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 multiplicative function 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. Dirichlet convolution 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 k : R ! R for all integers k ≥ 1 (called the Adams operations), such that: 1. 1 = id 2. m ◦ n = mn 3. p(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 k := 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. f1; 1; 5; 3; a; a; ag 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: f5; 3g f5; 3; 4; 4g and ? f1; 2; 6g 2 The following are examples of cancellation: f5; 5; 6g f5; 6g f5; 5; 3g f5; 3g ! and ! f4; 3; 5g f4; 3g f5g ? 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. f5; 3; 6g f5; 5; 7g f5; 5; 7; 5; 3; 6g f5; 5; 5; 7g ⊕ = = ? f3; 6g f3; 6g ? fb; b; cg fb; b; cg fcg ? ⊕ = = fa; b; bg fa; ag fa; a; a; b; bg fa; a; ag 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 fag=? ⊗ fbg=? = fa · bg=?. 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. f1; 4g {−1 · 0; −1 · 1g f0; −1g {−1g ⊗ ? = = = {−1g f0; 1g f1 · 0; 4 · 0; 1 · 1; 4 · 1g f0; 0; 1; 4g f0; 1; 4g f2; 4g f4; 1g f2 · 4; 2 · 1; 4 · 4; 4 · 1g f8; 2; 16; 4g f2; 16g ⊗ = = = ? f2g f2 · 2; 4 · 2g f4; 8g ? 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 f1g=?, in which 1 is the identity of whatever monoid one is working with. Definition 2.6. We define k 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. f1; 3g f12; 32g f1; 9g f9g 2 = = = {−1; −1g f(−1)2; (−1)2g f1; 1g f1g f2; −2g f23; (−2)3g f8; −8g 3 = = f1g f13g f1g 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 k 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 fα ; α ; : : : α g = 1 2 m B fβ1; β2; : : : ; βng 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 fxg .