I posted this question on Math Stack Exchange (MSE) on 21 Nov 2018 but it attracted only a single answer. This single answer agreed with my view that Proposition 5.1 in Choie et al 2007 appears to be false. However this paper had four co-authors and was published in a peer-reviewed journal which suggests any apparent problem with proposition 5.1 may be illusory, so I would prefer more than a single answer about its validity.
The question on MSE is, https://math.stackexchange.com/questions/3008367/is-proposition-5-1-in-choie-et-al-2007-correct
My motive in posting the question on MathOverflow is obviously I am hoping to get answers from those who look at MathOverflow but who may not look at MSE.
Why don’t I just send the question to the authors of the paper? I tried that with a different and simpler question but did not get a reply. I doubted it would be any different with this question so I have not tried.
The focus of Choie et al 2007 (references below) is Theorem 1 in Robin 1984 which states that the Riemann Hypothesis is true if and only if there are no numbers $ \gt$ 5040 that violate Robin’s Inequality, $ \sigma(n)\lt e^\gamma n\log\log n$ .
There are some minor errors in Choie et al 2007, for example, on p.357 the inclusion of n = 1 in the domain of Robin’s Inequality when Robin’s Inequality is undefined at n = 1, or in Theorem 1.4 where 72 is omitted from the list of squareful numbers that violate Robin’s Inequality.
However my question is about Proposition 5.1 on page 367 which for brevity I will refer to as just “5.1”.
When 5.1 is reformulated in terms of the sets involved then it appears to be equivalent to a statement of the form, $ A\cap B = \emptyset $ then $ B = \emptyset $ .
The statement of 5.1 in Choie et al 2007 is, “If Robin’s inequality holds for all Hardy–Ramanujan integers n with 5041 $ \leq $ n $ \leq $ x it holds for all integers in this same range.”
Hardy–Ramanujan integers are defined in Choie et al 2007, p.367 but it is not necessary to know their definition to understand my question.
Define the following sets,
H = set of all Hardy-Ramanujan numbers in ℕ
X = set of integers n such that 5041 ≤ n ≤ x
V = set of numbers in ℕ that violate Robin’s Inequality
$ \emptyset $ = empty set
then 5.1 may be re-stated as,
5.1a : If $ (H\cap X)\cap V = \emptyset $ then $ X\cap V = \emptyset $
and so 5.1 iff 5.1a (equivalence 1).
Set intersection is associative so 5.1a may be re-stated,
5.1b : If $ H\cap (X\cap V) = \emptyset $ then $ X\cap V = \emptyset $
and so 5.1b iff 5.1a (equivalence 2).
Equivalences(1,2) imply 5.1 iff 5.1b (equivalence 3).
5.1b has the form, $ A\cap B = \emptyset $ then $ B = \emptyset $ , and hence 5.1b is false and so equivalence 3 implies 5.1 is false.
Corollary : If 5.1 is false, and since any argument that claims to prove a false statement must itself be false, then the particular proof of 5.1 in Choie et al 2007 is false.
Note that the conclusions of 5.1a and 5.1b, $ X\cap V = \emptyset $ , is equivalent to the Riemann Hypothesis via Theorem 1 in Robin 1984 because 5.1 does not impose any upper bound on x in the definition of X and therefore x can be arbitrarily large $ \geq $ 5041.
Is the above correct? Does it show 5.1 is equivalent to 5.1b and hence 5.1 is false?
The reformulation of 5.1 as a statement about the sets involved and the steps that lead to 5.1 being equivalent to 5.1b, seem straight forward.
On the other hand, there is some reason for skepticism as 5.1 appears in a paper co-authored by four professional mathematicians and published in a peer-reviewed journal. This makes it seem probable that if there was a problem with 5.1 then it would have been noticed.
Robin,G., “Grandes Valeurs de la Fonction Somme des Diviseurs et Hypothèse de Riemann”, J. de Math. pures et appl., 63 (1984) 187-213.
YoungJu Choie, Nicolas Lichiardopol, Pieter Moree and Patrick Solé, “On Robin’s Criterion for the Riemann Hypothesis” “Journal de Théorie des Nombres de Bordeaux” 19 (2007) 357-372.