Approximations by special values of multiple cosine and sine functions

TL;DR

This paper demonstrates that real numbers can be strongly approximated by special values of multiple cosine, sine, eta, and beta functions, with rational or polynomial derivative coefficients, extending approximation theory using trigonometric integrals.

Contribution

It introduces new approximation properties of special function values, linking multiple cosine, sine, eta, and beta functions with rational and polynomial derivative coefficients.

Findings

Real numbers can be approximated by values in sets B and C.

Set D of special eta and beta function values also has approximation properties.

Approaches are inspired by recent works on trigonometric integrals.

Abstract

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 $$B=\left\{\frac{\log\mathcal{C}_{r}(x)}{\pi}~~\bigg|~~r=1,2,3,\ldots\right\}$$ and $$C=\left\{\frac{\log S_r(x)}{\pi}~~\bigg|~~r=1,2,3,\ldots\right\}$$ 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 $$D=\left\{\frac{\zeta_{E}(3)}{\pi^2},\frac{\zeta_{E}(5)}{\pi^4}, \ldots, \frac{\zeta_{E}(2k+1)}{\pi^{2k}},\ldots; \frac{\beta(4)}{\pi^3},\frac{\beta(6)}{\pi^5}, \ldots, \frac{\beta(2k+2)}{\pi^{2k+1}},\ldots\right\}$$ 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.