= Solution
The <Von Mangoldt function> is supported on <prime powers>:
$$
\Lambda(n)=\begin{cases}\log p,&n=p^k\text{ for a prime }p,\ k\geq1,\\0,&\text{otherwise}.\end{cases}
$$
Write $e(t)=\exp(2\pi it)$. A convenient form of the <Siegel–Walfisz theorem> is: for every fixed $A,B>0$, uniformly for $q\leq(\log x)^B$ and $(a,q)=1$,
$$
\psi(x;q,a):=\sum_{\substack{n\leq x\\n\equiv a\pmod q}}\Lambda(n)
=\frac{x}{\varphi(q)}+O_{A,B}\left(\frac{x}{(\log x)^A}\right).
$$
The implied constant can be ineffective because of possible <Siegel zeros>; the modulus bound is logarithmic, not an unrestricted uniformity assertion.
Apply this to modulus five and separate the reduced <residue classes>:
$$
\sum_{n\leq N}\Lambda(n)e(2n/5)
=\sum_{a=1}^4e(2a/5)\psi(N;5,a)
+\sum_{5^k\leq N}\log5.
$$
The last sum is $O(\log N)$: only powers of five can contribute in the zero <residue class>. By the <orthogonality of roots of unity>,
$$
\sum_{a=1}^4e(2a/5)=-1,
$$
since two is coprime to five and the sum including $a=0$ vanishes. As $\varphi(5)=4$, the <Siegel–Walfisz theorem> yields
$$
\boxed{\sum_{n\leq N}\Lambda(n)e(2n/5)
=-\frac N4+O_A\left(\frac{N}{(\log N)^A}+\log N\right)
\sim-\frac N4,\qquad c=-\frac14.}
$$
The sum is complex-valued, but the displayed error controls its absolute difference from the real main term.
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