For $q$ near $1$ this type of reasoning fails, but then we may use the other form of the function. Adding the two arguments all the result may be proved.

@Malik Younsi I have added some ideas to finish the argument. I have proved that the function decrease for 0<q< 0.29... I have some problems with the TeX. May somebody edit this?

I think a better idea is to take log. But then the coefficients are also positive and negative, and interesting.

I have changed the problem in other I think more tractable. But I see that still there is something missing.

I have a little confusion with increasing decreasing, but this must be the solution

But the uniform distribution of Rademacher-Hlawka refers only to the behavior of numbers mod 1. So that the gaps between the zeros has nothing to do with the uniform distribution os these zeros. See the references: H. Rademacher [in Collected works, 434--458, Cornell Univ. Press, Ithaca, NY; per bibl.], E. Hlawka [Österreich. Akad. Wiss. Math.-Naturwiss. Kl. S.-B. II 184 (1975), no. 8-10, 459--471; MR0453661 (56 #11921)] and A. Fujii [Comment. Math. Univ. St. Paul. 42 (1993), no. 2, 161--187; MR1241296 (94i:11065)]

Yes, but I found them while trying to locate the zeros in general. There are an infinite number of zeros that run mainly on a line parallell to the imaginary axis. Of course these zeros are separated one of other approximately in 2 pi i. For example the function for N = 8 has zeros at the points 27.26763038 + 163.7377084 i, 27.52677947 + 170.0540751 i, 27.77668695 + 176.3684616 i, 28.01799005 + 182.6810464 i, 28.25126216 + 188.9919872 i, 28.47702109 + 195.301423 i.

Not mentioning the zero at z=0 and complex conjugates: For N = 3 the smallest solution is 3.838602048 + 8.366815507 i. For N = 4 the smallest solution is 5.439213999 + 9.129463691 i. For N = 5 the smallest solution is 6.952562475 + 9.800729397 i. For N = 6 the smallest solution is 8.407369863 + 10.40707148 i. for N = 7 the smallest solution is 8.28937986 + 8.687044071 i for N = 8 the smallest solution is 9.536388591 + 8.053389728 i but there is a smaller real solution at -9.773234001

I just used Mathematica and the definition you give in this post. I give you here my definition, because certainly 100 is not the limit of my definition. And what I do not see in the numbers other can see. w[n_] := w[n] = Module[{Y, x, y, k}, Y = 1/(n + 1); Clear[x, y]; For[k = n - 1, k >= 1, k--, Y = (Y /. {x -> y}); (Print["Y = ", Y];) Y = Integrate[Y, {y, x, k/(k + 1)}]; (Print[k/(k+1)])]; n! (Y /. x -> 0) ] The last integral is also interesting when you do not compute it to 0. I do not saw your query before.