The Robin Inequality On Certain Numbers

. In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 12 . In 1915, Ramanujan proved that under the assumption of the Riemann hypothesis, the inequality σ ( n ) < e γ × n × log log n holds for all suﬃciently large n , where σ ( n ) is the sum-of-divisors function and γ ≈ 0 . 57721 is the Euler-Mascheroni constant. In 1984, Guy Robin proved that the inequality is true for all n > 5040 if and only if the Riemann hypothesis is true. Since then, this is called as the Robin inequality. It is known that the Robin inequality is satisﬁed for many classes of numbers. We show more classes of numbers for which the Robin inequality is always satisﬁed.


Introduction
As usual σ(n) is the sum-of-divisors function of n [1]: d|n d.
Define f (n) to be σ(n) n . Say Robins(n) holds provided f (n) < e γ × log log n.
The constant γ is the Euler-Mascheroni constant, and log is the natural logarithm. The importance of this property is: Theorem 1.1 Robins(n) holds for all n > 5040 if and only if the Riemann hypothesis is true [3].
It is known that Robins(n) holds for many classes of numbers n. We recall that an integer n is said to be square free if for every prime divisor q of n we have q 2 n [1]. Robins(n) holds for all n > 5040 that are square free [1]. Let core(n) denotes the square free kernel of a natural number n [1]. In addition, we show that: Let π 2 6 × log log core(n) ≤ log log n for some n > 5040. Then Robins(n) holds.

Theorem 1.3
Suppose that n > 5040. Then Robins(n) holds if every prime q that divides n satisfies q < β log n.

Known Results
We use that the following are known:

Constants
Let α = π 2 6 and β = e γ where γ is the Euler-Mascheroni constant. These constants play a key role in the next proofs. Lemma 1.6 The following hold:

A Central Lemma
The following is a key lemma. It gives an upper bound on f (n) that holds for all n. The bound is too weak to prove Robins(n) directly, but is critical because it holds for all n. Further the bound only uses the primes that divide n and not how many times they divide n. This is a key insight.
Proof Let n > 5040. Specifically, let core(n) be the product of the primes q 1 , . . . , q m , such that {q 1 , . . . , q m } ⊆ {2, 3, 5}. We need to prove that is true, because of lemma 1.4. Then, we have that However, for n > 5040 e γ × log log(5040) < e γ × log log n and hence, the proof is finished.

Main Insight
The next theorem is a main insight. It extends the class of n so that Robins(n) holds. The key is that the class on n depend on how close n is to core(n).
The usual classes of such n are defined by their prime structure not by an inequality. This is perhaps one of the main insights.
Proof Let n = core(n). Let n be the product of the distinct primes q 1 , . . . , q m . By assumption we have that × log log n ≤ log log n.
We claim that π 2 6 × m i=1 q i + 1 q i > e γ × log log n.
Since otherwise we would have a contradiction. This shows that × e γ × log log n .

Implications of Main Insight
The next lemma is a consequence of the theorem 4.1.
Lemma 5.1 Let n > 5040 and let n = core(n). Also let n be the product of the primes q 1 < · · · < q m . Then α × log log n > log log n =⇒ q β m ≥ log n where α = π 2 6 and β = e γ .
Proof Suppose by way of contradiction that We have that α × log log n > log log n holds and so does log n > (log n) 1 α .
We need some basic results from number theory. As usual define θ(x) as where the sum is over all primes q. Note that θ(q m ) = log q 1 + · · · + log q m .
The function θ(x) is x with small error, but we will need that for x > 0 it follows that where c = 1.01624 [4]. We note that θ(q m ) is greater than or equal to the logarithm of n and so by assumptions θ(q m ) ≥ log n > (log n) 1 α . This shows that c × (log n) 10454. This shows that we have a contradiction since 1 α − 1 β > 0.0464.

Proof of Main Theorem
Theorem 6.1 Suppose that n > 5040. Then Robins(n) holds if every prime q that divides n satisfies q < β log n.
Proof Let n > 5040. There are two cases: Case I: In this case π 2 6 × log log core(n) ≤ log log n.
This case follows from theorem 4.1, since Robins(n) is true in this case.
Case II: In this case π 2 6 × log log core(n) > log log n.
By lemma 5.1 we see that q β ≥ log n. This proves the theorem.