Short Note on the Riemann Hypothesis

Robin criterion states that the Riemann hypothesis is true if and only if the inequality σ ( n ) < e γ × n × loglog n holds for all natural numbers n > 5040, where σ ( n ) is the sum-of-divisors function of n and γ ≈ 0 . 57721 is the Euler-Mascheroni constant. Let q 1 = 2 , q 2 = 3 ,..., q m denote the first m consecutive primes, then an integer of the form ∏ mi = 1 q a i i with a 1 ≥ a 2 ≥ ··· ≥ a m ≥ 0 is called an Hardy-Ramanujan integer. If the Riemann hypothesis is false, then there are infinitely many Hardy-Ramanujan integers n > 5040 such that Robin inequality does not hold and we prove that n (cid:16) 1 − 0 . 6253 log qm (cid:17) < N m , where N m = ∏ mi = 1 q i is the primorial number of order m and q m is the largest prime divisor of n . In addition, we show that q m will not have an upper bound by some positive value for these counterexamples and therefore, the value of q m tends to infinity as n goes to infinity.


Introduction
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 1 2 [7]. Let N m = 2 × 3 × 5 × 7 × 11 × · · · × q m denotes a primorial number of order m such that q m is the m th prime number [5]. As usual σ (n) is the sum-of-divisors function of n [1]: ∑ d|n d F. Vega CopSonic, 1471 Route de Saint-Nauphary 82000 Montauban, France ORCiD: 0000-0001-8210-4126 E-mail: vega.frank@gmail.com where d | n means the integer d divides n and d ∤ n means the integer d does not divide n. Define f (n) to be σ (n) n . Say Robins(n) holds provided f (n) < e γ × log log n.
The constant γ ≈ 0.57721 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 natural numbers n > 5040 if and only if the Riemann hypothesis is true [7]. Moreover, if the Riemann hypothesis is false, then there are infinitely many natural numbers n > 5040 such that Robins(n) does not hold [7].
It is known that Robins(n) holds for many classes of numbers n. Robins(n) holds for all natural numbers n > 5040 that are not divisible by 2 [1]. 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].
Theorem 1.2 Robins(n) holds for all natural numbers n > 5040 that are square free [1].
Let q 1 = 2, q 2 = 3, . . . , q m denote the first m consecutive primes, then an integer of the form ∏ m i=1 q a i i with a 1 ≥ a 2 ≥ · · · ≥ a m ≥ 0 is called an Hardy-Ramanujan integer [1]. Based on the theorem 1.1, we know this result: Theorem 1.3 If the Riemann hypothesis is false, then there are infinitely many natural numbers n > 5040 which are an Hardy-Ramanujan integer and Robins(n) does not hold [1].
We prove if the Riemann hypothesis is false, then there are infinitely many Hardy-Ramanujan integers n > 5040 such that Robins(n) does not hold and n is the primorial number of order m and q m is the largest prime divisor of n. Furthermore, we show that q m will not have an upper bound by some positive value for these counterexamples and thus, the value of q m tends to infinity as n goes to infinity.
Theorem 2.5 [9]. For q m ≥ 20000, we have log q m < log log N m + 0.1253 log q m .

A Central Theorem
The following is a key theorem. It gives an upper bound on f (n) that holds for all natural numbers n. The bound is too weak to prove Robins(n) directly, but is critical because it holds for all natural numbers n. Further the bound only uses the primes that divide n and not how many times they divide n.
Theorem 3.1 Let n > 1 and let all its prime divisors be q 1 < · · · < q m . Then, Proof Putting together the theorems 2.1 and 2.2 yields the proof:

A Particular Case
We can easily prove that Robins(n) is true for certain kind of numbers.
Theorem 4.1 Robins(n) holds for n > 5040 when q ≤ 5, where q is the largest prime divisor of n.
Proof Let n > 5040 and let all its prime divisors be q 1 < · · · < q m ≤ 5, then we need to prove However, we know for n > 5040 e γ × log log(5040) < e γ × log log n and therefore, the proof is complete when q 1 < · · · < q m ≤ 5.

Robin on Divisibility
The next theorem implies that Robins(n) holds for a wide range of natural numbers n > 5040.
Theorem 5.1 Robins(n) holds for all natural numbers n > 5040 when a prime q ≤ 1771559 complies with q ∤ n.
Proof Note that f (n) < n ϕ(n) = ∏ q|n q q−1 from the theorem 2.1, where ϕ(x) is the Euler's totient function. We have that f (n) < 1771561 1771560 × e γ × log log(n) for any number n > 10 10 13.11485 . Suppose that n is not divisible by a prime q for q less than or equal to some prime bound Q and n > N = 10 10 13.11485 . Then, for n > N = 10 10 13.11485 . The right hand side is less than 1 for Q ≤ 1771559. Moreover, note that the inequality 10 10 13.11485 > e e 23.762143 is satisfied. Therefore, Robins(n) holds as a consequence of the theorems 2.3 and 2.4.

A Main Insight
The next theorem is a main insight. Theorem 6.1 Let π 2 6 × log log n ′ ≤ log log n for some natural number n > 5040 such that n ′ is the square free kernel of the natural number n. Then Robins(n) holds.
Proof Let n ′ be the square free kernel of the natural number n, that is the product of the distinct primes q 1 , . . . , q m . By assumption we have that × log log n ′ ≤ log log n.
We claim that Since otherwise we would have a contradiction. This shows that This is a contradiction since f (n ′ ) is equal to according to the formula f (x) for the square free numbers [1]. < N m , where N m = ∏ m i=1 q i is the primorial number of order m and q m is the largest prime divisor of n. In addition, q m will not have an upper bound by some positive value for these counterexamples and therefore, the value of q m tends to infinity as n goes to infinity.
Proof Let ∏ m i=1 q a i i be the representation of some natural number n > 5040 as a product of primes q 1 < · · · < q m with natural numbers as exponents a 1 , . . . , a m . The primes q 1 < · · · < q m must be the first m consecutive primes and a 1 ≥ a 2 ≥ · · · ≥ a m ≥ 0 since the natural number n > 5040 will be an Hardy-Ramanujan integer. We assume that Robins(n) does not hold. Indeed, we know there are infinitely many Hardy-Ramanujan integers such as n > 5040 when the Riemann hypothesis is false according to the theorem 1.3. From the theorem 5.1, we know that necessarily q m ≥ 1771559. So, because of the theorems 2.1 and 2.6. Hence, log log n < log q m + 0.5 log(q m ) .
From the theorem 2.5, we have that log log n < log log N m + 0.1253 log q m + 0.5 log(q m ) .
That is the same as log log n − log log N m < 0.6253 log q m . Then, ).
In addition, we know that Finally, we obtain that n 1− 0.6253 log qm < N m . Moreover, we know that q m will not have an upper bound by some positive value for these counterexamples because of the theorem 6.1. Certainly, if there is a possible upper bound for q m , then it cannot exist infinitely many Hardy-Ramanujan integers n > 5040 such that Robins(n) does not hold as a consequence of the theorem 6.1.