Since I know some math, I'll assume you're talking to me.

Your latest revision is much better, but there's still a problem with the conclusion drawn from (16). Let S denote the LHS of (14), and B denote the RHS of (16).

You want to show that, for all eps > 0, there exists x_0 s.t. x > x_0 implies S/x < eps. In this statement, x_0 must be a fixed constant (i.e., it can depend on NOTHING but eps).

To accomplish this, you try to show that, for all eps > 0, x > x_0 = c_2(exp(c_2)) implies S/x < B/x = 5c_3/ln(2c_2) < eps. This would be fine if x_0 and 5c_3/ln(2c_2) were not functionally related. However, they clearly ARE related -- through c_2. To try to avoid this problem, you argue that the c_2 in 5c_3/ln(2c_2) doesn't matter, because c_1 can be chosen arbitrarily large. However, choosing c_1 arbitrarily large is equivalent to choosing c_2 arbitrarily large, which changes the value of x_0 you started with.

In this case, epsilon = (5c_3)/(log (2c_2)), so x_0 = c_{2}exp(c_2) is indeed dependent only on epsilon.

The derivation of (16) can actually be done by replacing "for x > (c_2)exp(c_2)" by "for all large enough x".

By the way, please educate your colleagues why the left-hand side of (12) is o(x).

For the moment, let's focus on the proof.

What is the argument for going from (15) to (16)?

Thanks for being only interested in facts.

Okay, the update is, the sum on the extreme right-hand side of (15) is actually >>1 for any given c_2. This, unfortunately, invalidates the proof. That being said, the bound in (12) remains true, and it could be interesting in its own right.

However, I have temporarily withdrawn this paper.

May we now shift attention to the first paper:

https://figshare.com/articles/preprint/Untitled_Item/14776146

By arguing only for sigma in C, one is able to invoke the Mean-Value Theorem for integrals in the evaluation of f(sigma) and g(sigma), thus circumventing the issues raised by some people on MJR and here. Kindly see the latest version.