Menon-type identities concerning additive characters
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Arabian Journal of Mathematics
László Tóth
Menon-type identities concerning additive characters
Received: 21 December 2018 / Accepted: 22 May 2019 © The Author(s) 2019
Abstract By considering even functions (mod n), we generalize a recent Menon-type identity by Li and Kim, involving additive characters of the group Zn . We use a different approach, based on certain convolutional identities. Some other applications, including related formulas for Ramanujan sums, are discussed as well. Mathematics Subject Classfication
11A07 · 11A25 · 65T50
1 Introduction Menon’s identity [5] states that for every n ∈ N, n
(a − 1, n) = ϕ(n)τ (n),
(1.1)
a=1 (a,n)=1
the notations used here and throughout the paper being fixed in Sect. 2. There are many generalizations and analogs of identity (1.1) in the literature. See, e.g., the papers [3,6–9,11] and the references therein. Let χ be a Dirichlet character (mod n) with conductor d, where n, d ∈ N, d | n. Zhao and Cao [11] derived the formula n (a − 1, n)χ (a) = ϕ(n)τ (n/d), (1.2) a=1
which recovers (1.1) if χ is the principal character (mod n), that is d = 1. The author [8, Theorem 2.4] generalized identity (1.2) into n a=1
f n (a − s)χ (a) = ϕ(n)χ ∗ (s)
(μ ∗ f n )(δd) , ϕ(δd)
(1.3)
δ|n/d (δ,s)=1
where f n is an even function (mod n), s ∈ Z, and χ ∗ is the primitive character (mod d) that induces χ . We recall that a function f n : Z → C, a → f n (a) is said to be an even function (mod n) if f n ((a, n)) = f n (a) holds for every a ∈ Z, where n ∈ N is fixed. Examples of even functions (mod n) are f n (a) = (a, n), more generally f n (a) = F((a, n)), where F : N → C is an arbitrary arithmetic function, and f n (a) = cn (a), representing Ramanujan’s sum. L. Tóth (B) Department of Mathematics, University of Pécs, Ifjúság útja 6, Pécs 7624, Hungary E-mail: [email protected]
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In the case f n (a) = cn (a), (1.3) gives ([8, Cor. 2.6]) n
cn (a − s)χ (a) = dϕ(n)χ ∗ (s)
δμ(n/(δd)) . ϕ(δd)
(1.4)
δ|n/d (δ,s)=1
a=1
Notice that in (1.2), (1.3) and (1.4), the sums on the left hand sides are, in fact, over 1 ≤ a ≤ n with (a, n) = 1, since χ (a) = 0 for (a, n) > 1. Here, (1.4) is a generalization of the first identity, due to Cohen [2, Eq. (5.1)], of formulas n δμ(n/δ) = μ(n)cn (s), cn (a − s) = ϕ(n) (1.5) ϕ(δ) δ|n (δ,s)=1
a=1 (a,n)=1
the second formula of (1.5) being the Brauer–Rademacher identity. See [2, Cor. 34] and [4, Ch. 2]. Recently, Li and Kim [3] investigated the sums S(n, k) :=
n
(a − 1, n)e(ak/n)
a=1
and
n
S ∗ (n, k) :=
(a − 1, n)e(ak/n),
(1.6)
a=1 (a,n)=1
by considering the additive characters a → e(ak/n) := e2πiak/n of the group Zn . According to [3, Theorem 2.1], for every n ∈ N and every k ∈ Z one has the identity S(n, k) = e(k/n) ϕ(n/), |(n,k)
previously known in the literature, as mentioned by the authors, which is related to the discrete Fourier transform (DFT) of the gcd function. To investigate the sums S ∗ (n, k) defined by (1.6), Li and Kim [3] showed first that these sums enjoy the modifie
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