# [seqfan] Is this correct paper: Oscillations in Mertens Theorems and Other Finite Sums and Products

Georgi Guninski guninski at guninski.com
Sun May 13 13:09:44 CEST 2012

```Oscillations in Mertens Theorems and Other Finite Sums and Products [1]
gives exact formulas some prime sums in certain cases.

It defines the set E:
The exceptional subset of real numbers E ⊂ R is defined by
E = { x ∈ R : π ( x) − li ( x) = Ω± ( x ^(1/ 2) log log log x / log x) }.

For x in E the paper claims p. 8:
θ ( x)=x + Ω± ( x^(1/2) log log log x) [A]

where theta is the Chebyshev theta.

This is exact and efficiently computable formula (if E is known).

On the other hand
li[θ(x)] − π(x) > 0 [B] is equivalent to RH by G. Robin at [2]

Substituting the definition of E and A in B gives:
li[x + Ω± ( x^(1/2) log log log x)] - (li(x)+Ω± ( x ^(1/ 2) log log log x / log x))>0 [C]

[C] doesn't hold in the range [20,10^300] which seemingly contradicts RH
( using the approximation of of li(x) ~ x/log(x) gives analytic reason
for the failure).

If the papers are correct probably I am missing something.

Thank you.

[1] http://arxiv.org/abs/1005.4708
[2] http://arxiv.org/abs/arXiv:1109.6489

```