Landau–Ramanujan comparison conjecture for gcd parameter 3

Let $k$ be a positive integer and $q$ a prime with $(k,q-1)=3$. Let $S_{k,q}(x)$ and $S'_{k,q}(x)$ denote the two sums considered in the paper, and compare their Landau and Ramanujan approximations. Landau–Ramanujan comparison conjecture for $r=3$. The Landau approximation for $S_{k,q}(x)$ is better than the Ramanujan approximation for every prime $q$ except

$$q \in \{7,13,19,31,37,61,67,79,97,103,109,127,181\},$$

for which the Ramanujan approximation is better. The Landau approximation for $S'_{k,q}(x)$ is better than the Ramanujan approximation for every prime $q$ except

$$q\in\{7,13,19,31,61,67,97,109\}.$$

These assertions are based on numerical experiments because the authors could not test all remaining primes for the case $r=3$; the relevant comparisons are expected to require extensive computation.