[QUOTE=flouran;187557]Maybe they are the same, and you're wrong (after all, neither of us are as mathematically fluent as Polyakov. You're an amateur, and I'm an amateur)[/QUOTE]

I'm an amateur, true, but I know the subject much better than you. :smile:

It's clear that the first R(n) is integer-valued. The second is the sum of logarithms of primes. You can collect these together by the definition of the logarithm to get R(n) = log(m) for some integer m > 0. You know that this can't be an integer, right? So the functions are clearly different. In fact I don't even think they can ever have the same value.

[QUOTE=CRGreathouse;187560]
It's clear that the first R(n) is integer-valued. The second is the sum of logarithms of primes. You can collect these together by the definition of the logarithm to get R(n) = log(m) for some integer m > 0. You know that this can't be an integer, right? So the functions are clearly different. In fact I don't even think they can ever have the same value.[/QUOTE]
Even in [5] Polyakov uses the same definition (for \Omega(n). Note: \Omega(N,n) is my (2.0.3)).

[QUOTE=flouran;187559]I need more time to verify this.[/QUOTE]

Perfectly understandable -- I needed some time just to put it together! :smile:

[QUOTE=flouran;187559]No. (19) only considers the absence of \beta (which assumes GRH??). My upper-bound considers both cases.

However, he gives two lower bounds for both cases.[/QUOTE]

Polyakov's (18) applies in both cases. So is your (3.0.30) better than his (18)? Is your (3.0.31) better than his (18)?

[QUOTE=CRGreathouse;187562]Perfectly understandable -- I needed some time just to put it together! :smile:
[/QUOTE]
Give me a week :smile: Classes just started so I have way less time.
[QUOTE=CRGreathouse;187562]
Polyakov's (18) applies in both cases.
[/QUOTE]
He doesn't explicitly show this.
[QUOTE=CRGreathouse;187562]
So is your (3.0.30) better than his (18)? Is your (3.0.31) better than his (18)?[/QUOTE]
Yes.

[QUOTE=flouran;187561]Even in [5] Polyakov uses the same definition (for \Omega(n). Note: \Omega(N,n) is my (2.0.3)).[/QUOTE]

I saw that, of course. It doesn't ameliorate my concern, especially when Polyakov's other paper was known to be wrong.

But suppose he's using some abuse of notation by which he is correct. If I don't understand that, I'm liable to make mistakes in reading his or your paper. More importantly, if *you* don't understand it, you're liable to have made mistakes in authoring yours.

For example, suppose I didn't understand that the "=" in
x log x = O(x^2)
was actually not expressing an equality, and I substituted "x log x" for "O(x^2)" at some later point in the paper. I would be wrong even though the source stating the above was right.

[QUOTE=flouran;187564][QUOTE=CRGreathouse;187562]Polyakov's (18) applies in both cases.[/QUOTE]
He doesn't explicitly show this.[/QUOTE]

He writes[INDENT]If there is an exceptional zero, then in the right-hand side of (18) there appears
the factor [TEX](1-\beta)\log n,[/TEX] but, in view of (4), the estimate (18) remains valid.
[/INDENT]
[QUOTE=flouran;187564][QUOTE=CRGreathouse;187562]So is your (3.0.30) better than his (18)? Is your (3.0.31) better than his (18)?[/QUOTE]
Yes.[/QUOTE]

That's yes to both?

[QUOTE=CRGreathouse;187566]He writes[INDENT]If there is an exceptional zero, then in the right-hand side of (18) there appears
the factor [TEX](1-\beta)\log n,[/TEX] but, in view of (4), the estimate (18) remains valid.
[/INDENT][/QUOTE]
I had misinterpreted what you had said before. My bad. :smile:

[QUOTE=CRGreathouse;187562]
That's yes to both?[/QUOTE]
Yes.

[QUOTE=CRGreathouse;187565]I saw that, of course. It doesn't ameliorate my concern, especially when Polyakov's other paper was known to be wrong.
[/QUOTE]
The estimate in (14.5) of [5] was wrong. That doesn't mean the notation is necessarily wrong.

An upper-bound on R(n) is certainly notable (my exceptional set is contained within BPP's). Miech proved an equality for R(n) but with more exceptions than my upper-bound. In [5], Polyakov proved an equality for R(n) which had more exceptions to my upper-bound (but was proven wrong). BPP remark that a proof of H & W's conjecture is beyond the current mathematical knowledge.

I also consider both cases (in the absence or existence of \beta). So if the ref finds any mistakes in the proof and I am able to fix those, then it would be worth publishing. If I can't fix any errors in the proof, I'll just throw out the paper altogether. Most likely the latter will occur :smile:

[QUOTE=flouran;187568]That applies to (18), not (19).[/QUOTE]

In fairness, that's what I said.

[QUOTE=flouran;187568]Besides an upper-bound on R(n) has not been proven before in the literature.[/QUOTE]

I can't think of any sense in which
1. An upper bound on R(n) has not been proven in the literature, and
2. You proved an upper bound on R(n)
would hold.

You prove an upper bound on almost all values of R(n). But then again so do the whole list of papers I PM'd you weeks ago: Davenport-Heilbronn, Miech, Polyakov, Brunner-Perelli-Pintz, Wang, and Li. For example, Davenport & Heilbronn prove that, except for a set of density 0,
[TEX]R(n)<\frac{\sqrt n}{\log n}\mathscr{P}(n)+o(\text{stuff})[/TEX] (sorry, not enough references on hand, had to work off memory).

In fact essentially all of the main results can be expressed in the form
"Outside a set of density oO(A(n)), S(n) = ωΩ(B(n)) and S(n) = oO(C(n))"
where oO is either o or O, ωΩ is either ω or Ω, and [TEX]S(n)=R(n)-\frac{\sqrt n}{\log n}\mathscr{P}(n)[/tex]. The C(n) in each of these results is the upper bound.

[QUOTE=flouran;187569]An upper-bound on R(n) is certainly notable (my exceptional set is contained within BPP's). Miech proved an equality for R(n) but with more exceptions than my upper-bound.[/QUOTE]

Wang and Li have fewer exceptions to their upper bounds.



:ermm:



Hang on, I need to check something. I may have been interpreting something in your paper wrongly.

[QUOTE=CRGreathouse;187573]Wang and Li have fewer exceptions to their upper bounds.
[/QUOTE]
Could you show me *exactly* where this is in both papers please.