Multiplicative functions arising from the study of mutually unbiased bases

We embed the somewhat unusual multiplicative function, which was serendipitously discovered in 2010 during a study of mutually unbiased bases in the Hilbert space of quantum physics, into two families of multiplicative functions that we construct as generalizations of that particular example. In addition, we report yet another multiplicative function, which is also suggested by that example; it can be used to express the squarefree part of an integer in terms of an exponential sum.


Multiplicative functions
We begin by recalling a few basic definitions and results. A real or complex valued function defined on the positive integers is called an arithmetic function or a numbertheoretic function.
For any two positive integers m and n, we use (m, n) to denote the greatest common divisor of m and n. An arithmetic function f is called multiplicative if f is not identically zero and if f (mn) = f (m)f (n) whenever (m, n) = 1.
It follows that f (1) = 1 if f is multiplicative.
In general, given an arithmetic function f , it is impossible to determine f (n) for a large positive integer n. However, if f is a multiplicative function, then we can evaluate f (n) provided that the factorization of n into prime powers and the formulas for the multiplicative function f at prime powers are known.
Euler's totient function defined by which is the count of positive integers that are less than n and coprime with n, is an important example of a multiplicative function. For a proof of the multiplicative property of ϕ(n) using the Chinese Remainder Theorem, see [4,Section 5.5]. It is known that ϕ(p α ) = p α − p α−1 for α ≥ 1 and consequently, if Theorem 1. If f and g are multiplicative, then f * g is multiplicative.
For a proof, see [1,Theorem 2.14]. As an application of Theorem 1, we have the following corollary, which appears as an exercise in [1, p. 49]: Corollary 1. Let f be a multiplicative function. Then the function and, therefore, Since f and ϕ are both multiplicative, we deduce by Theorem 1 that ξ f = f * ϕ is multiplicative, too.

A curious multiplicative function
In [3, Appendix C], T. Durt, B.-G. Englert, I. Bengtsson, and K.Życzkowski observed, during their study of the properties of mutually unbiased bases, the following interesting identity associated with the Gauss sum: Theorem 2. For any two integers m and n with 0 < m < n, and ζ n = e 2πi/n , one has They arrived at this result by linear-algebra arguments and without relying on the properties of Jacobi symbols and contour integrals that are usually employed when evaluating Gauss sums. This connection between linear algebra and number theory is noteworthy and, therefore, we revisit the matter in Section 7. In [3], they also indicated that is a multiplicative function. It is possible to simplify the expression of h(n). We first introduce v p (n), the count of prime number factors p contained in the positive integer n, defined by v p (n) = α whenever p α |n but p α+1 ∤ n.
Then we can, after applying Theorem 2, rewrite h(n) as Note that by Corollary 1, ξ s (n) = n k=1 (k, n) is multiplicative because s(n) = √ n has this property.
The function h(n) appears to be a new multiplicative function. One question we can ask is whether we can construct multiplicative functions when the prime 2 that is privileged in the definition of h(n) is replaced by any odd prime p and whether the function s(n) can be replaced by other arithmetic functions. It turns out that this is possible.

First generalization
As we shall confirm in Section 5, one generalization of h(n) is the following: Choose a multiplicative function f and a prime number p, as well as a sequence of complex numbers κ 0 = 0, κ 1 , κ 2 , . . . , and a sequence of positive integers a 1 , a 2 , . . . with a α ≤ α. Then the function Proof. If (m, n) = 1 and (mn, p) = 1, then by Corollary 1, In order to complete the proof, we need to show that if α > 0 and (ν, p) = 1, then We evaluate the sum in in a few steps, proceeding from where ⌊x⌋ denotes the floor of x, the largest integer that does not exceed x. Now, it is known that with the Möbius function where the p i s are distinct primes, 0 otherwise, which we exploit in Only d = 1 and d = p contribute to the sum, so that where it is crucial that a α ≥ 1, and we arrive at where the last step recognizes that h

Second generalization.
Here is another generalization of h(n), also confirmed in Section 5: Choose a multiplicative function f and a prime number p, as well as a sequence of complex numbers κ 0 = 0, κ 1 , κ 2 , . . . , and a sequence of nonnegative integers a 1 , a 2 , . . . with a α ≤ α. Then the function for α > 0 and (ν, p) = 1. We have where we recall, from the proof of Theorem 3, that

Generalizations confirmed
We now confirm that h (1) f,p (n) and h (2) f,p (n), introduced in Theorems 3 and 4, are generalizations of h(n) in (2).
In view of the important role played by the privileged prime p, we write n = p vp(n) p −vp(n) n and note that Therefore, we obtain h for all α ≥ 0. Since these assignments work for all permissible choices for the a α s, we have multiple generalizations of h(n) from both h (1) f,p (n) and h (2) (2) because a α = 0 is not allowed in Theorem 3. Clearly, then, (2) is just one of many ways of rewriting h(n) of (1).

f,p and vice versa
The f,p are both characterized by a sequence of κ α s and a sequence of a α s, and-for given f and p-one can always adjust the sequences of one of them to the sequences of the other such that the right-hand sides in (4) are the same, h f,p p α . In this sense, all the multiplicative functions in the h (1) f,p family are also contained in the h (2) f,p family, and vice versa, although the two mappings are really different.
To justify this remark, we shall write κ (1) α and a (1) α for the parameters that specify h (1) f,p and κ (2) α and a (2) α for those of h (2) f,p . Then, for a particular choice of the κ

A linear-algebra proof of Theorem 2
We revisit here the linear-algebra proof of Theorem 2. While we follow the reasoning in [3], where Theorem 2 is a side issue and the ingredients are widely scattered, the presentation here is self-contained and adopts somewhat simpler conventions, in particular for the phase factor in the definition of C m in (7) below.
Getting started: Columns and rows, matrices, eigenvector bases. We consider column vectors with n complex entries (n ≥ 2), their adjoint row vectors, and the n × n matrices that implement linear mappings of columns to columns and rows to rows. For any two columns x and y , we denote the adjoint rows by x † and y † ; a row-times-column product such as x † y is a complex number that can be understood as the inner product of the columns x and y , or of the rows x † and y † , whereby x † x ≥ 0 with "=" only for x = 0 . The row-times-column products such as yx † are n × n matrices with tr yx † = x † y for the matrix trace. We recall that (x † ) † = x , (x † y ) † = y † x , and (yx † ) † = xy † .
Following H. Weyl [6, Sec. IV.D.14] and J. Schwinger [5, Sec. 1.14], our basic ingredients are two related unitary n × n matrices A and B of period n, that is A k = 1 n , B k = 1 n if k ≡ 0 (mod n) and only then, where 1 n is the n × n unit matrix. The eigenvalues of A and B are the powers of ζ n , the basic nth root of unity that appears in Theorem 2. We denote the jth eigencolumn of A by a j and the kth eigencolumn of B by b k , where we regard the labels as modulo-n integers, so that a j+n = a j and b k+n = b k .
The sets of eigencolumns are orthonormal and complete, j,k is the modulo-n version of the Kronecker delta symbol, and the projection matrices associated with the eigenvectors are The unitary matrices A and B are related to each other by the discrete Fourier transform that turns one set of eigenvectors into the other, As a consequence, we have the following identities: we leave their verification to the reader as an exercise.
More unitary matrices and their eigenstate bases. For m = 1, 2, . . . , n − 1, we define which are unitary matrices of period n, Upon denoting the jth eigencolumn of C m by c m,j = c m,j+n , we have C m c m,j = c m,j ζ j n and c † m,j C m = ζ j n c † m,j as well as To establish how the c mj s are related to the a j s and b k s, we first infer c † m,j b k from the recurrence relation where we adopt c † m,j b 0 = 1/ √ n by convention. Then we exploit the completeness of the b k s in This Gauss sum is our first ingredient. Next we utilize x † y = tr yx † and the linearity of the trace as well as statements in (5) Now writing d = (m, n), n = dν, m = dµ with (µ, ν) = 1, we have because ml ≡ 0 (mod n) requires l ≡ 0 (mod ν). Therefore, only the terms with l = 0, ν, 2ν, . . . , (d − 1)ν contribute to the final sum in (10), and we arrive at Upon combining this second ingredient with the first in (9), we conclude that For j = k, this is the statement in Theorem 2.
In passing, we found the following identity between the absolute value of a Gauss sum and a particular partial sum: does not change if we replace m by m ± n, in conjunction with replacing k by k + 1 2 n if n is even, extends the permissible m values to all integers.
The identity can, of course, also be verified directly. We note that ζ kl n ζ (n−l)lm 2n does not change when l is replaced by l ± n and, therefore, the summation over l can cover any range of n successive integers. Accordingly, we can replace l by l + a for any integer a without changing the value of the sum. We exploit this when writing the left-hand side as a double sum and then processing it, Remark: Unbiased bases. The eigenvector bases for the matrices A and B are such that a † j b k has the same value for all rows a † j and columns b k , which is the defining property of a pair of unbiased bases. Further, for each m, the basis C m = {c m,k } n k=1 is unbiased with the basis B = {b k } n k=1 . When n is prime, each C m is also unbiased with the basis A = {a j } n j=1 . When n is not prime, however, then some of the C m s are unbiased with A, namely those with (m, n) = 1, and the others are not. Further, two bases C m and C m ′ with m ′ > m are unbiased if (m ′ − m, n) = 1, and only then, because c † m,j c m ′ ,k = a † j c m ′ −m,k . We refer the reader to [3] for a detailed discussion of unbiased bases.

Yet another multiplicative function
Here we report one more multiplicative function that is suggested by h(n) in (2), but is not a generalization in the spirit of h (1) f,p (n) and h (2) f,p (n) in Theorems 3 and 4.
In preparation, and for the record, we note that the Gauss sum in Theorem 2, can be evaluated. We write ν = n/(m, n) and µ = m/(m, n) as above and distinguish three cases: (a) If v 2 (n) = 0 or v 2 (m) > v 2 (n) > 0, then ν is odd and (n − 1)µ is even, and we have , where we separate the terms with ζ (n−l)l 2n = 1 from the others. Next, we note that n,0 δ We now proceed to show that h # (n) is multiplicative. We first consider the case of n odd. Since holds when n is odd, and (n − l)l ≡ −l 2 (mod n) for all n, we have δ Now write n = λν 2 , where λ is squarefree, or equivalently that |µ(λ)| = 1. The fact that n|l 2 implies that ν 2 |l 2 or ν|l. Therefore, if we write l = νs then n|l 2 implies that λ|s 2 . But if p|λ then p|s 2 and this implies that p|s. Since λ is squarefree, we conclude that λ|s and consequently, l = νλω for some positive integer ω. Since l ≤ n, we conclude that ω ≤ ν. This implies that n l=1 δ (n) We remark that (13) is true for any positive integer n. Combining (13) and (12), we deduce that Turning to n even now, we write n = 2m with m > 0 and observe that Next suppose either a or b is even. We may assume that a is even and b is odd. Write a/2 = λ a ν 2 a and b = λ b ν 2 b with squarefree integers λ a and λ b ; then and this completes the proof that h # (n) is multiplicative.
Three remarks. (i) Although the multiplicative function h # is constructed differently, and is another multiplicative function in this sense, it is also contained in the families h (1) f,p and h (2) f,p of Theorems 3 and 4, because every multiplicative function is in these families for a corresponding f . This follows from the fact that the mapping f → ξ f = f * ϕ is invertible, and we can choose κ α = 0 for all α. In particular, we have h # = ξ f # with the multiplicative function f # (n) specified by its prime-power values, that is for powers of 2, and h # p α = p α/2+⌊α/2⌋ , f # p α = 1 + p − p 1/2 p + 1 (−p) α − 1 for powers of odd primes, where α ≥ 0.
(ii) It is striking that the m-sums of both the absolute value and the real part of S(m, n) yield multiplicative functions after a suitable modification for even n. This makes us wonder if there are other functions of the pair m, n with this property.
(iii) Although, right now, we do not know truly useful applications of any particular multiplicative functions in the families h (1) f,p and h (2) f,p , it is worth recalling that h(n) of (1) is closely related to a prime-distinguishing function [3]. Similarly, the value of h # (n) tells us the squarefree factor λ in n = λν 2 , if both n and v 2 (n) are even, n h # (n) 2 otherwise, which is a corollary to the proof of Theorem 5. However, it is only the current lack of an efficient algorithm for the evaluation of the Gauss sum in (11) that prevents h(n) and h # (n) from being practical tools.

Summary
Inspired by the peculiar multiplicative function h(n) in (2), we found two mappings that turn a given multiplicative function into other multiplicative functions, with each image function specified by a privileged prime number, a sequence of complex numbers, and a sequence of nonnegative integers. In addition, we reported one more multiplicative function, of a different kind, also suggested by the structure of h(n).