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In this short note, we show an analogue of Dawsey's formula on Chebotarev densities for finite Galois extensions of Q with respect to the Riemann zeta function ζ(ms), m 2. Her formula may be viewed as the limit version of our formula as m → ∞.

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We investigate the maximal solid tubes around short simple closed geodesics in hyperbolic three‐manifolds and how the complex length of curves relates to closed least area incompressible minimal surfaces. As applications, we prove the existence of closed hyperbolic three‐manifolds fibering over the circle which are not foliated by closed incompressible minimal surfaces isotopic to the fiber. We also show the existence of quasi‐Fuchsian manifolds containing arbitrarily many embedded closed incompressible minimal surfaces. Our strategy is to prove main theorems under natural geometric conditions on the complex length of closed curves on a fibered hyperbolic three‐manifold, then by computer programs, we find explicit examples where these conditions are satisfied.