertible functions both with an without square roots. As an example of a non-invertible function with a square root, consier the class of functions δ k given by, for each n, k N, { 1 if n = k, δ k (n = 0 otherwise. Then the function δ k 2, which is only invertible if k = 1, has δ k as a square root uner convolution. As an example of a non-invertible function without a square root, consier the function 1 ε, which evaluates to 0 if n = 1 an 1 otherwise. Suppose for contraiction that this function has a square root s A. Since s(1s(1 = (s s(1 = (1 ε(1 = 0, we must have s(1 = 0. But this implies 1 = (1 ε(2 = (s s(2 = s(1s(2 + s(2s(1 = 0, a contraiction. Hence s cannot exist.

10 10 MARK SCHACHNER Theorem 5.7 (Quaratic Formula. Let A, B, C A such that A an B 2 4A C are invertible. Then there exist exactly two roots f 1, f 2 A of the polynomial (5.8 A X 2 + B X + C, an these functions are given by f 1 = ( B + (B 2 4A C 1/2 (2A 1, f 2 = ( B (B 2 4A C 1/2 (2A 1. Proof. The majority of the proof is completely analogous to the usual erivation of the quaratic formula, rearranging (5.8 to obtain (5.9 (f B (2A 1 2 = (B 2 4A C (4A 2 1. Since the right sie is invertible, so is the left sie, so by Lemma 5.4 we can take the Dirichlet square root of both sies, an arrive at the formula specifie by the theorem. We now turn to proving the general case, a partial analog of the Funamental Theorem of Algebra for A. Definition Let F A[X] be given by n F(X = A i X i = A 0 + A 1 x + + A n X n. i=0 The formal erivative of F is enote F an given by n F (X = ia i X n 1 = A 1 + 2A 2 X + + na n X n 1. i=1 Definition Let F A[X]. The base polynomial of F, enote P F, is the polynomial over C whose coefficients are the coefficient functions of F evaluate at 1, i.e., if F(X = A 0 + A 1 X + + A n X n, then P F (x = A 0 (1 + A 1 (1x + + A n (1x n. The formal erivative an base polynomial of a polynomial over A interact in the following important way: Lemma Let F A[X]. Then P F = (P F, where the prime on the right-han sie enotes the erivative in the usual sense. Proof. Suppose F(X = A 0 + A 1 X + + A n X n. The coefficients of F are then A 1, 2A 2, 3A 3,..., na n, so we have (5.13 P F (x = A 1 (1 + 2A 2 (1x + + na n (x n 1. Also, since P F (x = A 0 (1 + A 1 (1x + + A n (1x n, taking the erivative as usual yiels P F(x = A 1 (1 + 2A 2 (1x + + na n (1x n 1, which is ientical to eq. (5.13. Given this fact, we make the following statement, the main theorem of this section: Theorem Let F A[X] be a polynomial over A of egree at least 1 such that P F has a root in C of multiplicity 1. Then F has a root g A.

11 ALGEBRAIC AND ANALYTIC PROPERTIES OF ARITHMETIC FUNCTIONS 11 Proof. The proof runs similarly to those of Lemmas 3.4 an 5.4, in that we will show that for any n N we can well-efine g(n such that F(g(n = 0, inucting on L(n. Base Case. Let L(n = 0, so n = 1. By hypothesis P F has a simple root r C; let g(1 = r. Now, we have F(g(1 = A 0 (1 + A 1 (1g(1 + + A n (1g(1 n = P F (g(1 = 0. Inuctive Step. Suppose that for some n N we have that if L(s n then g(s is well-efine. Fix some m N such that L(m = n + 1 an let (5.15 (( 1 n g(m = F A j (mg(1 j (g(1 j=0 ( n+1=m 1, 2,..., n+1 m A 0 ( 1 g( n+1 τ n (m/ 1 + A 1( 1 g( 2 g( 3... n+1 τ n 1 (m/( A 0( 1 g( 2 g( 3...g( n+1 τ 1 (1, where τ k (n counts the number of ways to write n as a prouct of k natural numbers. We first justify that this lengthy efinition is well-efine. By Lemma 5.12, F (g(1 = P F (g(1 = (P F (g(1. Since g(1 is a root of P F of multiplicity 1, we can not have that (P F (g(1 = 0. Next, observe that, in the secon summation, g is always evaluate at a proper ivisor of m; since L( < L(m for all proper ivisors of m, by our inuctive hypothesis g is well-efine wherever it is evaluate in eq. (5.15. Lastly, since τ k (n > 0 for all n, k N, each of the fractions in the secon summation is well efine. Thus g(m exists an is uniquely efine for each m N. We now show that, uner the efinition provie by eq. (5.15, F(g(m = 0 for all m N. We begin by multiplying both sies of eq. (5.15 by F (g(1 an aing g(m F (g(1 to both sies. This yiels

12 12 MARK SCHACHNER ( n n ( = ia i (1g(1 i 1 g(m + A j (mg(1 j + i= n+1=m 1, 2,..., n+1 m A 0 ( 1 g( n+1 τ n (m/ 1 j=0 + + A 0( 1 g( 2 g( 3...g( n+1 τ 1 (1 Now, by using the fact that τ k (n counts the number of ways to write n as a prouct of k natural numbers, we can represent the thir sum by 1 2=m 1, 2 m A 1 ( 1 g( =m 1, 2, 3 m + + A 1 ( 1 g( 2 g( n+1=m 1,..., n+1 m A 1 ( 1 g( 2...g( n+1. Lastly, observe that each of the terms in the first an secon summations of eq. (5.16 correspons to exactly one of the factorizations 1, 2,..., n+1 where k = m for some 1 k n + 1; that is, the first an secon sums are exactly the extra terms we exclue when we specifie that 1, 2,... n+1 m. Combining these facts, we can rewrite eq. (5.16 as ( ( 0 = A 0 (m + A 1 ( 1 g( 2 + A 1 ( 1 g( 2 g( 3 1 2=m =m n+1=m. A 1 ( 1 g( 2...g( n+1 = A 0 (m + (A 1 g(m + (A 2 g 2 (m + + (A n g n (m, which is our esire result. By repeate application of Theorem 5.14, we also have the following corollary: Corollary Let F A[X] such that P F has n simple roots. Then F has at least n roots, counte with multiplicity. Note that we have shown that the existence of a simple root of P F is a sufficient conition for F to have a root, but this conition is by no means necessary. The complexity in fining a necessary an sufficient conition for F to have a root stems largely from the existence of non-invertible arithmetic functions, as we state in Section 3.

13 ALGEBRAIC AND ANALYTIC PROPERTIES OF ARITHMETIC FUNCTIONS The Growth of Arithmetic Functions Having examine the elements of A algebraically, we now consier them analytically, an explore several techniques for quantifying their behavior. We begin with a efinition which will form the core of this section: Definition 6.1. Let f, g A. We say f is an average orer of g if f(n lim x g(n = 1. Average orers allow one to smooth out the erratic behavior of an arithmetic function, in the sense that arithmetic functions with unpreictable growth often have average orers with preictable growth. As a classical example, we present the following proof of an average orer of τ, the ivisor-counting function: Theorem 6.2. The function log(n is an average orer of τ. Proof. We wish to fin a function f A which satisfies the asymptotic formula τ(n. f(n Since τ(n = 1, this can be rewritten as f(n ab x We will follow Dirichlet s proof, a metho now commonly calle the Dirichlet hyperbola metho. Observe that the right-han sie can be interprete as counting the number of lattice points uner the hyperbola ab = x (See Fig. 1. In orer to enumerate these points, we ivie the figure into three overlapping sectors: where a x, where b x, an where a, b x. We then subtract off the overlap to obtain the formula: 1 Since x log x = τ(n = ab x = a x b x a = x + a a x a x 1. b x a x b b x = 2x 1 x 2 a = O(x log x. log x, we obtain that lim x log(x τ(n = 1. a,b x x b x 2 1

14 14 MARK SCHACHNER Figure 1. The sum of τ(n over all n x can be interprete as counting the lattice points uner the hyperbola ab = x. Here x = 12 an the number of points is 35. Hence log n is an average orer for τ. The above argument is in fact generalizable to a class of arithmetic functions which we now efine: Definition 6.3. The k-th Piltz function τ k, efine for each k N, counts the number of ways to express n as an orere prouct of k natural numbers. For example, τ 3 (6 = 9 because 6 = = = = = = = = = (To justify the reuse of notation, observe that τ as efine in Definition 2.4(iii is equal to τ 2. Aitionally, for each k N, the k-th ivisor summatory function

15 ALGEBRAIC AND ANALYTIC PROPERTIES OF ARITHMETIC FUNCTIONS 15 is enote D k an given by the sum D k (x = τ k (n. By consiering the sum as an enumeration of the number of k-imensional lattice points in R k uner the hyperbolic surface a 1 a 2...a k = x, it can be shown that D k (x = xp k (log x + k (x, where P k is a polynomial of egree k 1 an k is an error term. Thus (log n k 1 is an average orer for τ k. One can use similar techniques to boun the values of arithmetic functions, as emonstrate by the following analysis of the behavior of ϱ: Theorem 6.4. (i For all x > 3, ϱ(n (x 1 log 2 (x. (ii For all x N, ϱ(n x(log x 1. Proof. (i For x N, let P (x = ϱ(n. First, observe that since ϱ(n = n <n = 1 n ϱ( ϱ( <n 2< 1 =... =... 1 n 1, 2 1 1<n 2< 1 s 1 s s 1< s we may characterize ϱ combinatorially as the number of orere factorizations of n. Thus, P (x counts the number of orere tuples of natural numbers (excluing 1 in each tuple an incluing the empty tuple {Ø} whose prouct oes not excee x. Since the maximum length of such a tuple is log 2 (x (as long as x 4 an each natural number in such a tuple must be less than or equal to x, we may thus boun P (x above by (x 1 log 2 (x. (ii As above, we will use the combinatorial interpretation of ϱ an P. Fix x N. Of length 0, there is one tuple whose prouct is at most x, an of length 1 there are x 1 tuples whose prouct is at most x. Also, for each n x, there are x n 1 tuples of length 2 whose prouct is at most x. Summing these

16 16 MARK SCHACHNER three terms together, we obtain that P (x 1 + (x 1 + x n ( x 1 n = x x n x log x x = x(log x 1. While these are loose bouns, this example emonstrates the basic methos which can be use to treat arithmetic functions analytically. Acknowlegments. I am very grateful to my mentors, Billy Lee an Karl Schaefer, for their help an for encouraging my pursuit of these topics. I also want to thank Daniil Ruenko for a thoroughly engaging Apprentice curriculum, as well as for his input on the combinatorial sie of my research. References [1] Apostol, T. M. (2010. Introuction to analytic number theory. New York: Springer. [2] Tenenbaum, G. (2015. Introuction to analytic an probabilistic number theory. Provience: American Mathematical Society. [3] Snellman, J. (2002. The ring of arithmetical functions with unitary convolution: Divisorial an topological properties. Research Reports in Mathematics, 4. [4] Huxley, M. N. (2003. Exponential sums an lattice points III. Proceeings of the Lonon Mathematical Society, 87(3, [5] Hary, G. H. (1917. On Dirichlet s ivisor problem. Proceeings of the Lonon Mathematical Society, 2(1, 1-25.