Motivic Symbols and Classical Multiplicative Functions1

Ane Espeseth, Torstein Vik

1 Introduction

The theory of multiplicative functions is a central part of number theory, and has been so for a long time.

They have been extensively studied not only by Ramanujan, but also by both Gauss and Euler. One can think of a as a machine that tells you some fixed property of the number you give it. One example is the square indicator function, which tells you whether a number is a square number.

Another example is the Euler totient function, which when given a number, tells you how many smaller numbers that share no factors with the original number.

Although multiplicative functions are often studied for the sake of what they do to numbers, another central point of study is how multiplicative functions relate to one another. is a mathematical operation that takes two multiplicative functions and combines them into one, somewhat akin to multiplication of numbers. As the name suggest, this operation is quite convoluted and hard to understand. This is not unique to Dirichlet convolution, it is very usual for concepts in the theory of multiplicative functions to be hard to understand.

In this paper we look at a specific class of multiplicative functions, which we have called classical multiplicative functions. These multiplicative functions can be expressed in a certain way using the Riemann zeta function, a function central not only in number theory, but also in many other areas of mathematics.

Just to illustrate the importance of this function, a 1 million US dollar price awaits you if you can describe its zeroes. However, this is unfortunately unrelated to our project. The reason we call our functions “classical’ is because most functions that have been traditionally studied fall into this class.

Our project constructs a new language for this class of multiplicative functions that illuminates the theory of multiplicative functions by quite a bit. For instance, calculation of Dirichlet convolution is essen- tially reduced to a calculation virtually as simple as concatenating a pair of lists. Furthermore, this new language describes normal function multiplication (under a certain hypothesis), and it describes the norm of a multiplicative function (a definition will come later).

Acknowledgments

We would like to give a huge thanks to our math teacher Andreas Holmstrom, whose constant guidance and ideas have been an integral part of this project.

1This is a ten-page summary. Full 66-page report available at multiplicativelab.wordpress.com

1 2 Motivic Symbols & Lambda-rings

In this section we will develop the underlying theory behind the main theorem. We begin by defining lambda-rings and multisets (two relatively well-known concepts). We then move onto our main idea, namely motivic symbols. We also define central operations, and state one of our main theorems, namely that motivic symbols form a lambda-ring. We also define corresponding sequences, which are what allows us to apply this theory to the realm of number theory. In later sections, we explain how motivic symbols correspond to multiplicative functions of a certain type. In this 10-page summary, we assume that basic notions from commutative algebra are known, such as groups, rings and homomorphisms.

Definition 2.1. For the purposes of this article, one can define a lambda-ring as a commutative ring R together with ring homomorphisms ψk : R → R for all integers k ≥ 1 (called the Adams operations), such that:

1. ψ1 = id

2. ψm ◦ ψn = ψmn

3. ψp(x) ≡ xp (mod pR) for all primes p

Remark 1. A more general definition is available in [1], using lambda-operations instead of Adams operations.

Example 2.1. The integers Z together with homomorphisms ψk := id for all k is lambda-ring. The first two axioms follow trivially, while axiom three is equivalent to Fermat’s little theorem.

Definition 2.2. A multiset is a set in which the same element may appear multiple times. Formally, one can think of a multiset as a function from a set to the natural numbers, where the function assigns a multiplicity to each element of the set.

Example 2.2. {1, 1, 5, 3, a, a, a} is an example of a multiset.

Definition 2.3. A motivic symbol is an ordered pair of disjoint finite multisets A and B, written in fraction

A notation, i.e. A/B or B .

Remark 2. A motivic symbol where the multisets are not disjoint can be made into an actual motivic symbol by removing common elements. This cancellation-process is always applied after manipulating motivic symbols. Formally, one could define this with an equivalence relation, but it is more convenient to do it in the way we have when it comes to number theory.

Example 2.3. The following are motivic symbols:

{5, 3} {5, 3, 4, 4} and ∅ {1, 2, 6}

2 The following are examples of cancellation:

{5, 5, 6} {5, 6} {5, 5, 3} {5, 3} → and → {4, 3, 5} {4, 3} {5} ∅

Definition 2.4. We define the direct sum of two motivic symbols, written A/B ⊕ C/D, to be the pairwise disjoint union of the multisets, with addition of multiplicity. This can also be interpreted as the pairwise addition of the multiplicity functions. We also include cancellation of common elements in the calculation.

Example 2.4. {5, 3, 6} {5, 5, 7} {5, 5, 7, 5, 3, 6} {5, 5, 5, 7} ⊕ = = ∅ {3, 6} {3, 6} ∅ {b, b, c} {b, b, c} {c} ∅ ⊕ = = {a, b, b} {a, a} {a, a, a, b, b} {a, a, a}

Proposition 2.1. This operation is commutative, associative, and has the identity ∅/∅. It also admits inverses, as the inverse of A/B is B/A due to cancellation.

Definition 2.5. We define the tensor product of two motivic symbols, written A/B ⊗C/D, to be the unique operation that distributes over direct sum, and also satisfies for all a, b that {a}/∅ ⊗ {b}/∅ = {a · b}/∅. Explicitly, this can be written out as

A C A • C B • D ⊗ = ⊕ B D A • D B • C

where we by X • Y denote the multiset of products one can make from one element of X and one element of

Y , counted with multiplicity.

Example 2.5. Let’s say we are working with integers under normal multiplication.

{1, 4} {−1 · 0, −1 · 1} {0, −1} {−1} ⊗ ∅ = = = {−1} {0, 1} {1 · 0, 4 · 0, 1 · 1, 4 · 1} {0, 0, 1, 4} {0, 1, 4}

{2, 4} {4, 1} {2 · 4, 2 · 1, 4 · 4, 4 · 1} {8, 2, 16, 4} {2, 16} ⊗ = = = ∅ {2} {2 · 2, 4 · 2} {4, 8} ∅

Remark 3. This requires that the multisets of the motivic symbols contain elements that are part of some magma, a set equipped with an operation. For various reasons, we normally require this magma to be a commutative monoid, an abelian group without the “existence of inverses”-axiom.

Proposition 2.2. This operation is commutative, associative, and has the identity {1}/∅, in which 1 is the identity of whatever monoid one is working with.

Definition 2.6. We define ψk of A/B to be a function that raises every element of A/B to the power of k individually.

3 Example 2.6. Let’s say we are working with integers under normal multiplication.

 {1, 3}  {12, 32} {1, 9} {9} ψ2 = = = {−1, −1} {(−1)2, (−1)2} {1, 1} {1}

{2, −2} {23, (−2)3} {8, −8} ψ3 = = {1} {13} {1}

Remark 4. This also presumes that the multisets of the motivic symbols contain elements that are part of some commutative monoid. For all commutative monoids, we define

k ak = az · a}|··· a{

Theorem 2.3. The set of motivic symbols with elements in a commutative monoid M, written MS(M), forms a lambda-ring with direct sum as addition, tensor product as multiplication and the ψk operations as

Adams operations. The identities and inverses are described in Propositions 2.1 and 2.2.

Remark 5. Furthermore, this association is functorial, meaning that a monoid homomorphism f : M → N will turn into a lambda-ring homomorphism MS(f): MS(M) → MS(N), where we define MS(f) as applying the monoid homomorphism on each element.

Definition 2.7. Given a motivic symbol

A {α , α , . . . α } = 1 2 m B {β1, β2, . . . , βn}

We associate a formal power series (with formal symbol t)

(1 − β1t)(1 − β2t) ··· (1 − βnt) (1 − α1t)(1 − α2t) ··· (1 − αmt)

The coefficients of this power series is by definition the corresponding sequence of A/B, written Corr(A/B).

We write the formal power series as Corrt(A/B).

Remark 6. This only makes sense if our monoid is a submonoid of the multiplicative monoid of some ring, preferably an integral domain. 1 of course will denote the multiplicative identity, which is guaranteed to exist in a submonoid. When computing the formal power series, we of course work in our entire ring.

Proposition 2.4. The corresponding sequence exists for all motivic symbols, and furthermore is a special power series (meaning it has constant coefficient 1).

Theorem 2.5. If our submonoid (see Remark 6) is the submonoid of an integral domain, then the corre- sponding sequence is unique, provided the submonoid does not contain 0.

4 Theorem 2.6. The Corr-function maps direct sum to Cauchy product. In other words:

 A C   A   C  Corr ⊕ = Corr · Corr t B D t B t D

Theorem 2.7. The Corr-function will map tensor product to pointwise product, provided one factor is of the form {x} . In other words ∅

{x} C  {x}  C  Corr ⊗ = Corr × Corr ∅ D ∅ D where × denotes pointwise product.

3 Multiplicative Functions

In this section we define the concept of a multiplicative function together with two different types of generat- ing series associated to such a function. We also explain how to construct motivic symbols from multiplicative functions and provide some simple examples. When no other reference is given, the definitions and basic facts are taken from Wikipedia and the book “Introduction to Arithmetical Functions” by Paul J. McCarthy

[2].

Definition 3.1. An arithmetical function is a function that takes a positive integer as input, and gives a complex number as output.

Definition 3.2. An arithmetical function f is said to be multiplicative if f(1) = 1, and if f(mn) = f(m)f(n) for all m and n which have no common factors. A multiplicative function f is called completely multiplicative if f(mn) = f(m)f(n) for all values of m and n.

Example 3.1. The is denoted by λ(n). If n is a positive integer, then λ(n) is defined as:

λ(n) = (−1)Ω(n), where Ω(n) is the number of prime factors of n, counted with multiplicity. The Liouville function is completely multiplicative because Ω(mn) = Ω(m) + Ω(n) whether gcd(m, n) = 1 or not.

Example 3.2. The number 50 can be factored into primes as 2 · 5 · 5. Here there are three prime factors, so λ(50) is equal to (−1)3, i.e. -1.

Definition 3.3. The delta function δ(n) is defined by

δ(1) = 1, δ(n) = 0 for n ≥ 2

Definition 3.4. If f and g are two arithmetical functions, one defines a new arithmetical function f ∗ g, the

5 Dirichlet convolution of f and g, by

X n (f ∗ g)(n) = f(d) g d d | n where the sum extends over all positive divisors d of n. If f and g are multiplicative, then so is f ∗ g.

It is well-known that the set of multiplicative functions is an abelian group under Dirichlet convolution, in which the delta function δ(n) is the identity element.

Definition 3.5. The norm of a multiplicative function f is the function N(f) defined by

X N(f)(n) = f(n2/d)λ(d)f(d) d|n2 where λ(d) is the Liouville function. Here the sum is taken over all positive divisors of n2.

Note 1. The norm operator was studied by Sivaramakrishnan and Redmond in their paper “Some Properties of Specially Multiplicative Functions” [3], in which they also defined “higher norm” N r by the recursive formula N r(f) = N(N r−1(f)). The relevance for us is that these higher norms correspond to the Adams operations ψ2, ψ4, ψ8 and so on.

Definition 3.6. Let f be a multiplicative function and p a prime. We define a formal power series fp(t), called the Bell series of f at the prime p as

∞ X n n fp(t) = f(p )t n=0

Definition 3.7. A function f is said to have rational Bell series if for each prime p the Bell series can be writ- ten as a rational expression. It is known that this is the equivalent to the sequence f(1), f(p), f(p2), f(p3), ... being linearly recursive.

Example 3.3. The constant function 1(n) is defined by 1(n) = 1 for all n. Its Bell coefficients are 1, 1, 1, ··· ,

2 3 1 giving us the Bell series 1 + t + t + t + ··· . This can also be written as 1−t ,therefore the function has a rational Bell series.

Note 2. For many multiplicative functions, you can compute their corresponding Bell series without much effort, by finding the function values f(pe) and manipulating geometric series (see example in Table 1).

Example 3.4. The Euler totient function denoted by ϕ counts the positive integers less than or equal to n that are relatively prime to n. For n = pe (e > 0), there will be exactly pe − pe−1 integers that satisfies this demand, so we will get ϕ(pe) = pe − pe−1.

6 Table 1: Euler totient function at p = 2 pe f(pe) f(pe)te e=0 1 1 1 e=1 2 1 t e=2 4 2 2t2 e=3 8 4 4t3

2 3 The Bell series for the Euler totient function at p = 2 will be f2(t) = 1 + t + 2t + 4t + ... (See Table 1).

1−t This can be written as the rational Bell series f2(t) = 1−2t .

Definition 3.8. Let f be an arithmetical function. We define the Dirichlet series Df (s) of f to be

∞ X f(n) D (s) = f ns n=1

Remark 7. If f is multiplicative, this can be rewritten as an Euler product. More specifically:

Y −s Df (s) = fp p p prime

where fp is the Bell series at p.

Remark 8. Although it is possible to view a Dirichlet series as a function of a complex variable s, we prefer to think of it as a formal series.

Definition 3.9. The Dirichlet series of the constant function will be

∞ X 1 Y 1 D (s) = = f ns 1 − p−s n=1 p prime

We define ζ(s) to be this Dirichlet series (when viewed as a function of a complex variable, this function is called the Riemann zeta function).

Definition 3.10. We choose to define a classical multiplicative function to be a multiplicative function whose Dirichlet series can be written as a quotient D(s)/E(s), where both D(s) and E(s) are finite products of factors of the form ζ(t · s − t · k), where in each factor t is either 1 or 2, and k is a non-negative integer, and s is the variable.

Remark 9. This definition my seem unmotivated, but is explained by the fact that almost all functions traditionally included in number theory textbooks (or on relevant Wikipedia pages) have precisely this property. We refer to the attached table of functions for more details. We are able to prove our main theorem for much larger classes of functions (in fact for all multiplicative functions with rational Bell series) but this work was done after submitting the original report, and will be the subject of a future article.

7 Theorem 3.1. If h(t) is a rational expression in t with complex coefficients, satisfying h(0) = 1, then there exists a unique motivic symbol A/B such that h(t) = G(Corr(A/B); t). Such a rational expression can be written u(t)/v(t) where u and v are polynomials with constant term 1.

Definition 3.11. Let f be a multiplicative function with rational Bell series and let p be a . We

∗ define the motivic symbol of f at p as the motivic symbol A/B ∈ MS(C ) satisfying G(Corr(A/B); t) = fp(t)

2 n 1 Example 3.5. The constant function 1(n) has the Bell series 1 + t + t + ··· + t = 1−t at all primes. We have u(t) = 1 and v(t) = 1 − t. From v(t) = 1 − t we get the set {1} upstairs. From u(t) = 1 we get an empty set downstairs. The motivic symbol becomes {1} at all primes. ∅

Example 3.6. The Bell series of the Euler totient function varies with the prime at which we are working at, but p = 2 and p = 3 represented in the table below. It has the general Bell series

1 + (p1 − p0)t + (p2 − p1)t2 + (p3 − p2)t3 + ··· = (1 + p1t + p2t2 + p3t3 + ··· ) − (p0t + p1t2 + p2t3 + ··· ) =

1 t 1 − t − = 1 − pt 1 − pt 1 − pt

We find the Dirichlet series of the function through the following computation:

−s Q −s Y Y 1 − p p(1 − p ) ζ(s − 1) f (p−s) = = = p 1 − p · p−s Q (1 − p−(s−1)) ζ(s) p p p

In general for the Euler totient function, we have u(t) = 1−t and v(t) = 1−pt. This gives us the motivic

{p} symbol {1} .

Bell coefficients Bell series Motivic symbol

e=0 1 2 3 4 5 Num. Den. Upst. Downst.

p=2 1 1 2 4 8 16 1 − t 1 − 2t {2} {1}

p=3 1 2 6 18 54 162 1 − t 1 − 3t {3} {1}

4 The Main Theorem

Theorem 4.1. The set of all classical multiplicative functions can be equipped with a lambda-ring structure in which

1. Addition corresponds to Dirichlet convolution (and additive inverse corresponds to Dirichlet inverse).

2. Multiplication corresponds to the multiplication of multiplicative functions, provided at least one of

the factors is completely multiplicative.

3. The second Adams operation ψ2 corresponds to taking the norm of multiplicative functions.

8 4. Furthermore, ψ2, ψ4, ψ8, ψ16 and so on correspond to the higher norm operators of multiplicative

functions. (see Note 1 for definition)

This lambda-ring is isomorphic to the lambda-ring MS(C2 × N), where N is the monoid of non-negative integers under addition, and C2 is the cyclic monoid of order 2.

5 Applications

Given this theorem, a large number of classical identities between multiplicative functions become trivial

(see the tables in the illustrations for function descriptions).

Example 5.1. The Dirichlet convolution identity (σk is the k’th and ϕ is the Euler totient function) X n ϕ(d) · σ = σ (n) 0 d 1 d|n which in the language of motivic symbols simply becomes

{p} {1, 1} {p, 1} ⊕ = {1} ∅ ∅ which is obvious.

Example 5.2. Another example is

X n dkJ (d) · J = J r k d k+r d|n

where Jk is the k’th Jordan function. Note that this can be rewritten as

X n (Id J )(d) · J = J k r k d k+r d|n

where Idk is the k’th power function. Since Idk is completely multiplicative, we get in the language of motivic symbols: {pk} {pr} {pk} {pk+r} ⊗ ⊕ = ∅ {1} {1} {1}

k r k+r From the definition of tensor product, {p } ⊗ {p } becomes {p } . Now the identity is obvious. ∅ {1} {pk}

Example 5.3. By computing, we can see that

 {pk}  {p2k} ψ2 = {−1} {1}

9 In the language of multiplicative functions, this gives us the identity:

X 2 ψk(n /d)λ(d)ψk(d) = J2k(n) d|n2

where ψk is the k’th Dedekind’s ψ function and Jk is the k’th Jordan function.

Although these applications are interesting in themselves, they barely skim the surface of what we can do with motivic symbols. In the months after submitting our article, we have generalised our results to more classes of multiplicative functions, including the class of all multiplicative functions, and we have managed to describe a lot more of the theory of multiplicative functions using the language of motivic symbols, such as other improved descriptions of functional multiplication and similar operations. We have also discovered another lambda-ring structure on the class of all multiplicative functions. In the future we hope to continue this work, but also look at other applications of motivic symbols, for instance within the

fields of representation theory and algebraic geometry.

References

[1] Donald Yau, Lambda-rings, World Scientific, 2010.

[2] Paul J. McCarthy, Introduction to Arithmetical Functions, Springer-Verlag, 1986.

[3] Redmond, Sivaramakrishnan, “Some properties of specially multiplicative functions”, Journal of Number

Theory, Volume 13, Issue 2, 1981.

[4] Richard J. Mathar, “Survey of Dirichlet Series of Multiplicative Arithmetical Functions”, 2012, ArXiv.

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