Dear Professor Kouba,

Up to what n did you check the inequality numerically?

Best, Vladimir

I would look at the sum of cos(k)^2. For a given n, you get (3 + 2 n + Csc[1] Sin[1 + 2 n])/(4 n) as the result. This sum converges to 1/2 from above. Then I would relate the two sums, possibly using inequalities relating absolute values and squares.

If the inequality is true, this is a surprise for me. The reason is that the sequence (n mod 2π)n behaves on (0, 2π) more or less like results of repeating independent experiment, so I would expect that the partial sums of the series |cos(1)| + |cos(2)| + ... behave like a random walk, having in many instances of n the values greater than the mathematical expectation (2n+1)/π , but also in many instances values smaller than (2n+1)/π. The deviation from the mathematical expectation (as I would expect) should be unbounded, so for big n in many instances the values of partial sums are smaller than 2n/π. Of course, this is not a demonstration, but just a feeling what should happen.

Vladimir Kadets It takes about 12 minutes on Mathematica (i3 laptop) to check that the difference remains larger than 1/2 for 0