Definition 6.1.1. Anarithmetic functionis a complex-valued function defined on the
positive integers. ♣
Examples. 𝑑(𝑛)is the number of positive divisors of𝑛(see Theorem 1.6.3) Euler’s function𝜑(see Definition 2.2.7 and Theorem 2.3.1)
𝑓(𝑛) = (−1)𝑛,𝑔(𝑛) = √𝑛2+ 5 + 𝑖 sin 𝑛, etc.
We shall discuss some important arithmetic function in Section 6.2.
165
The following properties often play an important role:
Definition 6.1.2. An arithmetic function𝑓ismultiplicativeif𝑓(𝑎𝑏) = 𝑓(𝑎)𝑓(𝑏)for
every coprime𝑎and𝑏. ♣
Definition 6.1.3. An arithmetic function𝑓iscompletely multiplicative(ortotally mul-
tiplicative), if𝑓(𝑎𝑏) = 𝑓(𝑎)𝑓(𝑏)for every𝑎and𝑏. ♣
Examples. Euler’s function𝜑is multiplicative (this was verified in the first proof of Theorem 2.3.1), but it is not completely multiplicative, as𝜑(8) ≠ 𝜑(2)𝜑(4). The same holds for𝑑(𝑛)(see Exercise 6.1.1).
If𝛼is a fixed real number, then𝑓(𝑛) = 𝑛𝛼is completely multiplicative (hence it is multiplicative.
𝑔(𝑛) = 3𝑛 − 2is not multiplicative, since(2, 3) = 1, but𝑔(6) ≠ 𝑔(2)𝑔(3).
Requiring similar conditions for the sum of the values instead of their product, we get the notion of additive and completely additive arithmetic functions, resp.:
Definition 6.1.4. An arithmetic function𝑓isadditiveif𝑓(𝑎𝑏) = 𝑓(𝑎) + 𝑓(𝑏)for every
coprime𝑎and𝑏. ♣
Definition 6.1.5. An arithmetic function𝑓iscompletely additive(ortotally additive),
if𝑓(𝑎𝑏) = 𝑓(𝑎) + 𝑓(𝑏)forevery𝑎and𝑏. ♣
The definitions both of additivity and complete additivity refer to the values of 𝑓(𝑎𝑏)(and not of𝑓(𝑎 + 𝑏)).
Examples. The logarithm function (with any base) is completely additive.
𝑓(𝑛) = 1 + (−1)𝑛is additive, but not completely additive.
𝑔(𝑛) = 1 + log2𝑛is not additive (hence it cannot be completely additive either).
The identically zero function𝑓 = 0is both completely multiplicative and com- pletely additive, but no other function can be both multiplicative and additive (this follows from Theorem 6.1.6).
We show first that additive and non-zero multiplicative functions can assume only special values at1:
Theorem 6.1.6. If𝑓is multiplicative and𝑓 ≠ 0, then𝑓(1) = 1.
If𝑔is additive, then𝑔(1) = 0. ♣
Proof. Let𝑎be a positive integer satisfying𝑓(𝑎) ≠ 0. Then(𝑎, 1) = 1implies𝑓(𝑎) = 𝑓(𝑎 ⋅ 1) = 𝑓(𝑎)𝑓(1), and dividing by𝑓(𝑎) ≠ 0we get1 = 𝑓(1).
The other statement can be proved similarly. □
Theorem 6.1.6 gives a necessary (but not sufficient) condition for a function to be additive or multiplicative.
The definitions of additivity and multiplicativity imply that additive and (≠ 0) mul- tiplicative functions are uniquely determined by their values at prime powers:
Exercises 6.1 167
Theorem 6.1.7. Let𝑓be multiplicative,𝑔additive, and𝑛 = 𝑝𝛼11. . . 𝑝𝛼𝑟𝑟be the standard form of𝑛 > 1. Then
𝑓(𝑛) = 𝑓(𝑝𝛼11) . . . 𝑓(𝑝𝛼𝑟𝑟) and 𝑔(𝑛) = 𝑔(𝑝𝛼11) + ⋯ + 𝑔(𝑝𝛼𝑟𝑟). ♣ We used this fact deducing the formula for𝜑(𝑛)(in the first proof of Theorem 2.3.1).
Similarly, completely additive and (≠ 0) completely multiplicative functions are uniquely determined by their values at primes:
Theorem 6.1.8. Let 𝑓 be completely multiplicative, 𝑔 completely additive, and 𝑛 = 𝑝𝛼11. . . 𝑝𝛼𝑟𝑟be the standard form of𝑛 > 1. Then
𝑓(𝑛) = 𝑓(𝑝1)𝛼1. . . 𝑓(𝑝𝑟)𝛼𝑟 and 𝑔(𝑛) = 𝛼1𝑔(𝑝1) + ⋯ + 𝛼𝑟𝑔(𝑝𝑟). ♣ We can add to Theorem 6.1.7 that additivity or multiplicativity does not impose any restrictions on the values assumed at prime powers, these can be chosen freely.
This means that prescribing the values arbitrarily at prime powers, gives a multiplica- tive/additive function. An analogous statement holds with primes instead of prime powers for completely multiplicative/additive functions (see Exercise 6.1.4).
Exercises 6.1
1. Verify that𝑑(𝑛)is multiplicative but not completely.
2. Which of the following functions are multiplicative, completely multiplicative, ad- ditive, and completely additive?
(a) 𝑓(𝑛) = {0, if6 ∣ 𝑛 1, if6 ∤ 𝑛.
(b) 𝑔(𝑛) = {0, if3 ∣ 𝑛 1, if3 ∤ 𝑛.
(c) ℎ(𝑛) = {0, if3 ∣ 𝑛 2, if3 ∤ 𝑛.
(d) 𝑘(𝑛) = {2, if3 ∣ 𝑛 0, if3 ∤ 𝑛.
3. Does there exist an (a) additive (b) multiplicative functionℎsatisfyingℎ(6) = 0, ℎ(10) = 1, andℎ(15) = 3?
4. Consider the sequence of primes𝑝1,𝑝2, . . . =2,3,5,7, . . . , the sequence of prime powers𝑞1,𝑞2, . . . =2,3,4,5,7,8,9,11, . . . , and let𝑐1,𝑐2, . . . be arbitrary complex numbers.
(a) Prove that there exists exactly one multiplicative function𝑓 ≠ 0and exactly one additive function𝑔satisfying
𝑓(𝑞𝑖) = 𝑔(𝑞𝑖) = 𝑐𝑖, 𝑖 = 1, 2, . . . .
(b) Prove that there exists exactly one completely multiplicative function𝑠 ≠ 0 and exactly one completely additive function𝑡satisfying
𝑠(𝑝𝑖) = 𝑡(𝑝𝑖) = 𝑐𝑖, 𝑖 = 1, 2, . . . .
5. If 𝑔can assume only positive integer values, then we can define the composite functionℎ(𝑛) = (𝑓 ∘ 𝑔)(𝑛) = 𝑓(𝑔(𝑛))for any𝑓. True or false?
(a) If𝑓and𝑔are completely multiplicative, thenℎis completely multiplicative.
(b) If𝑓and𝑔are completely additive, thenℎis completely additive.
(c) If𝑓is multiplicative and𝑔is completely multiplicative, thenℎis multiplica- tive.
(d) If𝑓is completely multiplicative and𝑔is multiplicative, thenℎis multiplica- tive.
6. (a) Let 𝑓be completely additive. For which positive integers𝑘is the function 𝑔(𝑛) = 𝑓(𝑘𝑛)completely additive?
(b) Solve the problem for the case when we prescribe only additivity instead of complete additivity (for both of𝑓and𝑔).
(c) Investigate the variants for completely multiplicative and multiplicative func- tions.
S 7. (a) Show that if𝑓is completely additive, then
(A.6.1) 𝑓(𝑎) + 𝑓(𝑏) = 𝑓((𝑎, 𝑏)) + 𝑓([𝑎, 𝑏])holds for every𝑎and𝑏.
(b) Prove (A.6.1) for any additive𝑓.
* (c) Determine all functions𝑓satisfying (A.6.1).
* (d) Investigate also the corresponding equation𝑓(𝑎)𝑓(𝑏) = 𝑓((𝑎, 𝑏))𝑓([𝑎, 𝑏]).
8. Let𝑓be real valued and𝑔(𝑛) = 2𝑓(𝑛). Demonstrate that𝑔is multiplicative if and only if𝑓is additive.
Remark:This means that properties of additive functions assuming real values and of multiplicative functions assuming positive values can be mutually deduced from each other.
9. (a) Verify that both the sum and the difference of two additive functions are ad- ditive, and the same holds if “additive” is replaced by “completely additive.”
(b) Prove that the product of two completely additive functions is never com- pletely additive except in the trivial case when at least one of the factors is the0function.
(c) Give examples when the product of two≠ 0additive functions is (c1) additive (c2) not additive.
S* (d) Find all pairs of additive functions whose product is additive.
(e) Show that the product of two multiplicative functions is multiplicative, and the same holds if “multiplicative” is replaced by “completely multiplicative.”
(f) Verify that neither the sum nor the difference of two distinct≠ 0multiplicative functions can be multiplicative.
Exercises 6.1 169
10. (a) Show that the arithmetic mean of two additive or completely additive func- tions has the same property.
(b) Prove that if the arithmetic mean of two completely multiplicative functions is completely multiplicative then the two functions are equal. What happens if we require only multiplicativity instead of complete multiplicativity (for all three functions)?
11. Assume that𝑓is multiplicative,𝑔is additive, and𝑓 + 𝑔is constant. Show that 𝑓1000+ 𝑔1000is multiplicative and𝑓1000𝑔1000is additive.
* 12. Letℎbe an additive function.
(a) Prove that if ℎ is the difference of two multiplicative functions, then ℎ(𝑎)ℎ(𝑏)ℎ(𝑐) = 0for any pairwise coprime integers𝑎,𝑏, and𝑐.
(b) Ifℎhas only the trivial representation1 ⋅ ℎ = ℎas the product of a multiplica- tive and an additive function, thenℎ(𝑎)ℎ(𝑏)ℎ(𝑐) = 0for any pairwise coprime integers𝑎,𝑏, and𝑐.
S 13. (a) Assume that the range𝑅(𝑓)of an additive function𝑓is finite. Show that every 𝑐 ∈ 𝑅(𝑓)occurs infinitely often, i.e., there are infinitely many positive integers 𝑏satisfying𝑓(𝑏) = 𝑐.
(b) Give an example that shows that the same does not necessarily hold for mul- tiplicative functions.
(c) Assume that the range𝑅(𝑓)of a multiplicative function𝑓is finite and some 𝑑 ∈ 𝑅(𝑓)occurs only finitely many times, i.e.𝑓(𝑏) = 𝑑holds only for finitely many positive integers𝑏. Prove that there exists a𝐾such that𝑓(𝑛) = 0for every𝑛having a prime divisor greater than𝐾.
14. True or false?
(a) If𝑓is additive and𝑓(𝑎𝑏) = 𝑓(𝑎) + 𝑓(𝑏)for some𝑎and𝑏not coprime, then𝑓 is completely additive.
(b) If𝑓is additive and𝑓(𝑎𝑏) = 𝑓(𝑎) + 𝑓(𝑏)for some𝑎and𝑏not coprime, then there exist infinitely many such𝑎and𝑏.
(c) If𝑓is additive but not completely, then(𝑎, 𝑏) ≠ 1implies𝑓(𝑎𝑏) ≠ 𝑓(𝑎)+𝑓(𝑏).
(d) If𝑓is additive but not completely, then𝑓(𝑎𝑏) ≠ 𝑓(𝑎) + 𝑓(𝑏)for infinitely many𝑎and𝑏.
(e) If𝑓is multiplicative but not completely, then𝑓(𝑎𝑏) ≠ 𝑓(𝑎)𝑓(𝑏)for infinitely many𝑎and𝑏.
S* 15. Let 𝜑2(𝑛) denote the number of integers 𝑖 ∈ {1, 2, . . . , 𝑛} satisfying (𝑖, 𝑛) = (𝑖 + 1, 𝑛) = 1. Give a formula for𝜑2(𝑛)based on the standard form of𝑛.
* 16. Prove
∑
1≤𝑘≤𝑛 (𝑘,𝑛)=1
(𝑘 − 1, 𝑛) = 𝜑(𝑛)𝑑(𝑛).