Errata for the Paper “Weighted Gcd-Sum Functions”

László Tóth Department of Mathematics University of Pécs Ifjúság u. 6 7624 Pécs Hungary

                  ltoth@gamma.ttk.pte.hu

The paper [1] contains some errors. Let

P_{binom}(n) := ∑_{k=1}^{n} \binom{n}{k} gcd(k, n) (n ∈ ℕ).

In identity (26) of [1, Prop. 9], in its proof (second to last line of page 6) and in identity (28) of [1] the exponent of −1 is ℓn/d instead of ℓ. I thank Max Alekseyev for pointing this out.

The correct form of (26) is the following. For every n ∈ ℕ,

P_{binom}(n) = 2^{n} ∑_{d|n} \frac{ϕ(d)}{d} ∑_{ℓ=1}^{d}(−1)^{ℓn/d} cos^{n}(ℓπ/d) − n.

The correct form of (28) is

R_{binom}(n) := ∑_{\begin{matrix}k=1 \\ gcd(k,n)=1 \end{matrix}}^{n} \binom{n}{k} = 2^{n} ∑_{d|n} \frac{μ(d)}{d} ∑_{ℓ=1}^{d}(−1)^{ℓn/d} cos^{n}(ℓπ/d) (n > 1).

Furthermore, in the right hand side of identity (35) of [1, Prop. 13] the term n is missing. The correct form of (35) is the following. For every n ∈ ℕ and α ∈ ℝ,

P_{floor}(n) := ∑_{k=1}^{n} ⌊α + \frac{k}{n}⌋ gcd(k, n) = ∑_{d|n} ϕ(d) ⌊\frac{nα}{d}⌋ + n.

Also, in the right hand side of identity (39) the term 1 is missing. The correct form of (39) is

∑_{k=1}^{n} ⌊α + \frac{k}{n}⌋ = ⌊nα⌋ + 1 (n ∈ ℕ).

References

[1] L. Tóth, Weighted gcd-sum functions, J. Integer Seq. 14 (2011), Article 11.7.7.