# On some Féjer-type trigonometric sums

R. B. Paris111E-mail address<EMAIL_ADDRESS>

Division of Computing and Mathematics,
Abertay University, Dundee DD1 1HG, UK

###### Abstract

We examine the four Féjer-type trigonometric sums of the form

$S_{n}(x)=\sum_{k=1}^{n}\frac{f(g(kx))}{k}\qquad(0<x<\pi)$

where $f(x)$, $g(x)$ are chosen to be either $\sin x$ or $\cos x$. The
analysis of the sums with $f(x)=g(x)=\cos x$, $f(x)=\cos x$, $g(x)=\sin x$ and
$f(x)=\sin x$, $g(x)=\cos x$ is reasonably straightforward. It is shown that
these sums exhibit unbounded growth as $n\to\infty$ and also present ‘spikes’
in their graphs at certain $x$ values for which we give an explanation.

The main effort is devoted to the case $f(x)=g(x)=\sin x$, where we present
arguments that strongly support the conjecture made by H. Alzer that
$S_{n}(x)>0$ in $0<x<\pi$. The graph of the sum in this case presents a jump
in the neighbourhood of $x=\mbox{${\textstyle\frac{2}{3}}$}\pi$. This jump is
explained and is quantitatively estimated when $n\to\infty$.

MSC: 33B10, 41A30, 42A32

Keywords: Féjer sums, sine and cosine sums, trigonometric sums, asymptotics

1\. Introduction

In 1910 Féjer conjectured that all partial sums of the series
$\sum_{k=1}^{\infty}\frac{\sin kx}{k}=\mbox{${\textstyle\frac{1}{2}}$}(\pi-x)$
in the interval $0<x\leq\pi$ are positive, namely

$s_{n}(x)=\sum_{k=1}^{n}\frac{\sin kx}{k}>0\qquad n\geq 1,\ 0<x<\pi.$ (1.1)

Proofs of this result were given by Jackson [4], Gronwall [3], Landau [5],
Féjer [2] and Turán [6]. A similar result for the cosine series is [8]

$c_{n}(x)=\sum_{k=1}^{n}\frac{\cos kx}{k}>-1\qquad n\geq 1,\ 0<x<\pi.$ (1.2)

Extensions of the above trigonometric sums considered here have the form

$S_{n}^{(1)}(x)=\sum_{k=1}^{n}\frac{\cos(\cos kx)}{k},\qquad
S_{n}^{(2)}(x)=\sum_{k=1}^{n}\frac{\cos(\sin kx)}{k},$
$S_{n}^{(3)}(x)=\sum_{k=1}^{n}\frac{\sin(\cos kx)}{k},\qquad
S_{n}^{(4)}(x)=\sum_{k=1}^{n}\frac{\sin(\sin kx)}{k}.$ (1.3)

Recently, Alzer [1] made the following conjecture concerning $S_{n}^{(4)}(x)$
based on numerical evidence:

Conjecture. For all positive integer $n$ and $0<x<\pi$ we have

$S_{n}^{(4)}(x)=\sum_{k=1}^{n}\frac{\sin(\sin kx)}{k}>0,$ (1.4)

where the constant lower bound is sharp.

The investigation of the sums $S_{n}^{(j)}(x)$, $1\leq j\leq 3$ is relatively
straightforward and will be dealt with in Section 3. Our main effort is
concentrated on the sum $S_{n}^{(4)}(x)$ and is considered in Section 4. We do
not claim to establish a proof of the above conjecture but we present
arguments that strongly suggest its validity. For this purpose, we shall make
use of the identities [7, p. 226]

$\left.\begin{array}[]{l}\sin(z\cos\theta)={\displaystyle
2\sum_{r=0}^{\infty}(-)^{r}J_{2r+1}(z)\cos((2r+1)\theta)},\\\ \\\
\sin(z\sin\theta)={\displaystyle
2\sum_{r=0}^{\infty}J_{2r+1}(z)\sin((2r+1)\theta)}\end{array}\right\\}$ (1.5)

valid for $z$, $\theta\in{\bf C}$, where $J_{\nu}(z)$ denotes the Bessel
function of the first kind, together with some basic properties of the sine
and cosine series $s_{n}(x)$ and $c_{n}(x)$ in (1.1) and (1.2), which we
present in the next section.

2\. Properties of the basic series $s_{n}(x)$ and $c_{n}(x)$

In this section we present some basic properties of the simple sine and cosine
series $s_{n}(x)$ and $c_{n}(x)$ defined in (1.1) and (1.2). Both these series
are periodic functions of $x$ with period $2\pi$. Typical plots of these
functions are illustrated in Fig. 1.

($a$) ($b$)

($c$) ($d$)

Figure 1: Plots of (a) $s_{n}(x)$, (b) $s_{n}(3x)$ for $0\leq x\leq\pi$, (c)
$c_{n}(x)$ for $0\leq x\leq 2\pi$ and (d) $c_{n}(3x)$ for $0\leq x\leq\pi$
when $n=20$.

2.1 Properties of $s_{n}(x)$

Some straightforward properties of $s_{n}(x)$ are:

(i) Since $s_{n}^{\prime}(x)=\sum_{k=1}^{n}\cos kx$, we have the derivatives

$s_{n}^{\prime}(0)=n,\qquad
s_{n}^{\prime}(\pi)=\sum_{k=1}^{n}(-)^{k}=\left\\{\begin{array}[]{ll}0&(n\
\mbox{even})\\\ -1&(n\ \mbox{odd}).\end{array}\right.$ (2.1)

It is seen that the curves $s_{n}(x)$ approach the endpoint $x=\pi$ in
different manners according to the parity of $n$.

(ii) With $p$ a positive integer, the functions $s_{n}(px)$ approach the
sawtooth function as $n\to\infty$, with positive and negative parts in
$(0,\pi)$ when $p\geq 2$. The extreme values are approximately $\pm 1.852$ as
$n\to\infty$ due to the well-known Gibbs phenomenon. In Section 4, we shall
take the extreme values to be

$\xi:=\sup\\{\pm s_{n}(x):x\in(0,2\pi)\\}=2.$ (2.2)

(iii) For odd integer $p\geq 3$, the functions $s_{n}(px)$ are positive in the
intervals

$(0,\pi/p),\ \ (2\pi/p,\,3\pi/p),\ \
(4\pi/p,\,5\pi/p),\,\ldots,\,((p-1)\pi/p,\,\pi).$ (2.3)

More recondite properties resulting from detailed examination of the curves
$s_{n}(x)$ for different $n$ (which we do not show here) are:

(iv) For $n\geq 2$, the minimum value of $s_{n}(x)$ in the interval
$[\mbox{${\textstyle\frac{1}{3}}$}\pi,\mbox{${\textstyle\frac{2}{3}}$}\pi]$ is
given by

$s_{2}(\mbox{${\textstyle\frac{2}{3}}$}\pi)=s_{3}(\mbox{${\textstyle\frac{2}{3}}$}\pi)=\frac{\sqrt{3}}{4}~{}.$
(2.4)

(v) For $n\geq 2$, the lowest curve in the intervals $(0,\pi\lambda)$ and
$(\mbox{${\textstyle\frac{2}{3}}$}\pi,\pi)$ is

$s_{2}(x)=\sin x(1+\cos x).$ (2.5)

The quantity $\lambda$, which results from the first intersection (measured
from $x=0$) of the curves corresponding to $n=2$ and $n=9$, has the value
$\lambda\doteq 0.207685$.

2.2 Properties of $c_{n}(x)$

From the periodicity of $c_{n}(x)$ we have

$c_{n}(2\pi k)=\sigma_{n}\qquad(k=0,1,2,\ldots),$ (2.6)

where

$\sigma_{n}:=\sum_{k=1}^{n}\frac{1}{k}=\psi(n+1)+\gamma,$ (2.7)

with $\psi(x)=\Gamma^{\prime}(x)/\Gamma(x)$ and $\gamma=0.5772156\ldots$ being
the Euler-Mascheroni constant. In addition, we have the symmetry property

$c_{n}(x)=c_{n}(2\pi-x)\qquad(0\leq x\leq 2\pi).$ (2.8)

The cosine series is then easily seen to satisfy the inequality

$-1<c_{n}(x)\leq\sigma_{n}\qquad(x>0).$ (2.9)

Typical plots of $c_{n}(x)$ and $c_{n}(3x)$ are shown in Fig. 1.

From the large-$x$ behaviour $\psi(x)\sim\log\,x-\frac{1}{2x}+O(x^{-2})$ [7,
p. 140], we see that

$\sigma_{n}=\log\,n+\gamma+\frac{1}{2n}+O(n^{-2})\qquad(n\to\infty).$ (2.10)

Hence it follows that

$c_{n}(2\pi k)=\log\,n+\gamma+O(n^{-1})\qquad(k=0,1,2,\ldots;\ n\to\infty).$

Some specific evaluations of $c_{n}(x)$ will be useful in what follows. When
$x=\pi$ we have the value

$c_{n}(\pi)=\sum_{k=1}^{n}\frac{(-)^{k}}{k}=\psi(\mbox{${\textstyle\frac{1}{2}}$}n+1-\mbox{${\textstyle\frac{1}{2}}$}r)-\psi(n+1),$

where $r=0$ ($n$ even) and $r=1$ ($n$ odd). The value of
$c_{n}(\mbox{${\textstyle\frac{2}{3}}$}\pi)$ is given by

$c_{n}(\mbox{${\textstyle\frac{2}{3}}$}\pi)=\sum_{k=1}^{n}\frac{\cos(\mbox{${\textstyle\frac{2}{3}}$}\pi
k)}{k},$

which, when $n=3m+r$, $r=0,1,2$, yields

$\displaystyle c_{n}(\mbox{${\textstyle\frac{2}{3}}$}\pi)$ $\displaystyle=$
$\displaystyle\frac{1}{3}\biggl{(}1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{m}\biggr{)}-\frac{1}{2}\biggl{(}1+\frac{1}{2}+\frac{1}{4}+\frac{1}{5}+\cdots+\frac{1}{3m+r}\biggr{)}$
(2.11) $\displaystyle=$
$\displaystyle\frac{1}{2}\biggl{(}1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{m}\biggr{)}-\frac{1}{2}\biggl{(}1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{3m+r}\biggr{)}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\biggl{\\{}\psi(\mbox{${\textstyle\frac{1}{3}}$}n+1-\mbox{${\textstyle\frac{1}{3}}$}r)-\psi(n+1)\biggr{\\}}.$

Consequently, we find the large-$n$ behaviour

$c_{n}(\mbox{${\textstyle\frac{2}{3}}$}\pi)=-\frac{1}{2}\log\,3+O(n^{-1}),\qquad
c_{n}(\pi)=-\log\,2+O(n^{-1})\qquad(n\to\infty).$ (2.12)

3\. The trigonometric sums $S_{n}^{(j)}(x)$ ($j=1,2,3$)

In Fig. 2 we show typical plots of the trigonometric sums $S_{n}^{(j)}(x)$ for
$1\leq j\leq 4$ and $0\leq x\leq\pi$. We note that the sums $S_{n}^{(1)}(x)$
and $S_{n}^{(2)}(x)$ are symmetrical about
$x=\mbox{${\textstyle\frac{1}{2}}$}\pi$, whereas the other sums do not possess
this symmetry. In Fig. 2(a, b) the appearance of a ‘spike’ at
$x=\mbox{${\textstyle\frac{1}{2}}$}\pi$ is just visible when $n=200$; this
‘spike’ becomes more pronounced as $n$ increases. A ‘spike’ is also visible
near $x=\mbox{${\textstyle\frac{2}{3}}$}\pi$ in the graph of $S_{n}^{(3)}(x)$
in Fig. 2(c). The graph of $S_{n}^{(4)}(x)$, however, exhibits a finite jump
near $x=\mbox{${\textstyle\frac{2}{3}}$}\pi$; this is considered in detail in
Section 4. All the curves consist of tiny undulations that are too small to be
visible on the scale of the figures.

($a$) ($b$)

($c$) ($d$)

Figure 2: Plots of the Féjer-type sums for $0\leq x\leq\pi$: (a)
$S_{n}^{(1)}(x)$, (b) $S_{n}^{(2)}(x)$, (c) $S_{n}^{(3)}(x)$ when $n=200$ and
(d) $S_{n}^{(4)}(x)$ when $n=100$.

3.1 The sum $S_{n}^{(1)}(x)$

The sum $S_{n}^{(1)}(x)$ satisfies $S_{n}^{(1)}(0)=\sigma_{n}\cos(1)$ and,
since $\cos kx\in[-1,1]$, it readily follows that

$S_{n}^{(1)}(x)=\sum_{k=1}^{n}\frac{\cos(\cos
kx)}{k}\geq\cos(1)\sum_{k=1}^{n}\frac{1}{k}=\sigma_{n}\cos(1).$ (3.1)

It is found from numerical calculations when $n$ is even and $n\geq 18$ that
the maximum value of $S_{n}^{(1)}(x)$ occurs at
$x=\mbox{${\textstyle\frac{1}{2}}$}\pi$; the maximum value also occurs at
$x=\mbox{${\textstyle\frac{1}{2}}$}\pi$ for all odd values of $n$. When $n$ is
even and $n\leq 16$ the maximum value of $S_{n}^{(1)}(x)$ occurs at the
neighbouring local maxima situated at

$x=\frac{1}{2}\pi\pm\frac{\pi}{2n+2}.$

In Fig. 3(a) we show details of the ‘spike’ occurring in $S_{n}^{(1)}(x)$ near
$x=\mbox{${\textstyle\frac{1}{2}}$}\pi$ that becomes progreesively more
pronounced as $n$ increases.

($a$) ($b$)

Figure 3: Plots of $S_{n}^{(j)}(x)$, $j=1,2$ for $x$ near
$\mbox{${\textstyle\frac{1}{2}}$}\pi$ when $n=1000$.

To determine the height of the ‘spike’ we consider the value of
$S_{n}^{(1)}(x)$ at $x=\mbox{${\textstyle\frac{1}{2}}$}\pi$. When $n=2m$ is an
even integer, we have

$\displaystyle S_{n}^{(1)}(\mbox{${\textstyle\frac{1}{2}}$}\pi)$
$\displaystyle=$ $\displaystyle
1+\frac{1}{3}+\frac{1}{5}+\cdots+\frac{1}{2m-1}+\frac{\cos(1)}{2}\biggl{(}1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{m}\biggr{)}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\psi(\mbox{${\textstyle\frac{1}{2}}$}n+\mbox{${\textstyle\frac{1}{2}}$})+\frac{1}{2}\gamma+\log\,2+\frac{\cos(1)}{2}\biggl{\\{}\psi(\mbox{${\textstyle\frac{1}{2}}$}n+1)+\gamma\biggr{\\}}\qquad(n\
\mbox{even})$

and in a similar manner when $n$ is an odd integer

$S_{n}^{(1)}(\mbox{${\textstyle\frac{1}{2}}$}\pi)=\frac{1}{2}\psi(\mbox{${\textstyle\frac{1}{2}}$}n+1)+\frac{1}{2}\gamma+\log\,2+\frac{\cos(1)}{2}\biggl{\\{}\psi(\mbox{${\textstyle\frac{1}{2}}$}n+\mbox{${\textstyle\frac{1}{2}}$})+\gamma\biggr{\\}}\qquad(n\
\mbox{odd}).$

Thus we have the large-$n$ behaviour

$S_{n}^{(1)}(\mbox{${\textstyle\frac{1}{2}}$}\pi)\sim\frac{(1+\cos(1))}{2}\,\log\,(\mbox{${\textstyle\frac{1}{2}}$}n)\qquad(n\to\infty).$
(3.2)

Thus the height of the ‘spike’ increases without limit as $n\to\infty$ and
consequently there is no finite upper bound for the sum $S_{n}^{(1)}(x)$.

3.2 The sum $S_{n}^{(2)}(x)$

The sum $S_{n}^{(2)}(x)$ satisfies $S_{n}^{(2)}(0)=\sigma_{n}$ and, since
$\sin kx\in[-1,1]$, it readily follows that

$S_{n}^{(2)}(x)=\sum_{k=1}^{n}\frac{\cos(\sin
kx)}{k}>\cos(1)\sum_{k=1}^{n}\frac{1}{k}=\sigma_{n}\cos(1).$ (3.3)

Plots of $S_{n}^{(2)}(x)$ are shown in Figs. 2(b) and 3(b), where when $n=200$
the central spike at $x=\mbox{${\textstyle\frac{1}{2}}$}\pi$ is just visible,
whereas when $n=1000$ it is more prominent. When $n=2m$ is an even integer,
the value at $x=\mbox{${\textstyle\frac{1}{2}}$}\pi$ is given by

$\displaystyle S_{n}^{(2)}(\mbox{${\textstyle\frac{1}{2}}$}\pi)$
$\displaystyle=$
$\displaystyle\cos(1)\biggl{\\{}1+\frac{1}{3}+\frac{1}{5}+\cdots+\frac{1}{2m-1}\biggr{\\}}+\frac{1}{2}\biggl{\\{}1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{m}\biggr{\\}}$
$\displaystyle=$
$\displaystyle\cos(1)\biggl{\\{}\frac{1}{2}\psi(\mbox{${\textstyle\frac{1}{2}}$}n+\mbox{${\textstyle\frac{1}{2}}$})+\frac{1}{2}\gamma+\log\,2\biggr{\\}}+\frac{1}{2}\biggl{\\{}\psi(\mbox{${\textstyle\frac{1}{2}}$}n+1)+\gamma\biggr{\\}}\qquad(n\
\mbox{even})$

and similarly

$S_{n}^{(2)}(\mbox{${\textstyle\frac{1}{2}}$}\pi)=\cos(1)\biggl{\\{}\frac{1}{2}\psi(\mbox{${\textstyle\frac{1}{2}}$}n+1)+\frac{1}{2}\gamma+\log\,2\biggr{\\}}+\frac{1}{2}\biggl{\\{}\psi(\mbox{${\textstyle\frac{1}{2}}$}n+\mbox{${\textstyle\frac{1}{2}}$})+\gamma\biggr{\\}}\qquad(n\
\mbox{odd}).$

Thus, we have the large-$n$ behaviour of the central spike given by

$S_{n}^{(2)}(\mbox{${\textstyle\frac{1}{2}}$}\pi)\sim\frac{(1+\cos(1))}{2}\,\log\,(\mbox{${\textstyle\frac{1}{2}}$}n)\qquad(n\to\infty),$
(3.4)

which is asymptotically less than the value
$S_{n}^{(2)}(0)=\sigma_{n}\sim\log\,n$. This leads us to the conjecture based
on the above estimate of the central spike and numerical evidence that

$\sigma_{n}\cos(1)<S_{n}^{(2)}(x)\leq\sigma_{n}\qquad(0\leq x\leq\pi).$ (3.5)

3.3 The sum $S_{n}^{(3)}(x)$

The sum $S_{n}^{(3)}(x)$ satisfies $S_{n}^{(3)}(0)=\sigma_{n}\sin(1)$ and

$S_{n}^{(3)}(\pi)=\sum_{k=0}^{n}\frac{\sin(\cos\pi
k)}{k}=\sin(1)\sum_{k=1}^{n}\frac{(-)^{k}}{k}=\sin(1)\biggl{\\{}\psi(\mbox{${\textstyle\frac{1}{2}}$}n+1-\mbox{${\textstyle\frac{1}{2}}$}r)-\psi(n+1)\biggr{\\}},$

where $r=0$ ($n$ even) and $r=1$ ($n$ odd). Thus for large $n$ we find

$S_{n}^{(3)}(\pi)\sim-\sin(1)\,\log\,2\doteq-0.583263.$

A typical plot of $S_{n}^{(3)}(x)$ is shown in Fig. 2(c), where it is seen
that there is a negative spike situated near
$x=\mbox{${\textstyle\frac{2}{3}}$}\pi$.

Routine algebra shows that when $n=3m+r$, $m=0,1,2,\ldots$ and for $r=0,1,2$

$\displaystyle S_{n}^{(3)}(\mbox{${\textstyle\frac{2}{3}}$}\pi)$
$\displaystyle=$
$\displaystyle\frac{1}{3}\sin(1)\biggl{(}1+\frac{1}{2}+\cdots+\frac{1}{m}\biggr{)}-\sin(\mbox{${\textstyle\frac{1}{2}}$})\biggl{(}1+\frac{1}{2}+\frac{1}{4}+\frac{1}{5}+\cdots+\frac{1}{3m-1}+\sum_{k=1}^{r}\frac{1}{3m+k}\biggr{)}$
$\displaystyle=$
$\displaystyle\frac{1}{3}(\sin(\mbox{${\textstyle\frac{1}{2}}$})+\sin(1))\biggl{(}1+\frac{1}{2}+\cdots+\frac{1}{m}\biggr{)}-\sin(\mbox{${\textstyle\frac{1}{2}}$})\sum_{k=1}^{3m+r}\frac{1}{k}$
$\displaystyle=$
$\displaystyle\frac{1}{3}(\sin(\mbox{${\textstyle\frac{1}{2}}$})+\sin(1))\biggl{(}\psi(\frac{n}{3}+1-\frac{r}{3})+\gamma\biggr{)}-\sin(\mbox{${\textstyle\frac{1}{2}}$})\biggl{(}\psi(n+1)+\gamma\biggr{)}.$

Thus, we obtain the large-$n$ behaviour

$S_{n}^{(3)}(\mbox{${\textstyle\frac{2}{3}}$}\pi)\sim\frac{1}{3}\biggl{\\{}\sin(1)-2\sin(\mbox{${\textstyle\frac{1}{2}}$})\biggr{\\}}\log\,n\sim-0.0391267\log\,n\qquad(n\to\infty).$
(3.6)

This result shows that the negative spike grows without bound as $n\to\infty$.
For example, when $n=4\times 10^{5}$ we find the value
$S_{n}^{(3)}(\mbox{${\textstyle\frac{2}{3}}$}\pi)\doteq-1.011006$.

We now give a qualitative explanation for the appearance of the negative spike
in $S_{n}^{(3)}(x)$ at $x=\mbox{${\textstyle\frac{2}{3}}$}\pi$, which is just
visible in Fig. 2(c) when $n=200$. From the first relation in (1.5) we have

$S_{n}^{(3)}(x)=2\sum_{r=0}^{\infty}(-)^{r}J_{2r+1}(1)\,c_{n}((2r+1)x),$ (3.7)

where $c_{n}(x)$ is defined in (1.2). The Bessel function satisfies
$J_{\nu}(1)>0$ for $\nu\geq 0$ and decreases rapidly with increasing $\nu$
given by the asymptotic formula [7, (10.19.1)]

$J_{\nu}(1)\sim\frac{1}{\sqrt{\pi
e}}\biggl{(}\frac{2\nu}{e}\biggr{)}^{\\!\\!-\nu-1/2}\qquad(\nu\to+\infty).$

The first few values of $J_{2r+1}(1)$ are:

$J_{1}(1)\doteq 0.4400510,\quad J_{3}(1)\doteq 0.0195634,\quad J_{5}(1)\doteq
2.497577\times 10^{-4},$ $J_{7}(1)\doteq 1.502326\times 10^{-6},\quad
J_{9}(1)\doteq 5.249250\times 10^{-9},\ldots\ .$

Whenever the argument of the $c_{n}(x)$ function in (3.7) equals a multiple of
$2\pi$ (where its value is $\sigma_{n}$) we can expect a ‘spike’ in the graph
of $S_{n}^{(3)}(x)$ as $n\to\infty$. Thus when
$x=\mbox{${\textstyle\frac{2}{3}}$}\pi$, we have
$c_{n}(3x)=c_{n}(9x)=c_{n}(15x)=\ldots=\sigma_{n}$ and, using the periodic and
symmetry properties of $c_{n}(x)$ in (2.8), we obtain

$S_{n}^{(3)}(\mbox{${\textstyle\frac{2}{3}}$}\pi)=-2\sigma_{n}\biggl{\\{}J_{3}(1)\\!-\\!J_{9}(1)\\!+\\!J_{15}(1)\\!-\cdots\biggr{\\}}+2c_{n}(\mbox{${\textstyle\frac{2}{3}}$}\pi)\biggl{\\{}J_{1}(1)\\!+\\!J_{5}(1)\\!-\\!J_{7}(1)\\!-\\!J_{11}(1)\\!+\cdots\biggr{\\}}.$

From (2.12), $c_{n}(\mbox{${\textstyle\frac{2}{3}}$}\pi)=O(1)$ as
$n\to\infty$, so that

$S_{n}^{(3)}(\mbox{${\textstyle\frac{2}{3}}$}\pi)=-2\sigma_{n}\biggl{\\{}J_{3}(1)\\!-\\!J_{9}(1)\\!+\\!J_{15}(1)\\!-\cdots\biggr{\\}}+O(1)\sim-0.0386302\,\sigma_{n}\qquad(n\to\infty),$

which results in an increasingly negative spike as $n\to\infty$.

A similar argument shows that when $x=\mbox{${\textstyle\frac{2}{5}}$}\pi,\
\mbox{${\textstyle\frac{4}{5}}$}\pi$, where
$c_{n}(5x)=c_{n}(15x)=c_{n}(25x)=\ldots=\sigma_{n}$, we have

$\displaystyle S_{n}^{(3)}(\mbox{${\textstyle\frac{2}{5}}$}\pi)$
$\displaystyle=$ $\displaystyle
2\sigma_{n}\biggl{\\{}J_{5}(1)-J_{15}(1)+J_{25}(1)-\cdots\biggr{\\}}+O(1)$
$\displaystyle\sim$ $\displaystyle+0.000499515\sigma_{n}\qquad(n\to\infty).$

This results in the appearance of positive spikes in the neighbourhood of
$x=\mbox{${\textstyle\frac{2}{5}}$}\pi,\ \mbox{${\textstyle\frac{4}{5}}$}\pi$
as $n\to\infty$, which have been observed in numerical calculations (which we
do not present here). Other spikes will appear, for example at
$x=\mbox{${\textstyle\frac{2}{7}}$}\pi$, $\mbox{${\textstyle\frac{4}{7}}$}\pi$
and $\mbox{${\textstyle\frac{6}{7}}$}\pi$, but require increasingly large
values of $n$ for them to be visible.

4\. The trigonometric sum $S_{n}^{(4)}(x)$

We now examine the central case of $S_{n}^{(4)}(x)$ defined by

$S_{n}^{(4)}(x)=\sum_{k=1}^{n}\frac{\sin(\sin kx)}{k}\qquad(0<x<\pi).$

The aim is to present an argument that strongly supports the conjecture (1.4)
and also explains and quantifies the jump in the graphs of $S_{n}^{(4)}(x)$ in
the vicinity of $x=\mbox{${\textstyle\frac{2}{3}}$}\pi$; see Fig. 2(d). It is
evident that $S_{1}^{(4)}(x)>0$ for $0<x<\pi$ so that we can take $n\geq 2$
throughout. Application of the second equation in (1.5) shows that
$S_{n}^{(4)}(x)$ can be expressed as the infinite sum involving unit-argument
Bessel functions in the form

$S_{n}^{(4)}(x)=2\sum_{r=0}^{\infty}J_{2r+1}(1)\,s_{n}((2r+1)x),$ (4.1)

where the sine series $s_{n}(x)$ is defined in (1.1).

Let us define for non-negative integer $m$

$F_{m}(n;x)=\sum_{k=0}^{m}J_{2k+1}(1)s_{n}((2k\\!+\\!1)x),\qquad
T_{m}(n;x)=\sum_{k=m+1}^{\infty}J_{2k+1}(1)s_{n}((2k\\!+\\!1)x),$

so that

$S_{n}^{(4)}(x)=2\\{F_{m}(n;x)+T_{m}(n;x)\\}\qquad(m=0,1,2,\ldots).$ (4.2)

The tail $T_{m}(n;x)$ of the series consists of positive and negative
contributions. This can be majorated by taking the most negative value
possible for $s_{n}(x)$ given by $-\xi=-2$; see (2.2). Thus the most negative
value of $T_{m}(n;x)$ is

$\xi\\!\\!\sum_{k=m+1}^{\infty}J_{2k+1}(1)<\sum_{k=m+1}^{\infty}\frac{2^{-2k}}{(2k+1)!}$

upon use of the result [7, (10.14.4)] $J_{\nu}(1)<2^{-\nu}/\Gamma(1+\nu)$ for
$\nu>-\mbox{${\textstyle\frac{1}{2}}$}$.

If we define the bound $B_{m}$ by

$B_{m}:=\sum_{k=m+1}^{\infty}\frac{2^{-2k}}{(2k+1)!}=2\sinh\mbox{${\textstyle\frac{1}{2}}$}-\sum_{k=0}^{m}\frac{2^{-2k}}{(2k+1)!},$
(4.3)

then we obtain

$\hskip
85.35826ptS_{n}^{(4)}(x)>2\\{F_{m}(n;x)-B_{m}\\}\qquad(m=0,1,2,\ldots).$ (4.4)

The values of $B_{m}$ decrease monotonically with $m$ and are given
asymptotically by

$B_{m}\sim\frac{2^{-2m-2}}{(2m+3)!}\\{1+O(m^{-2})\\}\qquad(m\to\infty).$ (4.5)

Values of $B_{m}$ for $1\leq m\leq 10$ are shown in Table 1 from which it is
apparent that they decrease very rapidly.

We have the following two lemmas:

###### Lemma 1

$\\!\\!\\!.$ The quantity $B_{m}$ satisfies the upper bound

$B_{m-1}<\frac{{\cal H}_{m}}{2m+1},\qquad{\cal
H}_{m}:=\frac{2^{-2m}}{(2m)!}\,\frac{16m^{2}}{16m^{2}-1}\quad(m\geq 2).$ (4.6)

Furthermore we have the inequality

$J_{1}(1)\sin\frac{\pi}{2m+1}>2B_{m-1}\qquad(m\geq 2).$ (4.7)

Proof. From the definition (4.3) we have for $m\geq 2$

$\displaystyle B_{m-1}$ $\displaystyle=$
$\displaystyle\sum_{k=m}^{\infty}\frac{2^{-2k}}{(2k+1)!}=\frac{2^{-2m}}{(2m+1)!}\biggl{\\{}1+\frac{2^{-2}}{(2m\\!+\\!2)(2m\\!+\\!3)}+\frac{2^{-4}}{(2m\\!+\\!2)\ldots(2m\\!+\\!5)}+\cdots\biggr{\\}}$
$\displaystyle<$
$\displaystyle\frac{2^{-2m}}{(2m+1)!}\sum_{r=0}^{\infty}(4m^{2})^{-r}=\frac{2^{-2m}}{(2m+1)!}\,\frac{16m^{2}}{16m^{2}-1},$

thereby establishing (4.6).

The inequality (4.7) can be verified by using the lower bound $\sin\theta\geq
2\theta/\pi$ for $\theta\in[0,\mbox{${\textstyle\frac{1}{2}}$}\pi]$ and the
upper bound for $B_{m-1}$ in (4.6). Then (4.7) is satisfied if

$J_{1}(1)\,\frac{2}{\pi}\,\frac{\pi}{2m+1}>\frac{2{\cal H}_{m}}{2m+1},$

which yields $J_{1}(1)>{\cal H}_{m}$. Since $J_{1}(1)\doteq 0.44005$ and
${\cal H}_{m}\leq{\cal H}_{2}=\mbox{${\textstyle\frac{1}{378}}$}$ ($m\geq 2$)
this is seen to be satisfied.

###### Lemma 2

$\\!\\!\\!.$ When $m=1$ the sum $F_{1}(n;x)>0$ in $(0,\pi)$.

Proof. With $m=1$ we have

$F_{1}(n;x)=J_{1}(1)s_{n}(x)+J_{3}(1)s_{n}(3x).$ (4.8)

Now, since $s_{n}(x)>0$ in $(0,\pi)$ by (1.1) and $s_{n}(3x)>0$ in
$(0,\mbox{${\textstyle\frac{1}{3}}$}\pi)$ and
$(\mbox{${\textstyle\frac{2}{3}}$}\pi,\pi)$, it follows that $F_{1}(n;x)>0$ in
$(0,\mbox{${\textstyle\frac{1}{3}}$}\pi)\cup(\mbox{${\textstyle\frac{2}{3}}$}\pi,\pi)$.
In the interval
$[\mbox{${\textstyle\frac{1}{3}}$}\pi,\mbox{${\textstyle\frac{2}{3}}$}\pi]$,
the minimum value of $s_{n}(x)$ is $\sqrt{3}/4$ by (2.4) and $s_{n}(3x)\leq 0$
with the extreme value taken to be $-\xi$ by (2.2). Then

$F_{1}(n;x)<\frac{\sqrt{3}}{4}J_{1}(1)-\xi J_{3}(1)\doteq
0.151421\qquad(x\in[\mbox{${\textstyle\frac{1}{3}}$}\pi,\mbox{${\textstyle\frac{2}{3}}$}\pi])$

and hence $F_{1}(n;x)>0$ in $(0,\pi)$.

We now proceed with our positivity argument. From (4.4) when $m=1$, we have

$S_{n}^{(4)}(x)>2\\{F_{1}(n;x)-B_{1}\\},\qquad B_{1}=5.23944\times 10^{-4}.$

The roots of $F_{1}(n;x)-B_{1}=0$ will be close to $x=0$ and $x=\pi$, since
$B_{1}\ll 1$. Let ${\cal L}_{\lambda}$ denote the intervals
$(0,\pi\lambda)\cup(\mbox{${\textstyle\frac{2}{3}}$}\pi,\pi)$, where
$\lambda=0.207685$; see §2.1(v). Then for $x\in{\cal L}_{\lambda}$, we have
$s_{n}(3x)>0$ so that from (4.8) $F_{1}(n;x)>J_{1}(1)s_{n}(x)=F_{0}(n;x)$.
Furthermore, since the lowest curve of $s_{n}(x)$ in ${\cal L}_{\lambda}$
corresponds to $n=2$ by (2.5), we have

$F_{1}(n;x)>F_{0}(2;x)=J_{1}(1)\sin x(1+\cos x)\qquad(x\in{\cal L}_{\lambda})$

and consequently

$S_{n}^{(4)}(x)>2\\{F_{0}(2;x)-B_{1}\\}\qquad(x\in{\cal L}_{\lambda}).$

The roots of $F_{0}(2;x)-B_{1}=0$ are $x_{1}^{-}=1.895\times 10^{-4}\pi$ and
$x_{1}^{+}=0.957\pi$, which are situated in ${\cal L}_{\lambda}$. Hence

$S_{n}^{(4)}(x)>0\quad\mbox{for}\quad x\in[x_{1}^{-},x_{1}^{+}].$ (4.9)

When $m=2$, we have

$S_{n}^{(4)}(x)>2\\{F_{2}(n;x)-B_{2}\\},\qquad B_{2}=3.111\times 10^{-6}.$

In $(0,\mbox{${\textstyle\frac{1}{5}}$}\pi)$ and
$(\mbox{${\textstyle\frac{4}{5}}$}\pi,\pi)$ both $s_{n}(3x)$ and $s_{n}(5x)$
are positive. Hence $S_{n}^{(4)}(x)>2\\{F_{0}(n;x)-B_{2}\\}$ for
$x\in(0,\mbox{${\textstyle\frac{1}{5}}$}\pi)\cup(\mbox{${\textstyle\frac{4}{5}}$}\pi,\pi)$.
These intervals are contained in ${\cal L}_{\lambda}$ so that $F_{0}(n;x)$ can
be replaced by $F_{0}(2;x)$ to yield

$S_{n}^{(4)}(x)>2\\{F_{0}(2;x)-B_{2}\\}\qquad(x\in(0,\mbox{${\textstyle\frac{1}{5}}$}\pi)\cup(\mbox{${\textstyle\frac{4}{5}}$}\pi,\pi)).$

The roots of $F_{0}(2;x)-B_{2}=0$ are $x_{2}^{-}=1.125\pi\times 10^{-6}$ and
$x_{2}^{+}=0.992\pi$. Hence $S_{n}^{(4)}(x)>0$ in the intervals
$[x_{2}^{-},\mbox{${\textstyle\frac{1}{5}}$}\pi)\cup(\mbox{${\textstyle\frac{4}{5}}$}\pi,x_{2}^{+}]$.
Since $x_{2}^{-}<x_{1}^{-}<\mbox{${\textstyle\frac{1}{5}}$}\pi$ and
$\mbox{${\textstyle\frac{4}{5}}$}\pi<x_{1}^{+}<x_{2}^{+}$, and
$S_{n}^{(4)}(x)>0$ in $[x_{1}^{-},x_{1}^{+}]$ by (4.9), these intervals
overlap and so we deduce that

$S_{n}^{(4)}(x)>0\quad\mbox{for}\quad x\in[x_{2}^{-},x_{2}^{+}].$ (4.10)

This process can be continued. From (2.3) with $m\geq 2$, we have
$s_{n}((2r+1)x)>0$ for $1\leq r\leq m$ in the intervals

$L_{m}:=(0,\mbox{${\textstyle\frac{\pi}{2m+1}}$})\cup(\mbox{${\textstyle\frac{2m\pi}{2m+1}}$},\pi)\subset{\cal
L}_{\lambda}\qquad(m\geq 2).$

Then $S_{n}^{(4)}(x)>0$ for $x\in L_{m}$. It follows that
$F_{m}(n;x)>F_{0}(n;x)>F_{0}(2;x)$ for $x\in L_{m}$, so that from (4.4) we
obtain

$S_{n}^{(4)}(x)>2\\{F_{0}(2;x)-B_{m}\\}\qquad(x\in L_{m};\ m\geq 2).$

The roots of $F_{0}(2;x)-B_{m}=0$ are $x_{m}^{-}$ and $x_{m}^{+}$. From the
leading behaviour of $F_{0}(2;x)=J_{1}(1)\sin x(1+\cos x)$ given by

$F_{0}(2;x)\sim\left\\{\begin{array}[]{ll}2xJ_{1}(1)&(x\to 0)\\\
\mbox{${\textstyle\frac{1}{2}}$}(\pi-x)^{3}J_{1}(1)&(x\to\pi),\end{array}\right.$

we find that

$x_{m}^{-}\sim\frac{B_{m}}{2J_{1}(1)},\qquad
x_{m}^{+}\sim\pi-\biggl{(}\frac{2B_{m}}{J_{1}(1)}\biggr{)}^{\\!1/3}\quad(m\gg
1).$

Table 1: Values of the roots of $F_{0}(2,x)-B_{m}=0$ for $1\leq m\leq 10$. $m$ | $B_{m}$ | $x_{m}^{-}/\pi$ | $x_{m}^{+}/\pi$
---|---|---|---
1 | $5.2394432083\times 10^{-4}$ | $1.8949717051\times 10^{-4}$ | 0.9574201720
2 | $3.1109874947\times 10^{-6}$ | $1.1251639101\times 10^{-6}$ | 0.9923025565
3 | $1.0789082025\times 10^{-8}$ | $3.9021325986\times 10^{-9}$ | 0.9988349231
4 | $2.4504202963\times 10^{-11}$ | $8.8625379772\times 10^{-12}$ | 0.9998468538
5 | $3.9253367506\times 10^{-14}$ | $1.4196930249\times 10^{-14}$ | 0.9999820808
6 | $4.6717514135\times 10^{-17}$ | $1.6896519501\times 10^{-17}$ | 0.9999981010
8 | $3.1377910235\times 10^{-23}$ | $1.1348580548\times 10^{-23}$ | 0.9999999833
10 | $9.2262804088\times 10^{-30}$ | $3.3369075760\times 10^{-30}$ | 0.9999999999

We now establish a lemma concerning the roots $x_{m}^{\pm}$ and
$x_{m-1}^{\pm}$.

###### Lemma 3

$\\!\\!\\!.$ The roots $x_{m}^{\pm}$ and $x_{m-1}^{\pm}$ satisfy the
inequalities

$x_{m}^{-}<x_{m-1}^{-}<\frac{\pi}{2m+1},\qquad\frac{2m\pi}{2m+1}<x_{m-1}^{+}<x_{m}^{+}\qquad(m\geq
2).$ (4.11)

Proof. Since, from (4.5), $B_{m}$ is a rapidly decreasing function of $m$ it
follows that $x_{m}^{-}$ and $x_{m}^{+}$ are decreasing and increasing
sequences, respectively. Values of $x_{m}^{\pm}$ and $B_{m}$ for $1\leq m\leq
10$ are presented in Table 1.

The roots $x_{m}^{-}$ are determined from $\sin x(1+\cos x)=B_{m}/J_{1}(1)$,
from which it follows for
$x_{m}^{-}\in(0,\mbox{${\textstyle\frac{1}{3}}$}\pi)$ that

$\frac{B_{m}}{2J_{1}(1)}<\sin x_{m}^{-}<\frac{2B_{m}}{3J_{1}(1)}.$

The requirement that $x_{m-1}^{-}<\pi/(2m+1)$ is satisfied if
$\arcsin(2B_{m-1}/(3J_{1}(1))<\pi/(2m+1)$, or

$J_{1}(1)\sin\frac{\pi}{2m+1}>\frac{2}{3}B_{m-1}\qquad(m\geq 2).$

This is seen to be verified by (4.7) in Lemma 1.

For the roots $x_{m}^{+}$, we let ${\hat{x}}:=\pi-x$. Then
$\sin{\hat{x}}(1-\cos{\hat{x}})=B_{m}/J_{1}(1)$, whence for
${\hat{x}}_{m}^{+}\in(0,\mbox{${\textstyle\frac{1}{3}}$}\pi)$ we have

$\frac{B_{m}}{J_{1}(1)}<\sin{\hat{x}}_{m}^{+}<\frac{2B_{m}}{J_{1}(1)}.$

The second inequality in (4.11) $x_{m-1}^{+}>2m\pi/(2m+1)$ is therefore
satisfied if $\pi-\arcsin(2B_{m-1}/J_{1}(1))>2m\pi/(2m+1)$, or

$J_{1}(1)\sin\frac{\pi}{2m+1}>2B_{m-1}\qquad(m\geq 2),$

which is true by Lemma 1.

$S_{n}^{(4)}(x)>0$$x_{m-1}^{-}$$x_{m-1}^{+}$$x_{m}^{-}$$x_{m}^{+}$$\frac{\pi}{2m\\!+\\!1}$$\frac{2m\pi}{2m\\!+\\!1}$$0$$\pi$
Figure 4: Schematic representation of the roots $x_{m}^{\pm}$ and
$x_{m-1}^{\pm}$ and the interval of positivity $[x_{m-1}^{-},x_{m-1}^{+}]$.

Since the intervals $[x_{m}^{-},\pi/(2m\\!+\\!1))$ and
$(2m\pi/(2m\\!+\\!1),x_{m}^{+}]$ overlap with the interval of positivity
$[x_{m-1}^{-},x_{m-1}^{+}]$ from the previous step (see Fig. 4), it then
follows that

$S_{n}^{(4)}(x)>0\qquad x\in[x_{m}^{-},\ x_{m}^{+}]$

thereby establishing positivity for the $m$th step.

From Table 1, it is seen that when $m=10$, we have $x_{10}^{-}=3.337\pi\times
10^{-30}$ and $x_{10}^{+}=0.999\,999\,999\,9\pi$. Thus, with this positivity
argument we can approach the endpoints $x=0$ and $x=\pi$ as close as we
please.

5\. Concluding remarks

We explain and quantify the positive jump in the graph of $S_{n}^{(4)}(x)$
appearing in the vicinity of $x=\mbox{${\textstyle\frac{2}{3}}$}\pi$ that is
visible in Fig. 2(d). From (4.1), the functions $s_{n}(3x)$, $s_{n}(9x)$,
$s_{n}(15x),\ldots$ have arguments close to $2\pi$, $6\pi$, $10\pi,\ldots\ $,
where they undergo a rapid change from $-\Delta$ to $+\Delta$ (the Gibbs
phenomenon), where $\Delta\doteq 1.852$ as $n\to\infty$; see §2.1(ii). Thus we
expect a positive jump in the neighbourhood of
$x=\mbox{${\textstyle\frac{2}{3}}$}\pi$ equal to approximately

$2\\{J_{3}(1)+J_{9}(1)+J_{15}(1)+\cdots\\}2\Delta\doteq
0.1449\qquad(n\to\infty).$ (5.1)

Numerical calculations with $n$ large confirm this value for the jump.

Finally, since $d/dx\,S_{n}^{(4)}(x)=\sum_{k=1}^{n}\cos(\sin kx)\cos kx$, we
have the derivatives

$d/dx\,S_{n}^{(4)}(0)=n,\qquad
d/dx\,S_{n}^{(4)}(\pi)=\left\\{\begin{array}[]{ll}0&(n\ \mbox{even})\\\ -1&(n\
\mbox{odd}).\end{array}\right.$ (5.2)

Thus the gradient of $S_{n}^{(4)}(x)$ at $x=0$ becomes progressively steeper
as $n\to\infty$ and the gradient at $x=\pi$ is either 0 or $-1$ depending on
the parity of $n$. The expansions of $\sin(\sin kx)$ near $x=0$ and $x=\pi$
are

$\sin(\sin
kx)=\left\\{\begin{array}[]{ll}kx-\mbox{${\textstyle\frac{1}{3}}$}(kx)^{3}+\mbox{${\textstyle\frac{1}{10}}$}(kx)^{5}-\cdots&(x\to
0)\\\ \\\
(-1)^{k-1}(k{\hat{x}}-\mbox{${\textstyle\frac{1}{3}}$}(k{\hat{x}})^{3}+\mbox{${\textstyle\frac{1}{10}}$}(k{\hat{x}})^{5}-\cdots)&(x\to\pi),\end{array}\right.$

where ${\hat{x}}:=\pi-x$. This leads to the expansions

$S_{n}^{(4)}(x)=nx-\mbox{${\textstyle\frac{1}{6}}$}n(n+1)(2n+1)\biggl{(}\mbox{${\textstyle\frac{1}{3}}$}x^{3}-\mbox{${\textstyle\frac{1}{50}}$}(3n^{2}+3n-1)x^{5}+\cdots\biggr{)}\quad(x\to
0)$

and

$S_{n}^{(4)}(x)=\left\\{\\!\\!\begin{array}[]{l}0\\\
{\hat{x}}\end{array}\\!\\!\right\\}+(-)^{n}n(n+1)\biggl{(}\mbox{${\textstyle\frac{1}{6}}$}{\hat{x}}^{3}-\mbox{${\textstyle\frac{1}{20}}$}(n^{2}+n-1){\hat{x}}^{5}+\cdots\biggr{)}\qquad
n\left\\{\begin{array}[]{l}\\!\\!\mbox{even}\\\
\\!\\!\mbox{odd}\end{array}\right.\quad(x\to\pi),$

These results are readily seen to yield the derivatives in (5.2).

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