On a finite sum of cosecants appearing in various problems

TL;DR

The work addresses finite sums of trigonometric functions, centering on the general cosecant sum $S_n(\varphi,a)$ and the secant sum $C_n(\varphi,a)$. It delivers integral and series representations, sharp asymptotic expansions for large $n$, and tight bounds, while unveiling deep links to the digamma and polygamma functions and to Gauss’s digamma theorem. A key finding is the qualitative dependence of the leading asymptotics on the parameters $\varphi$ and $a$, including regimes with sporadically large terms. The paper also offers historical context, unifies various regimes into coherent asymptotic formulas, and shows how these finite sums relate to classical results of Euler and later reformulations, with potential applications in analysis, number theory, and physics. Overall, it provides a comprehensive framework to understand, bound, and compute these finite trigonometric sums and their relatives.

Abstract

In this paper we investigate the finite sum of cosecants $\sum\csc\big(\varphi+aπl/n\big),$ where the index $l$ runs through 1 to $n-1$ and $\varphi$ and $a$ are arbitrary parameters, as well as several closely related sums, such as similar sums of a series of secants, of tangents and of cotangents. These trigonometric sums appear in various problems in mathematics, physics, and a variety of related disciplines. Their particular cases were fragmentarily considered in previous works, and it was noted that even a simple particular case $\sum\csc\big(πl/n\big)$ does not have a closed-form, i.e. a compact summation formula. In the paper, we derive several alternative representations for the above-mentioned sums, study their properties, relate them to many other finite and infinite sums, obtain their complete asymptotic expansions for large $n$ and provide accurate upper and lower bounds (e.g. the typical relative error for the upper bound is lesser than $2\times10^{-9}$ for $n\geqslant10$ and lesser than $7\times10^{-14}$ for $n\geqslant50$, which is much better than the bounds we could find in previous works). Our researches reveal that these sums are deeply related to several special numbers and functions, especially to the digamma function (furthermore, as a by-product, we obtain several interesting summations formulae for the digamma function). Asymptotical studies show that these sums may have qualitatively different behaviour depending on the choice of $\varphi$ and $a$; in particular, as $n$ increases some of them may become sporadically large. Finally, we also provide several historical remarks related to various sums considered in the paper. We show that some results in the field either were rediscovered several times or can easily be deduced from various known formulae, including some formulae dating back to the XVIIIth century.

Figures (5)

• Figure 1: A fragment of page 212 of euler_04, where Euler provides both cotangent summation theorems and states that similar relationships can be established for higher powers as well.
• Figure 2: A fragment of page 211 of euler_04, where Euler evaluates the alternating finite sum of cosecants, equivalent to our \ref{['972ryhxded']}.
• Figure 3: The sum $S_n(\varphi,a)$ as a function of $n$ for various $\varphi$ and $a$. Top: $S_n(\varphi,1)$ for $\varphi=2\ln2$ and $\varphi=0$; buttom: $S_n(0,a)$ for $a=2\ln2$ and $a=\ln2$. The value of the sum is given by the red point, the dashed black line is shown only for better visualisation. One may clearly observe very different behaviour of this sum, depending on the choice of the parameters $\varphi$ and $a$. Note also that $\ln{n}\notin\mathbbm{Q}$ if $n\in\mathbbm{N}.$
• Figure 4: The absolute error between $S_n$ and the approximations given by Theorems \ref{['ordtj56jx']} and \ref{['ordtj56jx2']} for $N=3$ and $N=4$. The black solid line corresponds to the approximation given by Theorem \ref{['ordtj56jx']}, the black dashed line corresponds to that given by Watson's approximation (Theorem \ref{['ordtj56jx2']}). Notice that the corresponding relative errors are even smaller, since $S_{10}\approx15.4$ and $S_{50}\approx129.$
• Figure 5: The differences between four different upper bounds for $S_n$ and the value of $S_n$. One may readily note that the bounds given by Theorem \ref{['oiehrhg4w5h']} (solid and dashed lines) are much more accurate than \ref{['08934cnu4']} obtained by Pomerance in 2011 (dotted line) and than \ref{['9872xychn2389']} obtained by Tong et al. in 2023 (dash-dotted line). Moreover, in Theorem \ref{['oiehrhg4w5h']}, the bounds provided by inequality \ref{['o4uincf87nb4']}, are more accurate than those provided by \ref{['8438394yrxn']}. In particular, the difference between the upper bound given in \ref{['o4uincf87nb4']} and $S_n$ is lesser than $3\times10^{-8}$ for $n\geqslant10$ and lesser than $8\times10^{-12}$ for $n\geqslant50$ (the relative differences are lesser than $2\times10^{-9}$ and $7\times10^{-14}$ respectively).

Theorems & Definitions (38)

• Remark 1: A representation related to Euler--like product
• Lemma 1: Improper integral representation
• proof
• Lemma 2: Series representation via a finite sum of cotangents
• proof
• Theorem 1: Digamma finite series representation, particular case
• proof
• Theorem 2: Digamma infinite series representation
• proof
• Theorem 3: Digamma finite series representations