Approximations by special values of multiple cosine and sine functions

Kurokawa and Koyama’s multiple cosine function ${\mathcal{C}}_r(x)$ and Kurokawa’s multiple sine function $S_r(x)$ are generalizations of the classical cosine and sine functions from their infinite product representations, respectively. For any fixed $x\in [0,\frac{1}{2})$, let $$B=\{\frac{\log {\mathcal{C}}_r(x)}{\pi }\mid r=1,2,3,\dots \}$$ and $$C=\{\frac{\log S_r(x)}{\pi }\mid r=1,2,3,\dots \}$$ be the sets of special values of ${\mathcal{C}}_r(x)$ and $S_r(x)$ at $x$, respectively. In this paper, we will show that the real numbers can be strongly approximated by linear combinations of elements in $B$ and $C$ respectively, with rational coefficients. Furthermore, let $$D=\{\frac{{\zeta }_E(3)}{{\pi }^2},\frac{{\zeta }_E(5)}{{\pi }^4},\dots ,\frac{{\zeta }_E(2k+1)}{{\pi }^{2k}},\dots ;\frac{\beta (4)}{{\pi }^3},\frac{\beta (6)}{{\pi }^5},\dots ,\frac{\beta (2k+2)}{{\pi }^{2k+1}},\dots \}$$ be the set of special values of Dirichlet’s eta and beta functions. We will prove that the set $D$ has a similar approximation property, where the coefficients are values of the derivatives of rational polynomials. Our approaches are inspired by recent works of Alkan (Proc. Amer. Math. Soc. 143: 3743–3752, 2015) and Lupu-Wu (J. Math. Anal. Appl. 545: Article ID 129144, 2025) as applications of the trigonometric integrals.