 Solution

The Von Mangoldt function is supported on prime powers:Write $e(t)=\exp (2\pi it)$. A convenient form of the Siegel–Walfisz theorem is: for every fixed $A,B>0$, uniformly for $q\le (\log x{)}^B$ and $(a,q)=1$,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: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,since two is coprime to five and the sum including $a=0$ vanishes. As $\phi (5)=4$, the Siegel–Walfisz theorem yieldsThe sum is complex-valued, but the displayed error controls its absolute difference from the real main term.