On the properties of generalized cyclotomic binary sequences of period $2p^m$

Abstract

Xiao, Zeng, Li and Helleseth proposed new generalized cyclotomic binary sequences $s^{\infty }$ of period $p^m$ and showed that these sequences are almost balanced and have very large linear complexity if p is a non-Wieferich prime and $m=2$. Wu, Xu, Chen and Ke determined the values of the k-error linear complexity for $m=2$ in terms of the theory of Fermat quotients and the results indicated that sequences $s^{\infty }$ have good stability. Edemskiy, Li, Zeng and Helleseth studied the linear complexity of $s^{\infty }$ for general integers $m\ge 2$. Furthermore, Ouyang and Xie constructed new $2p^{m}$-periodic binary sequences ${\widehat{s}}^{\infty }$ and ${\widetilde{s}}^{\infty }$ and proved that the sequences ${\widehat{s}}^{\infty }$ and ${\widetilde{s}}^{\infty }$ are of high linear complexity when $m\ge 2$. In this paper we shall show that despite a high linear complexity the sequences $s^{\infty }$, ${\widehat{s}}^{\infty }$ and ${\widetilde{s}}^{\infty }$ have some undesirable features which may not suggest them for cryptography. The properties of multiplicative character sums modulo $p^m$ play an important role in the proof of this paper.

Appendices

Appendix A: Proof of Lemma 1

We only prove (9) since similarly we can get (10). Let $\chi ^*$ be a generator of the group of characters modulo $p^m$, and let $\displaystyle \chi _u=\left( \chi ^*\right) ^{\delta _u}$, where $1\le \delta _u\le \phi (p^m)-1$ for $u=1,2,3, 4$. We get

$$\begin{aligned}&\sum _{n=1}^{p^m}\chi _1\left( n\right) \chi _2\left( n+2p^{\frac{m}{2}}\right) \chi _3\left( n+4p^{\frac{m}{2}}\right) \chi _4\left( n+6p^{\frac{m}{2}}\right) \\&\quad =\sum _{n=1}^{p^m}\chi ^*\left( n^{\delta _1}\left( n+2p^{\frac{m}{2}}\right) ^{\delta _2} \left( n+4p^{\frac{m}{2}}\right) ^{\delta _3}\left( n+6p^{\frac{m}{2}}\right) ^{\delta _4}\right) \\&\quad =\mathop {\sum _{y=1}^{p^{\frac{m}{2}}}}_{(y,p)=1}\sum _{z=1}^{p^{\frac{m}{2}}} \chi ^*\left( (y+zp^{\frac{m}{2}})^{\delta _1}\left( y+zp^{\frac{m}{2}}+2p^{\frac{m}{2}}\right) ^{\delta _2}\right. \\&\qquad \times \left. \left( y+zp^{\frac{m}{2}}+4p^{\frac{m}{2}}\right) ^{\delta _3}\left( y+zp^{\frac{m}{2}}+6p^{\frac{m}{2}}\right) ^{\delta _4}\right) \\&\quad =\mathop {\sum _{y=1}^{p^{\frac{m}{2}}}}_{(y,p)=1}\sum _{z=1}^{p^{\frac{m}{2}}} \chi ^*\left( (y^{\delta _1}+\delta _1y^{\delta _1-1}zp^{\frac{m}{2}})(y^{\delta _2}+\delta _2y^{\delta _2-1}(z+2)p^{\frac{m}{2}}) \right. \\&\qquad \times \left. (y^{\delta _3}+\delta _3y^{\delta _3-1}(z+4)p^{\frac{m}{2}}) (y^{\delta _4}+\delta _4y^{\delta _4-1}(z+6)p^{\frac{m}{2}})\right) \\&\quad =\mathop {\sum _{y=1}^{p^{\frac{m}{2}}}}_{(y,p)=1}\chi ^*\left( y^{\delta _1+\delta _2+\delta _3+\delta _4}\right) \sum _{z=1}^{p^{\frac{m}{2}}}\chi ^*\left( (1+\delta _1y^{-1}zp^{\frac{m}{2}}) (1+\delta _2y^{-1}(z+2)p^{\frac{m}{2}})\right. \\&\qquad \times \left. (1+\delta _3y^{-1}(z+4)p^{\frac{m}{2}}) (1+\delta _4y^{-1}(z+6)p^{\frac{m}{2}}) \right) . \end{aligned}$$

Obviously $\chi ^*\left( 1+up^{\frac{m}{2}}\right)$ is a primitive additive character modulo $p^{\frac{m}{2}}$, so there is uniquely an integer $\beta$ such that $1\le \beta \le p^{\frac{m}{2}}$, $(\beta , p)=1$ and

$$\begin{aligned} \chi ^*\left( 1+up^{\frac{m}{2}}\right) =e\left( \frac{\beta u}{p^{\frac{m}{2}}}\right) , \end{aligned}$$

(20)

where $e(y)=\hbox {e}^{2\pi iy}$. Hence,

$$\begin{aligned}&\sum _{n=1}^{p^m}\chi _1\left( n\right) \chi _2\left( n+2p^{\frac{m}{2}}\right) \chi _3\left( n+4p^{\frac{m}{2}}\right) \chi _4\left( n+6p^{\frac{m}{2}}\right) \\&\quad =\mathop {\sum _{y=1}^{p^{\frac{m}{2}}}}_{(y,p)=1}\chi ^*\left( y^{\delta _1+\delta _2+\delta _3+\delta _4}\right) \\&\qquad \times \sum _{z=1}^{p^{\frac{m}{2}}}e\left( \frac{\beta \left( \delta _1y^{-1}z+\delta _2y^{-1}(z+2) +\delta _3y^{-1}(z+4)+\delta _4y^{-1}(z+6)\right) }{p^{\frac{m}{2}}}\right) . \end{aligned}$$

Note that

$$\begin{aligned} \sum _{z=1}^{p^{\frac{m}{2}}}e\left( \frac{\beta \left( \delta _1+\delta _2+\delta _3+\delta _4\right) y^{-1}z}{p^{\frac{m}{2}}}\right) =\left\{ \begin{array}{ll} p^{\frac{m}{2}}, &{}\quad \hbox {if } \delta _1+\delta _2+\delta _3+\delta _4\equiv 0\ \left( \bmod \ p^{\frac{m}{2}}\right) , \\ 0, &{}\quad \hbox {otherwise}. \end{array} \right. \end{aligned}$$

If $\delta _1+\delta _2+\delta _3+\delta _4\not \equiv 0\ \left( \bmod \ p^{\frac{m}{2}}\right)$, then

$$\begin{aligned} \sum _{n=1}^{p^m}\chi _1\left( n\right) \chi _2\left( n+2p^{\frac{m}{2}}\right) \chi _3\left( n+4p^{\frac{m}{2}}\right) \chi _4\left( n+6p^{\frac{m}{2}}\right) =0. \end{aligned}$$

(21)

If $\delta _1+\delta _2+\delta _3+\delta _4\equiv 0\ \left( \bmod \ p^{\frac{m}{2}}\right)$, we have

$$\begin{aligned}&\sum _{n=1}^{p^m}\chi _1\left( n\right) \chi _2\left( n+2p^{\frac{m}{2}}\right) \chi _3\left( n+4p^{\frac{m}{2}}\right) \chi _4\left( n+6p^{\frac{m}{2}}\right) \\&\quad =p^{\frac{m}{2}} \mathop {\sum _{y=1}^{p^{\frac{m}{2}}}}_{(y,p)=1}(\chi ^*)^{\delta _1+\delta _2+\delta _3+\delta _4}\left( y\right) e\left( \frac{\beta \left( 2\delta _2+4\delta _3+6\delta _4\right) y^{-1}}{p^{\frac{m}{2}}}\right) . \end{aligned}$$

Clearly $(\chi ^*)^{\delta _1+\delta _2+\delta _3+\delta _4}$ is a multiplicative character modulo $p^{\frac{m}{2}}$ since $\delta _1+\delta _2+\delta _3+\delta _4\equiv 0\ \left( \bmod \ p^{\frac{m}{2}}\right)$. By the properties of Gauss sums we get

$$\begin{aligned}&\sum _{n=1}^{p^m}\chi _1\left( n\right) \chi _2\left( n+2p^{\frac{m}{2}}\right) \chi _3\left( n+4p^{\frac{m}{2}}\right) \chi _4\left( n+6p^{\frac{m}{2}}\right) \nonumber \\&\quad =p^{\frac{m}{2}} \mathop {\sum _{y=1}^{p^{\frac{m}{2}}}}_{(y,p)=1}\overline{\chi ^*}^{\delta _1+\delta _2+\delta _3+\delta _4}\left( y\right) e\left( \frac{\beta \left( 2\delta _2+4\delta _3+6\delta _4\right) y}{p^{\frac{m}{2}}}\right) \nonumber \\&\quad =\left\{ \begin{array}{ll} p^{m}+O\left( p^{m-1}\right) , &{}\quad \hbox {if } \phi (p^m)\mid \delta _1+\delta _2+\delta _3+\delta _4 \hbox { and } p^{\frac{m}{2}}\mid \delta _2+2\delta _3+3\delta _4, \\ O\left( p^{m-1}\right) , &{}\quad \hbox {if } \phi (p^m)\mid \delta _1+\delta _2+\delta _3+\delta _4 \hbox { and } p^{\frac{m}{2}}\not \mid \delta _2+2\delta _3+3\delta _4, \\ O\left( p^{m-\frac{1}{2}}\right) , &{}\quad \hbox {if } \phi (p^m)\not \mid \delta _1+\delta _2+\delta _3+\delta _4. \end{array}\right. \end{aligned}$$

(22)

Now combining (21) and (22) we immediately have

$$\begin{aligned}&\sum _{n=1}^{p^m}\chi _1\left( n\right) \chi _2\left( n+2p^{\frac{m}{2}}\right) \chi _3\left( n+4p^{\frac{m}{2}}\right) \chi _4\left( n+6p^{\frac{m}{2}}\right) \nonumber \\&\quad =\left\{ \begin{array}{ll} p^{m}+O\left( p^{m-1}\right) , &{}\quad \hbox {if } \chi _1\chi _2\chi _3\chi _4 \hbox { and } \left( \chi _2\chi _3^2\chi _4^3\right) ^{\phi (p^{\frac{m}{2}})} \hbox { are trivial}, \\ O\left( p^{m-\frac{1}{2}}\right) , &{}\quad \hbox {otherwise}. \end{array}\right. \end{aligned}$$

This proves (9).

Appendix B: Proof of Lemma 2

We get

$$\begin{aligned} \varOmega _{d,m}=&\mathop {\mathop {\mathop { \sum _{|k_1|\le \frac{d}{2}p^{m-1}}}_{2\not \mid k_1} \mathop {\sum _{|k_2|\le \frac{d}{2}p^{m-1}}}_{2\not \mid k_2} \mathop {\sum _{|k_3|\le \frac{d}{2}p^{m-1}}}_{2\not \mid k_3} \mathop {\sum _{|k_4|\le \frac{d}{2}p^{m-1}}}_{2\not \mid k_4} }_{k_1+k_2+k_3+k_4\equiv 0 (\bmod dp^{m-1})} }_{k_2+2k_3+3k_4\equiv 0 (\bmod p^{\frac{m}{2}})}\\&\times \frac{1}{\left( 1-e\left( \frac{k_1}{dp^{m-1}}\right) \right) \left( 1-e\left( \frac{k_2}{dp^{m-1}}\right) \right) \left( 1-e\left( \frac{k_3}{dp^{m-1}}\right) \right) \left( 1-e\left( \frac{k_4}{dp^{m-1}}\right) \right) }. \end{aligned}$$

For $\theta \in {\mathbb {R}}\setminus {\mathbb {Z}}$ we have

$$\begin{aligned} \frac{1}{1-e(\theta )}=&\frac{1}{1-\hbox {e}^{2\pi i\theta }}=\frac{1}{1-\cos 2\pi \theta -i\sin 2\pi \theta } =\frac{1}{2}+\frac{i}{2}\cot \pi \theta . \end{aligned}$$

Hence,

$$\begin{aligned} \varOmega _{d,m}&=\mathop {\mathop { \mathop {\sum _{|k_1|\le \frac{d}{2}p^{m-1}}}_{2\not \mid k_1} \mathop {\sum _{|k_2|\le \frac{d}{2}p^{m-1}}}_{2\not \mid k_2} \mathop {\sum _{|k_3|\le \frac{d}{2}p^{m-1}}}_{2\not \mid k_3} \mathop {\sum _{|k_4|\le \frac{d}{2}p^{m-1}}}_{2\not \mid k_4} }_{k_1+k_2+k_3+k_4\equiv 0 (\bmod dp^{m-1})} }_{k_2+2k_3+3k_4\equiv 0 (\bmod p^{\frac{m}{2}})} \left( \frac{1}{2}+\frac{i}{2}\cot \pi \frac{k_1}{dp^{m-1}}\right) \\&\quad \times \left( \frac{1}{2}+\frac{i}{2}\cot \pi \frac{k_2}{dp^{m-1}}\right) \left( \frac{1}{2}+\frac{i}{2}\cot \pi \frac{k_3}{dp^{m-1}}\right) \left( \frac{1}{2}+\frac{i}{2}\cot \pi \frac{k_4}{dp^{m-1}}\right) \\&=\frac{1}{16}\mathop {\mathop { \mathop {\sum _{|k_1|\le \frac{d}{2}p^{m-1}}}_{2\not \mid k_1} \mathop {\sum _{|k_2|\le \frac{d}{2}p^{m-1}}}_{2\not \mid k_2} \mathop {\sum _{|k_3|\le \frac{d}{2}p^{m-1}}}_{2\not \mid k_3} \mathop {\sum _{|k_4|\le \frac{d}{2}p^{m-1}}}_{2\not \mid k_4} }_{k_1+k_2+k_3+k_4\equiv 0 (\bmod dp^{m-1})} }_{k_2+2k_3+3k_4\equiv 0 (\bmod p^{\frac{m}{2}})} \cot \pi \frac{k_1}{dp^{m-1}}\cot \pi \frac{k_2}{dp^{m-1}}\nonumber \\&\quad \times \cot \pi \frac{k_3}{dp^{m-1}}\cot \pi \frac{k_4}{dp^{m-1}} +O\left( d^3p^{3m-3}(\log p^m)^3\right) . \end{aligned}$$

It is not hard to show that

$$\begin{aligned}&\mathop {\mathop { \mathop {\sum _{\frac{p^{\frac{m}{2}}}{16}+1\le |k_1|\le \frac{d}{2}p^{m-1}}}_{2\not \mid k_1} \mathop {\sum _{|k_2|\le \frac{d}{2}p^{m-1}}}_{2\not \mid k_2} \mathop {\sum _{|k_3|\le \frac{d}{2}p^{m-1}}}_{2\not \mid k_3} \mathop {\sum _{|k_4|\le \frac{d}{2}p^{m-1}}}_{2\not \mid k_4} }_{k_1+k_2+k_3+k_4\equiv 0 (\bmod dp^{m-1})} }_{k_2+2k_3+3k_4\equiv 0 (\bmod p^{\frac{m}{2}})} \cot \pi \frac{k_1}{dp^{m-1}}\\&\qquad \times \cot \pi \frac{k_2}{dp^{m-1}}\cot \pi \frac{k_3}{dp^{m-1}}\cot \pi \frac{k_4}{dp^{m-1}}\\&\quad \ll dp^{m-1-\frac{m}{2}}\mathop {\mathop { \mathop {\sum _{\frac{p^{\frac{m}{2}}}{16}+1\le |k_1|\le \frac{d}{2}p^{m-1}}}_{2\not \mid k_1} \mathop {\sum _{|k_2|\le \frac{d}{2}p^{m-1}}}_{2\not \mid k_2} \mathop {\sum _{|k_3|\le \frac{d}{2}p^{m-1}}}_{2\not \mid k_3} \mathop {\sum _{|k_4|\le \frac{d}{2}p^{m-1}}}_{2\not \mid k_4} }_{k_1+k_2+k_3+k_4\equiv 0 (\bmod dp^{m-1})} }_{k_2+2k_3+3k_4\equiv 0 (\bmod p^{\frac{m}{2}})}\\&\qquad \times \frac{1}{\left<\frac{k_2}{dp^{m-1}}\right>\left<\frac{k_3}{dp^{m-1}}\right>\left<\frac{k_4}{dp^{m-1}}\right>}\\&\quad \ll dp^{m-1-\frac{m}{2}} \mathop {\sum _{|k_2|\le \frac{d}{2}p^{m-1}}}_{2\not \mid k_2} \mathop {\sum _{|k_3|\le \frac{d}{2}p^{m-1}}}_{2\not \mid k_3} \mathop {\sum _{|k_4|\le \frac{d}{2}p^{m-1}}}_{2\not \mid k_4} \frac{1}{\left<\frac{k_2}{dp^{m-1}}\right>\left<\frac{k_3}{dp^{m-1}}\right>\left<\frac{k_4}{dp^{m-1}}\right>}\\&\quad \ll d^4p^{\frac{7}{2}m-4}(\log p^m)^3. \end{aligned}$$

Similarly we get

$$\begin{aligned}&\mathop {\mathop { \mathop {\sum _{|k_1|\le \frac{d}{2}p^{m-1}}}_{2\not \mid k_1} \mathop {\sum _{\frac{p^{\frac{m}{2}}}{16}+1\le |k_2|\le \frac{d}{2}p^{m-1}}}_{2\not \mid k_2} \mathop {\sum _{|k_3|\le \frac{d}{2}p^{m-1}}}_{2\not \mid k_3} \mathop {\sum _{|k_4|\le \frac{d}{2}p^{m-1}}}_{2\not \mid k_4} }_{k_1+k_2+k_3+k_4\equiv 0 (\bmod dp^{m-1})} }_{k_2+2k_3+3k_4\equiv 0 (\bmod p^{\frac{m}{2}})}\cot \pi \frac{k_1}{dp^{m-1}}\\&\qquad \times \cot \pi \frac{k_2}{dp^{m-1}}\cot \pi \frac{k_3}{dp^{m-1}}\cot \pi \frac{k_4}{dp^{m-1}}\\&\quad \ll d^4p^{\frac{7}{2}m-4}(\log p^m)^3, \\&\mathop {\mathop { \mathop {\sum _{|k_1|\le \frac{d}{2}p^{m-1}}}_{2\not \mid k_1} \mathop {\sum _{|k_2|\le \frac{d}{2}p^{m-1}}}_{2\not \mid k_2} \mathop {\sum _{\frac{p^{\frac{m}{2}}}{16}+1\le |k_3|\le \frac{d}{2}p^{m-1}}}_{2\not \mid k_3} \mathop {\sum _{|k_4|\le \frac{d}{2}p^{m-1}}}_{2\not \mid k_4} }_{k_1+k_2+k_3+k_4\equiv 0 (\bmod dp^{m-1})} }_{k_2+2k_3+3k_4\equiv 0 (\bmod p^{\frac{m}{2}})} \cot \pi \frac{k_1}{dp^{m-1}}\\&\qquad \times \pi \frac{k_2}{dp^{m-1}}\cot \pi \frac{k_3}{dp^{m-1}}\cot \pi \frac{k_4}{dp^{m-1}}\\&\quad \ll d^4p^{\frac{7}{2}m-4}(\log p^m)^3, \\&\mathop {\mathop { \mathop {\sum _{|k_1|\le \frac{d}{2}p^{m-1}}}_{2\not \mid k_1} \mathop {\sum _{|k_2|\le \frac{d}{2}p^{m-1}}}_{2\not \mid k_2} \mathop {\sum _{|k_3|\le \frac{d}{2}p^{m-1}}}_{2\not \mid k_3} \mathop {\sum _{\frac{p^{\frac{m}{2}}}{16}+1\le |k_4|\le \frac{d}{2}p^{m-1}}}_{2\not \mid k_4} }_{k_1+k_2+k_3+k_4\equiv 0 (\bmod dp^{m-1})} }_{k_2+2k_3+3k_4\equiv 0 (\bmod p^{\frac{m}{2}})} \cot \pi \frac{k_1}{dp^{m-1}}\\&\qquad \times \cot \pi \frac{k_2}{dp^{m-1}}\cot \pi \frac{k_3}{dp^{m-1}}\cot \pi \frac{k_4}{dp^{m-1}}\\&\quad \ll d^4p^{\frac{7}{2}m-4}(\log p^m)^3. \end{aligned}$$

Therefore

$$\begin{aligned} \varOmega _{d,m}=&\frac{1}{16}\mathop {\mathop { \mathop {\sum _{|k_1|\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_1} \mathop {\sum _{|k_2|\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_2} \mathop {\sum _{|k_3|\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_3} \mathop {\sum _{|k_4|\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_4} }_{k_1+k_2+k_3+k_4\equiv 0 (\bmod dp^{m-1})} }_{k_2+2k_3+3k_4\equiv 0 (\bmod p^{\frac{m}{2}})} \cot \pi \frac{k_1}{dp^{m-1}}\cot \pi \frac{k_2}{dp^{m-1}}\nonumber \\&\qquad \times \cot \pi \frac{k_3}{dp^{m-1}}\cot \pi \frac{k_4}{dp^{m-1}} +O\left( d^4p^{\frac{7}{2}m-4}(\log p^m)^3\right) \\ =&\frac{1}{16}\mathop {\mathop { \mathop {\sum _{|k_1|\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_1} \mathop {\sum _{|k_2|\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_2} \mathop {\sum _{|k_3|\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_3} \mathop {\sum _{|k_4|\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_4} }_{k_1+k_2+k_3+k_4=0} }_{k_2+2k_3+3k_4=0} \cot \pi \frac{k_1}{dp^{m-1}}\cot \pi \frac{k_2}{dp^{m-1}}\nonumber \\&\qquad \times \cot \pi \frac{k_3}{dp^{m-1}}\cot \pi \frac{k_4}{dp^{m-1}} +O\left( d^4p^{\frac{7}{2}m-4}(\log p^m)^3\right) \\ =&-\frac{1}{16}\mathop {\sum _{|k_3|\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_3} \mathop {\sum _{|k_4|\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_4} \cot \pi \frac{k_3+2k_4}{dp^{m-1}}\cot \pi \frac{2k_3+3k_4}{dp^{m-1}}\nonumber \\&\qquad \times \cot \pi \frac{k_3}{dp^{m-1}}\cot \pi \frac{k_4}{dp^{m-1}} +O\left( d^4p^{\frac{7}{2}m-4}(\log p^m)^3\right) \\ =&-\frac{1}{8}\mathop {\sum _{1\le k_3\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_3}\mathop {\sum _{1\le k_4\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_4} \cot \pi \frac{k_3+2k_4}{dp^{m-1}}\cot \pi \frac{2k_3+3k_4}{dp^{m-1}}\\&\qquad \times \cot \pi \frac{k_3}{dp^{m-1}}\cot \pi \frac{k_4}{dp^{m-1}}\nonumber \\&+\frac{1}{8}\mathop {\sum _{1\le k_3\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_3} \mathop {\sum _{1\le k_4\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_4} \cot \pi \frac{k_3-2k_4}{dp^{m-1}}\cot \pi \frac{2k_3-3k_4}{dp^{m-1}}\\&\qquad \times \cot \pi \frac{k_3}{dp^{m-1}}\cot \pi \frac{k_4}{dp^{m-1}}+O\left( d^4p^{\frac{7}{2}m-4}(\log p^m)^3\right) . \end{aligned}$$

Note that

$$\begin{aligned}&\frac{1}{8}\mathop {\mathop {\sum _{1\le k_3\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_3} \mathop {\sum _{1\le k_4\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_4} }_{3k_4<2k_3<4k_4} \cot \pi \frac{k_3-2k_4}{dp^{m-1}}\cot \pi \frac{2k_3-3k_4}{dp^{m-1}} \cot \pi \frac{k_3}{dp^{m-1}}\cot \pi \frac{k_4}{dp^{m-1}}\\&=-\frac{1}{8}\mathop {\mathop {\sum _{1\le k_3\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_3} \mathop {\sum _{1\le k_4\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_4} }_{3k_4<2k_3<4k_4} \cot \pi \frac{2k_4-k_3}{dp^{m-1}}\cot \pi \frac{2k_3-3k_4}{dp^{m-1}}\\&\qquad \qquad \qquad \times \cot \pi \frac{k_3}{dp^{m-1}}\cot \pi \frac{k_4}{dp^{m-1}}\\&=-\frac{1}{8}\mathop {\mathop {\mathop {\sum _{1\le l_3\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid l_3} \mathop {\sum _{1\le l_4\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid l_4} }_{1\le 2l_3+3l_4\le \frac{p^{\frac{m}{2}}}{16}}}_{1\le l_3+2l_4\le \frac{p^{\frac{m}{2}}}{16}} \cot \pi \frac{l_4}{dp^{m-1}}\cot \pi \frac{l_3}{dp^{m-1}} \cot \pi \frac{2l_3+3l_4}{dp^{m-1}}\cot \pi \frac{l_3+2l_4}{dp^{m-1}}. \end{aligned}$$

Thus we have

$$\begin{aligned} \varOmega _{d,m}=&\frac{1}{8}\mathop {\mathop {\sum _{1\le k_3\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_3} \mathop {\sum _{1\le k_4\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_4}}_{k_3>2k_4\ \hbox {or}\ k_3<\frac{3}{2}k_4} \cot \pi \frac{k_3-2k_4}{dp^{m-1}}\cot \pi \frac{2k_3-3k_4}{dp^{m-1}}\\&\qquad \times \cot \pi \frac{k_3}{dp^{m-1}}\cot \pi \frac{k_4}{dp^{m-1}}\\&\qquad -\frac{1}{4}\mathop {\sum _{1\le k_3\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_3} \mathop {\sum _{1\le k_4\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_4} \cot \pi \frac{k_3+2k_4}{dp^{m-1}}\cot \pi \frac{2k_3+3k_4}{dp^{m-1}}\\&\qquad \times \cot \pi \frac{k_3}{dp^{m-1}}\cot \pi \frac{k_4}{dp^{m-1}}\\&\qquad +O\left( d^4p^{\frac{7}{2}m-4}(\log p^m)^3\right) \\ =&\frac{1}{8}\mathop {\mathop {\sum _{1\le k_3\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_3} \mathop {\sum _{1\le k_4\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_4}}_{k_3\ge 3k_4} \cot \pi \frac{k_4}{dp^{m-1}}\cot \pi \frac{k_3-2k_4}{dp^{m-1}}\\&\qquad \times \cot \pi \frac{k_3}{dp^{m-1}}\cot \pi \frac{2k_3-3k_4}{dp^{m-1}}\\&\qquad +\frac{1}{8}\mathop {\mathop {\sum _{1\le k_3\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_3}\mathop {\sum _{1\le k_4\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_4}}_{2k_4<k_3<3k_4} \cot \pi \frac{k_3-2k_4}{dp^{m-1}}\cot \pi \frac{k_4}{dp^{m-1}}\\&\qquad \times \cot \pi \frac{k_3}{dp^{m-1}}\cot \pi \frac{2k_3-3k_4}{dp^{m-1}}\\&\qquad +\frac{1}{8}\mathop {\mathop {\sum _{1\le k_3\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_3}\mathop {\sum _{1\le k_4\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_4}}_{k_3\le k_4} \cot \pi \frac{k_3}{dp^{m-1}}\cot \pi \frac{k_4}{dp^{m-1}}\\&\qquad \times \cot \pi \frac{2k_4-k_3}{dp^{m-1}}\cot \pi \frac{3k_4-2k_3}{dp^{m-1}}\\&\qquad +\frac{1}{8}\mathop {\mathop {\sum _{1\le k_3\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_3} \mathop {\sum _{1\le k_4\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_4}}_{k_4<k_3<\frac{3}{2}k_4} \cot \pi \frac{3k_4-2k_3}{dp^{m-1}}\cot \pi \frac{2k_4-k_3}{dp^{m-1}}\\&\qquad \times \cot \pi \frac{k_4}{dp^{m-1}}\cot \pi \frac{k_3}{dp^{m-1}}\\&\qquad -\frac{1}{4}\mathop {\mathop {\sum _{1\le k_3\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_3} \mathop {\sum _{1\le k_4\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_4}}_{k_3\le k_4} \cot \pi \frac{k_3}{dp^{m-1}}\cot \pi \frac{k_4}{dp^{m-1}}\\&\qquad \times \cot \pi \frac{k_3+2k_4}{dp^{m-1}}\cot \pi \frac{2k_3+3k_4}{dp^{m-1}}\\&\qquad -\frac{1}{4}\mathop {\mathop {\sum _{1\le k_3\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_3} \mathop {\sum _{1\le k_4\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_4}}_{k_4< k_3} \cot \pi \frac{k_4}{dp^{m-1}}\cot \pi \frac{k_3}{dp^{m-1}}\\&\qquad \times \cot \pi \frac{k_3+2k_4}{dp^{m-1}}\cot \pi \frac{2k_3+3k_4}{dp^{m-1}}\\&\qquad +O\left( d^4p^{\frac{7}{2}m-4}(\log p^m)^3\right) . \end{aligned}$$

Furthermore, we get

$$\begin{aligned}&\frac{1}{8}\mathop {\mathop {\sum _{1\le k_3\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_3}\mathop {\sum _{1\le k_4\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_4}}_{k_3\ge 3k_4} \cot \pi \frac{k_4}{dp^{m-1}}\cot \pi \frac{k_3-2k_4}{dp^{m-1}} \cot \pi \frac{k_3}{dp^{m-1}}\cot \pi \frac{2k_3-3k_4}{dp^{m-1}}\\&\quad =\frac{1}{8}\mathop {\mathop {\mathop {\sum _{1\le l_3\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid l_3} \mathop {\sum _{1\le l_4\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid l_4}}_{l_4\le l_3}}_{1\le l_3+2l_4\le \frac{p^{\frac{m}{2}}}{16}} \cot \pi \frac{l_4}{dp^{m-1}}\cot \pi \frac{l_3}{dp^{m-1}} \cot \pi \frac{l_3+2l_4}{dp^{m-1}}\cot \pi \frac{2l_3+l_4}{dp^{m-1}},\\&\frac{1}{8}\mathop {\mathop {\sum _{1\le k_3\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_3}\mathop {\sum _{1\le k_4\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_4}}_{2k_4<k_3<3k_4} \cot \pi \frac{k_3-2k_4}{dp^{m-1}}\cot \pi \frac{k_4}{dp^{m-1}} \cot \pi \frac{k_3}{dp^{m-1}}\cot \pi \frac{2k_3-3k_4}{dp^{m-1}}\\&\quad =\frac{1}{8}\mathop {\mathop {\mathop {\sum _{1\le l_3\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid l_3} \mathop {\sum _{1\le l_4\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid l_4}}_{l_4<l_3}}_{1\le 2l_3+l_4\le \frac{p^{\frac{m}{2}}}{16}} \cot \pi \frac{l_4}{dp^{m-1}}\cot \pi \frac{l_3}{dp^{m-1}} \cot \pi \frac{2l_3+l_4}{dp^{m-1}}\cot \pi \frac{l_3+2l_4}{dp^{m-1}},\\&\frac{1}{8}\mathop {\mathop {\sum _{1\le k_3\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_3}\mathop {\sum _{1\le k_4\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_4}}_{k_4<k_3<\frac{3}{2}k_4} \cot \pi \frac{3k_4-2k_3}{dp^{m-1}}\cot \pi \frac{2k_4-k_3}{dp^{m-1}} \cot \pi \frac{k_4}{dp^{m-1}}\cot \pi \frac{k_3}{dp^{m-1}}\\&\quad =\frac{1}{8}\mathop {\mathop {\mathop {\sum _{1\le l_3\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid l_3} \mathop {\sum _{1\le l_4\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid l_4}}_{l_3<l_4}}_{1\le 3l_4-2l_3\le \frac{p^{\frac{m}{2}}}{16}} \cot \pi \frac{l_3}{dp^{m-1}}\cot \pi \frac{l_4}{dp^{m-1}} \cot \pi \frac{2l_4-l_3}{dp^{m-1}}\cot \pi \frac{3l_4-2l_3}{dp^{m-1}}. \end{aligned}$$

Hence,

$$\begin{aligned} \varOmega _{d,m}&=\frac{1}{4}\mathop {\mathop {\sum _{1\le k_3\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_3} \mathop {\sum _{1\le k_4\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_4}}_{k_4< k_3} \cot \pi \frac{k_4}{dp^{m-1}}\cot \pi \frac{k_3}{dp^{m-1}}\cot \pi \frac{k_3+2k_4}{dp^{m-1}}\\&\quad \times \left( \cot \pi \frac{2k_3+k_4}{dp^{m-1}}-\cot \pi \frac{2k_3+3k_4}{dp^{m-1}}\right) \\&\quad +\frac{1}{4}\mathop {\mathop {\sum _{1\le k_3\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_3} \mathop {\sum _{1\le k_4\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k_4}}_{k_3< k_4} \cot \pi \frac{k_3}{dp^{m-1}}\cot \pi \frac{k_4}{dp^{m-1}}\\&\quad \times \left( \cot \pi \frac{2k_4-k_3}{dp^{m-1}}\cot \pi \frac{3k_4-2k_3}{dp^{m-1}}- \cot \pi \frac{k_3+2k_4}{dp^{m-1}}\cot \pi \frac{2k_3+3k_4}{dp^{m-1}}\right) \\&\quad +\frac{1}{8}\mathop {\sum _{1\le k\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k} \left( \cot \pi \frac{k}{dp^{m-1}}\right) ^2 \left( \cot \pi \frac{3k}{dp^{m-1}}\right) ^2\\&\quad +\frac{1}{8}\mathop {\sum _{1\le k\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k} \left( \cot \pi \frac{k}{dp^{m-1}}\right) ^4\\&\quad -\frac{1}{4}\mathop {\sum _{1\le k\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k}\left( \cot \pi \frac{k}{dp^{m-1}}\right) ^2 \cot \pi \frac{3k}{dp^{m-1}}\cot \pi \frac{5k}{dp^{m-1}}\\&\quad +O\left( d^4p^{\frac{7}{2}m-4}(\log p^m)^3\right) . \end{aligned}$$

Clearly,

$$\begin{aligned}&\cot \pi \frac{2k_3+k_4}{dp^{m-1}}-\cot \pi \frac{2k_3+3k_4}{dp^{m-1}}>0, \\&\cot \pi \frac{2k_4-k_3}{dp^{m-1}}\cot \pi \frac{3k_4-2k_3}{dp^{m-1}}- \cot \pi \frac{k_3+2k_4}{dp^{m-1}}\cot \pi \frac{2k_3+3k_4}{dp^{m-1}}>0, \end{aligned}$$

for $1\le k_3, k_4\le \frac{p^{\frac{m}{2}}}{16}$, and we also get

$$\begin{aligned} \cot \pi \frac{k}{dp^{m-1}}>\cot \pi \frac{3k}{dp^{m-1}}>\cot \pi \frac{5k}{dp^{m-1}}, \quad \hbox {for} \quad 1\le k\le \frac{p^{\frac{m}{2}}}{16}. \end{aligned}$$

Therefore

$$\begin{aligned} \varOmega _{d,m}\ge&\frac{1}{4}\mathop {\sum _{1\le k\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k} \left( \cot \pi \frac{k}{dp^{m-1}}\right) ^2 \cot \pi \frac{3k}{dp^{m-1}}\left( \cot \pi \frac{3k}{dp^{m-1}}-\cot \pi \frac{5k}{dp^{m-1}}\right) \\&+O\left( d^4p^{\frac{7}{2}m-4}(\log p^m)^3\right) \\ =&\frac{1}{4}\mathop {\sum _{1\le k\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k}\left( \cot \pi \frac{k}{dp^{m-1}}\right) ^2 \cot \pi \frac{3k}{dp^{m-1}}\cdot \frac{\sin \pi \frac{2k}{dp^{m-1}}}{\sin \pi \frac{3k}{dp^{m-1}}\sin \pi \frac{5k}{dp^{m-1}}}\\&+O\left( d^4p^{\frac{7}{2}m-4}(\log p^m)^3\right) \\ \ge&\frac{1}{4}\mathop {\sum _{1\le k\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k} \frac{\frac{2k}{dp^{m-1}}}{\left( \pi \frac{k}{dp^{m-1}}\right) ^2\left( \pi \frac{3k}{dp^{m-1}}\right) ^2\pi \frac{5k}{dp^{m-1}}} +O\left( d^4p^{\frac{7}{2}m-4}(\log p^m)^3\right) \\ =&\frac{d^4p^{4m-4}}{90\pi ^5}\mathop {\sum _{1\le k\le \frac{p^{\frac{m}{2}}}{16}}}_{2\not \mid k}\frac{1}{k^4} +O\left( d^4p^{\frac{7}{2}m-4}(\log p^m)^3\right) \\ \ge&\frac{d^4}{90\pi ^5}p^{4m-4}+O\left( d^4p^{\frac{7}{2}m-4}(\log p^m)^3\right) . \end{aligned}$$

This completes the proof of Lemma 2.