Robin’s Criterion on Divisibility (II)

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Robin’s Criterion on Divisibility (II)

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Frank Vega^*

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Submitted:

16 October 2024

Posted:

21 October 2024

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Abstract

Robin's criterion states that the Riemann hypothesis is true if and only if the inequality $\sigma(n) < e^{\gamma } \cdot n \cdot \log \log n$ holds for all natural numbers $n > 5040$, where $\sigma(n)$ is the sum-of-divisors function of $n$ and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. The possible smallest counterexample $n > 5040$ of the Robin inequality implies that $(N_{m})^{Y_{m}} > n$, $(\log q_{m}) \cdot \left(1 + \frac{1}{\log^{2}(q_{m})}\right) > \log \log n$ and $\left(1 + \frac{0.2}{\log^{3}(q_{m})}\right) > \frac{\log \log N_{m}}{\log q_{m}}$, where $N_{m} = \prod_{i = 1}^{m} q_{i}$ is the primorial number of order $m$, $q_{m}$ is the largest prime divisor of $n$ and $Y_{m} = \frac{e^{\frac{0.2}{\log^{2}(q_{m})}}}{\left(1 - \frac{1}{\log^{3}(q_{m})}\right)}$. By combining these results, we present a proof of the Riemann hypothesis. This work is an expansion and refinement of the article "Robin's criterion on divisibility", published in The Ramanujan Journal.

Keywords:

Subject: Computer Science and Mathematics - Algebra and Number Theory

MSC: Primary 11M26; Secondary 11A41; 11A25

1. 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

\frac{1}{2}

. As usual

σ (n)

is the sum-of-divisors function of n:

\sum_{d ∣ n} d

where

d ∣ n

means the integer d divides n. Define

f (n)

to be

\frac{σ (n)}{n}

. We say that

Robin (n)

holds provided that

f (n) < e^{γ} \cdot log log n .

The constant

γ \approx 0.57721

is the Euler-Mascheroni constant and log is the natural logarithm. An upper bound for

f (n)

can be derived from its multiplicity:

Proposition 1.1.

For

n > 1

([4] (2.7) pp. 362):

f (n) < \prod_{q ∣ n} \frac{q}{q - 1} .

The following inequality is based on natural logarithms:

Proposition 1.2.

For

t > 0

[5]:

log (1 + \frac{1}{t}) < \frac{1}{t} .

The Chebyshev function

θ (x)

is given by

θ (x) = \sum_{p \leq x} log p

where

p \leq x

means all the prime numbers p that are less than or equal to x. It is known that

Proposition 1.3.

For

x \geq 7232121212

([3] Lemma 2.7 (4) pp. 19):

θ (x) \geq (1 - \frac{0.01}{{log}^{3} (x)}) \cdot x .

Proposition 1.4.

For

x \geq 2278382

([3] [Lemma 2.7 (5) pp. 19]):

\prod_{q \leq x} \frac{q}{q - 1} \leq e^{γ} \cdot (log x) \cdot (1 + \frac{0.2}{{log}^{3} (x)}) .

Proposition 1.5.

For

x > 1

([8] [Corollary 1 (3.30) pp. 70]):

\prod_{q \leq x} \frac{q}{q - 1} < e^{γ} \cdot (log x) \cdot (1 + \frac{1}{{log}^{2} (x)}) .

The Ramanujan’s Theorem states that if the Riemann hypothesis is true, then

Robin (n)

holds for large enough n [6]. Next, we have the Robin’s Theorem:

Proposition 1.6.

Robin (n)

holds for all natural numbers

n > 5040

if and only if the Riemann hypothesis is true [7] [Theorem 1 pp. 188].

In 1997, Ramanujan’s old notes were published where it was defined the generalized highly composite numbers, which include the superabundant and colossally abundant numbers [6]. These numbers were also studied by Leonidas Alaoglu and Paul Erdos (1944) [2]. Let

q_{1} = 2, q_{2} = 3, \dots, q_{m}

denote the first m consecutive primes, then an integer of the form

\prod_{i = 1}^{m} q_{i}^{a_{i}}

with

a_{1} \geq a_{2} \geq \dots \geq a_{m} \geq 0

is called an Hardy-Ramanujan integer [4] [pp. 367]. A natural number n is called superabundant precisely when, for all natural numbers

m < n

f (m) < f (n) .

Proposition 1.7.

If n is superabundant, then n is a Hardy-Ramanujan integer [2] [Theorem 1 pp. 450].

Several analogues of the Riemann hypothesis have already been proved. Many authors expect (or at least hope) that it is true. However, there are some implications in case of the Riemann hypothesis could be false.

Proposition 1.8.

n > 5040

is the smallest integer such that

Robin (n)

does not hold, then n must be a superabundant number [1] [Theorem 3 pp. 273].

Proposition 1.9.

n > 5040

is the smallest integer such that

Robin (n)

does not hold, then

q < log n

where q is the largest prime factor of n [4] [Lemma 6.1 pp. 369].

Proposition 1.10.

n > 5040

is the smallest integer such that

Robin (n)

does not hold, then

q > e^{31.018189471}

where q is the largest prime factor of n [9] [Theorem 4.2 pp. 748].

By combining these results, we present a proof of the Riemann hypothesis.

2. Main Result

Definition 2.1.

For every prime number

p_{n} > 2

, we define the sequence

Y_{n} = \frac{e^{\frac{0.2}{{log}^{2} (p_{n})}}}{(1 - \frac{0.01}{{log}^{3} (p_{n})})} .

The following is a key Lemma.

Lemma 2.2.

Let

q_{1}, q_{2}, \dots, q_{m}

denote the first m consecutive primes such that

q_{1} < q_{2} < \dots < q_{m}

and

q_{m} > 7232121212

. Then

\prod_{i = 1}^{m} \frac{q_{i}}{q_{i} - 1} \leq e^{γ} \cdot log (Y_{m} \cdot θ (q_{m})) .

Proof.

By Proposition 1.3, we know that

θ (q_{m}) \geq (1 - \frac{0.01}{{log}^{3} (q_{m})}) \cdot q_{m} .

In this way, we can show that

\begin{matrix} log (Y_{m} \cdot θ (q_{m})) & \geq log (Y_{m} \cdot (1 - \frac{0.01}{{log}^{3} (q_{m})}) \cdot q_{m}) \\ = log q_{m} + log (Y_{m} \cdot (1 - \frac{0.01}{{log}^{3} (q_{m})})) . \end{matrix}

We notice that

\begin{matrix} log (Y_{m} \cdot (1 - \frac{0.01}{{log}^{3} (q_{m})})) & = log (\frac{e^{\frac{0.2}{{log}^{2} (q_{m})}}}{(1 - \frac{0.01}{{log}^{3} (q_{m})})} \cdot (1 - \frac{0.01}{{log}^{3} (q_{m})})) \\ = log (e^{\frac{0.2}{{log}^{2} (q_{m})}}) \\ = \frac{0.2}{{log}^{2} (q_{m})} . \end{matrix}

Consequently, we obtain that

log q_{m} + log (Y_{m} \cdot (1 - \frac{0.01}{{log}^{3} (q_{m})})) \geq (log q_{m} + \frac{0.2}{{log}^{2} (q_{m})}) .

By Proposition 1.4, we can prove that

\prod_{i = 1}^{m} \frac{q_{i}}{q_{i} - 1} \leq e^{γ} \cdot (log q_{m} + \frac{0.2}{{log}^{2} (q_{m})}) \leq e^{γ} \cdot log (Y_{m} \cdot θ (q_{m}))

when

q_{m} > 7232121212

. □

This is the main insight.

Lemma 2.3.

n > 5040

is the smallest integer such that

Robin (n)

does not hold, then

{(N_{m})}^{Y_{m}} > n

(log q_{m}) \cdot (1 + \frac{1}{{log}^{2} (q_{m})}) > log log n

and

(1 + \frac{0.2}{{log}^{3} (q_{m})}) > \frac{log log N_{m}}{log q_{m}}

, where

N_{m} = \prod_{i = 1}^{m} q_{i}

is the primorial number of order m and

n = \prod_{i = 1}^{m} q_{i}^{a_{i}}

Proof.

By Propositions 1.7 and 1.8, the primes

q_{1} < \dots < q_{m}

must be the first m consecutive primes and

a_{1} \geq a_{2} \geq \dots \geq a_{m} \geq 0

. In addition, we know that

q_{m} > e^{31.018189471}

by Proposition 1.10. If

n > 5040

is the smallest integer such that

Robin (n)

does not hold, then we deduce that

f (n) \geq e^{γ} \cdot log log n

and

\prod_{i = 1}^{m} \frac{q_{i}}{q_{i} - 1} > f (n)

by Proposition 1.1. In addition, we know that

\prod_{i = 1}^{m} \frac{q_{i}}{q_{i} - 1} \leq e^{γ} \cdot log log ({(N_{m})}^{Y_{m}})

for all

q_{m} > e^{31.018189471}

by Lemma 2.2 since

log ({(N_{m})}^{Y_{m}}) = Y_{m} \cdot θ (q_{m})

. As result, we obtain that

{(N_{m})}^{Y_{m}} > n

since

e^{γ} \cdot log log ({(N_{m})}^{Y_{m}}) > e^{γ} \cdot log log n

by transitivity. By Proposition 1.1 and 1.5, we can see that

\begin{matrix} e^{γ} \cdot (log q_{m}) \cdot (1 + \frac{1}{{log}^{2} (q_{m})}) & > \prod_{q \leq q_{m}} \frac{q}{q - 1} \\ > f (n) \\ \geq e^{γ} \cdot log log n \end{matrix}

under the assumption that

Robin (n)

does not hold. This implies that

(log q_{m}) \cdot (1 + \frac{1}{{log}^{2} (q_{m})}) > log log n .

We claim that

\prod_{q \leq q_{m}} \frac{q}{q - 1} > e^{γ} \cdot log log N_{m}

(1)

under the assumption that

Robin (n)

does not hold. Certainly, if we assume that

\prod_{q \leq q_{m}} \frac{q}{q - 1} \leq e^{γ} \cdot log log N_{m},

then we would have

\begin{matrix} e^{γ} \cdot log log N_{m} & \geq \prod_{q \leq q_{m}} \frac{q}{q - 1} \\ > f (n) \\ \geq e^{γ} \cdot log log n \end{matrix}

where this implies that

N_{m} > n

which is a trivial contradiction according to the Proposition 1.7. By Proposition 1.4, we can infer from (1) the following result:

(1 + \frac{0.2}{{log}^{3} (q_{m})}) \geq \frac{\prod_{q \leq q_{m}} \frac{q}{q - 1}}{e^{γ} \cdot log q_{m}} > \frac{log log N_{m}}{log q_{m}}

which directly implies that

(1 + \frac{0.2}{{log}^{3} (q_{m})}) > \frac{log log N_{m}}{log q_{m}} .

Therefore, the proof is done. □

This is the main Theorem.

Theorem 2.4.

The Riemann hypothesis is true.

Proof.

We will proceed by contradiction. Assume that

n > 5040

is the smallest integer such that

Robin (n)

does not hold. By Propositions 1.7 and 1.8, the primes

q_{1} < \dots < q_{m}

must be the first m consecutive primes and

a_{1} \geq a_{2} \geq \dots \geq a_{m} \geq 0

. By Proposition 1.10, this also implies that

q_{m} > e^{31.018189471}

. By Lemma 2.3, we deduce that

{(N_{m})}^{Y_{m}} > n

which is the same as

log Y_{m} > log log n - log log N_{m}

after of applying the logarithm and distributing the terms. Certainly, we get this inequaltiy following the next steps:

First, we obtain $Y_{m} \cdot log (N_{m}) > log n$ after of applying the logarithm to the both sides.
Next, we get $Y_{m} > \frac{log n}{log (N_{m})}$ when we distribute the terms.
Finally, we arrive at $log Y_{m} > log log n - log log N_{m}$ if we apply the logarithm to the both sides once again.

That is equivalent to

\frac{log Y_{m}}{log q_{m}} > 1 - \frac{log log N_{m}}{log log n}

after dividing both sides by

log log n

and under the assumption that

\frac{1}{log q_{m}} > \frac{1}{log log n}

since

q_{m} < log n

by Proposition 1.9. By Proposition 1.2, we obtain that

\begin{matrix} log Y_{m} & = \frac{0.2}{{log}^{2} (q_{m})} + log (\frac{{log}^{3} (q_{m})}{{log}^{3} (q_{m}) - 0.01}) \\ = \frac{0.2}{{log}^{2} (q_{m})} + log (1 + \frac{0.01}{{log}^{3} (q_{m}) - 0.01}) \\ < \frac{0.2}{{log}^{2} (q_{m})} + \frac{0.01}{{log}^{3} (q_{m}) - 0.01} \end{matrix}

for all

q_{m} > e^{31.018189471}

. So, we would have

\frac{log Y_{m}}{log q_{m}} < \frac{0.2}{{log}^{3} (q_{m})} + \frac{0.01}{(log (q_{m})) \cdot ({log}^{3} (q_{m}) - 0.01)} < \frac{1}{{log}^{3} (q_{m})}

for all

q_{m} > e^{31.018189471}

. We arrive at:

log (\frac{1}{{log}^{3} (q_{m})}) > log (1 - \frac{log log N_{m}}{log log n})

after of applying the logarithm. That would be

- log (\frac{1}{{log}^{3} (q_{m})}) < - log (1 - \frac{log log N_{m}}{log log n})

which is

log {log}^{3} (q_{m}) < log (\frac{log log n}{log log n - log log N_{m}})

and

{log}^{3} (q_{m}) < \frac{log log n}{log log n - log log N_{m}}

after of multiplying both sides by

- 1

and applying the exponentiation. By Lemma 2.3, we can further deduce that

\frac{log log n}{log log n - log log N_{m}} < \frac{(log q_{m}) \cdot (1 + \frac{1}{{log}^{2} (q_{m})})}{(log q_{m}) \cdot (1 + \frac{1}{{log}^{2} (q_{m})}) - log log N_{m}}

where

\frac{(log q_{m}) \cdot (1 + \frac{1}{{log}^{2} (q_{m})})}{(log q_{m}) \cdot (1 + \frac{1}{{log}^{2} (q_{m})}) - log log N_{m}} = \frac{(1 + \frac{1}{{log}^{2} (q_{m})})}{(1 + \frac{1}{{log}^{2} (q_{m})}) - \frac{log log N_{m}}{log q_{m}}} .

Furthermore, we can infer that

\frac{(1 + \frac{1}{{log}^{2} (q_{m})})}{(1 + \frac{1}{{log}^{2} (q_{m})}) - \frac{log log N_{m}}{log q_{m}}} < \frac{(1 + \frac{1}{{log}^{2} (q_{m})})}{(1 + \frac{1}{{log}^{2} (q_{m})}) - (1 + \frac{0.2}{{log}^{3} (q_{m})})}

where

\frac{(1 + \frac{1}{{log}^{2} (q_{m})})}{(1 + \frac{1}{{log}^{2} (q_{m})}) - (1 + \frac{0.2}{{log}^{3} (q_{m})})} = \frac{(1 + \frac{1}{{log}^{2} (q_{m})})}{(\frac{1}{{log}^{2} (q_{m})} - \frac{0.2}{{log}^{3} (q_{m})})} .

Putting all together yields the following inequality:

{log}^{3} (q_{m}) < \frac{(1 + \frac{1}{{log}^{2} (q_{m})})}{(\frac{1}{{log}^{2} (q_{m})} - \frac{0.2}{{log}^{3} (q_{m})})}

which is

({log}^{3} (q_{m})) \cdot (\frac{1}{{log}^{2} (q_{m})} - \frac{0.2}{{log}^{3} (q_{m})}) < (1 + \frac{1}{{log}^{2} (q_{m})}) .

Hence, it is enough to show that

log (q_{m}) - 0.2 < (1 + \frac{1}{{log}^{2} (q_{m})})

does not hold for all

q_{m} > e^{31.018189471}

since

({log}^{3} (q_{m})) \cdot (\frac{1}{{log}^{2} (q_{m})} - \frac{0.2}{{log}^{3} (q_{m})}) = log (q_{m}) - 0.2 .

Thus our original assumption that

Robin (n)

does not hold has led to a final contradiction. By reductio ad absurdum, we prove that the Riemann hypothesis is true by Proposition 1.6. □

References

Amir Akbary and Zachary Friggstad. Superabundant numbers and the Riemann hypothesis. The American Mathematical Monthly, 116(3):273–275, 2009. [CrossRef]
Leonidas Alaoglu and Paul Erdos. On Highly Composite and Similar Numbers. Transactions of the American Mathematical Society, 56(3):448–469, 1944. [CrossRef]
Safia Aoudjit, Djamel Berkane, and Pierre Dusart. On Robin’s criterion for the Riemann Hypothesis. Notes on Number Theory and Discrete Mathematics, 27(4):15–24, 2021. [CrossRef]
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(2):357–372, 2007. [CrossRef]
Jean-Louis Nicolas. The sum of divisors function and the Riemann hypothesis. The Ramanujan Journal, 58:1113–1157, 2022. [CrossRef]
Jean-Louis Nicolas and Guy Robin. Highly Composite Numbers by Srinivasa Ramanujan. The Ramanujan Journal, 1(2):119–153, 1997. [CrossRef]
Guy Robin. Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann. J. Math. pures appl, 63(2):187–213, 1984.
J. Barkley Rosser and Lowell Schoenfeld. Approximate Formulas for Some Functions of Prime Numbers. Illinois Journal of Mathematics, 6(1):64–94, 1962. [CrossRef]
Frank Vega. Robin’s criterion on divisibility. The Ramanujan Journal, 59(3):745–755, 2022. [CrossRef]

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Robin’s Criterion on Divisibility (II)

Abstract

1. Introduction

2. Main Result

References

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