Note for the Riemann Hypothesis

1. Introduction

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 $\frac{1}{2}$. It is considered by many to be the most important unsolved problem in pure mathematics. The hypothesis was proposed by Bernhard Riemann (1859). The Riemann hypothesis belongs to Hilbert’s eighth problem on David Hilbert’s list of twenty-three unsolved problems. This is one of the Clay Mathematics Institute’s Millennium Prize Problems. In recent years, there have been several developments that have brought us closer to a proof of the Riemann hypothesis. There are many approaches to the Riemann hypothesis based on analytic number theory, algebraic geometry, non-commutative geometry, etc [1].

The Riemann zeta function $\zeta \left(s\right)$ is a function under the domain of complex numbers. It has zeros at the negative even integers: These are called the trivial zeros. The zeta function is also zero for other values of s, which are called nontrivial zeros. The Riemann hypothesis is concerned with the locations of these nontrivial zeros. Bernhard Riemann conjectured that the real part of every nontrivial zero of the Riemann zeta function is $\frac{1}{2}$.

The Riemann hypothesis’s importance remains from its deep connection to the distribution of prime numbers, which are essential in many computational and theoretical aspects of mathematics. Understanding the distribution of prime numbers is crucial for developing efficient algorithms and improving our understanding of the fundamental structure of numbers. Besides, the Riemann hypothesis stands as a testament to the power and allure of mathematical inquiry. It challenges our understanding of the fundamental structure of numbers, inspiring mathematicians to push the boundaries of their field and seek ever deeper insights into the universe of mathematics.

2. Materials and methods

In mathematics, the Chebyshev function $\theta \left(x\right)$ is given by

$$\theta \left(x\right)=\sum _{q\le x}logq$$

with the sum extending over all prime numbers q that are less than or equal to x, where log is the natural logarithm. We know the following inequalities:

Proposition 1. 

For $x\ge -1$ and $0\le r\le 1$ (This is a useful variant of Bernoulli’s inequality [2]):

$${(1+x)}^r\le 1+r·x.$$

Proposition 2. 

For $x>-1$ [3](pp. 1):

$$\frac{x}{1+x}\le log(1+x).$$

Leonhard Euler studied the following value of the Riemann zeta function (1734) [4].

Proposition 3. 

We define [4]((1) pp. 1070):

$$\zeta \left(2\right)=\prod _{k=1}^{\infty }\frac{q_k^2}{q_k^2-1}=\frac{{\pi }^2}{6},$$

where $q_k$ is the $k^{\mathrm{th}}$ prime number (Mathematicians also use the notation $q_n$ to represent the $n^{\mathrm{th}}$ prime number). By definition, we have

$$\zeta \left(2\right)=\sum _{n=1}^{\infty }\frac{1}{n^2},$$

where n denotes a natural number. Leonhard Euler proved in his solution to the Basel problem that

$$\sum _{n=1}^{\infty }\frac{1}{n^2}=\prod _{k=1}^{\infty }\frac{q_k^2}{q_k^2-1}=\frac{{\pi }^2}{6},$$

where $\pi \approx 3.14159$ is a well-known constant linked to several areas in mathematics such as number theory, geometry, etc.

The number $\gamma \approx 0.57721$ is the Euler-Mascheroni constant which is defined as

$$\begin{array}{cc}\hfill \gamma & =\underset{n\to \infty }{lim}\left(-logn+\sum _{k=1}^n\frac{1}{k}\right)\hfill \\ \hfill & ={\int }_1^{\infty }\left(-\frac{1}{x}+\frac{1}{\lfloor x\rfloor }\right)dx.\hfill \end{array}$$

Here, $\lfloor \dots \rfloor$ represents the floor function. In number theory, $\Psi \left(n\right)=n·{\prod }_{q\mid n}\left(1+\frac{1}{q}\right)$ is called the Dedekind $\Psi$ function, where $q\mid n$ means the prime q divides n.

Definition 1. 

We say that $\mathsf{Dedekind}\left(q_n\right)$ holds provided that

$$\prod _{q\le q_n}\left(1+\frac{1}{q}\right)\ge \frac{e^{\gamma }}{\zeta \left(2\right)}·log\theta \left(q_n\right).$$

A natural number $N_n$ is called a primorial number of order n precisely when,

$$N_n=\prod _{k=1}^n{q}_k.$$

We define $R\left(n\right)=\frac{\Psi \left(n\right)}{n·loglogn}$ for $n\ge 3$. $\mathsf{Dedekind}\left(q_n\right)$ holds if and only if $R\left(N_n\right)\ge \frac{e^{\gamma }}{\zeta \left(2\right)}$ is satisfied.

Proposition 4. 

Unconditionally on Riemann hypothesis, we know that [5] (Proposition 3 pp. 3):

$$\underset{n\to \infty }{lim}R\left(N_n\right)=\frac{e^{\gamma }}{\zeta \left(2\right)}.$$

The well-known asymptotic notation $\Omega$ was introduced by Godfrey Harold Hardy and John Edensor Littlewood [6]. In 1916, they also introduced the two symbols ${\Omega }_R$ and ${\Omega }_L$ defined as [7]:

$$\begin{array}{cc}\hfill & f\left(x\right)={\Omega }_R\left(g\left(x\right)\right)\mathit{as}x\to \infty \mathit{if}\underset{x\to \infty }{lim\; sup}\frac{f\left(x\right)}{g\left(x\right)}>0;\hfill \\ \hfill & f\left(x\right)={\Omega }_L\left(g\left(x\right)\right)\mathit{as}x\to \infty \mathit{if}\underset{x\to \infty }{lim\; inf}\frac{f\left(x\right)}{g\left(x\right)}<0.\hfill \end{array}$$

After that, many mathematicians started using these notations in their works. From the last century, these notations ${\Omega }_R$ and ${\Omega }_L$ changed as ${\Omega }_{+}$ and ${\Omega }_{-}$, respectively. There is another notation: $f\left(x\right)={\Omega }_{\pm }\left(g\left(x\right)\right)$ (meaning that $f\left(x\right)={\Omega }_{+}\left(g\left(x\right)\right)$ and $f\left(x\right)={\Omega }_{-}\left(g\left(x\right)\right)$ are both satisfied). Nowadays, the notation $f\left(x\right)={\Omega }_{+}\left(g\left(x\right)\right)$ has survived and it is still used in analytic number theory as:

$$f\left(x\right)={\Omega }_{+}\left(g\left(x\right)\right)\mathit{if}\exists k>0\forall x_0\exists x>x_0:f\left(x\right)\ge k·g\left(x\right)$$

which has the same meaning to the Hardy and Littlewood older notation. For $x\ge 2$, the function f was introduced by Nicolas in his seminal paper as [8] ([Theorem 3 pp. 376]), [9] ((5.5) pp. 111):

$$f\left(x\right)=e^{\gamma }·log\theta \left(x\right)·\prod _{q\le x}\left(1-\frac{1}{q}\right).$$

Finally, we have Nicolas’ Theorem:

Proposition 5. 

If the Riemann hypothesis is false then there exists a real b with $0<b<\frac{1}{2}$ such that, as $x\to \infty$ [8] (Theorem 3 (c) pp. 376), [9] (Theorem 5.29 pp. 131):

$$logf\left(x\right)={\Omega }_{\pm }\left(x^{-b}\right).$$

Putting all together yields a proof for the Riemann hypothesis.

3. Results

The following inequality is a trivial result:

Lemma 1. 

Let ${ϵ}_1$ be a positive integer between 0 and $e-1$ (i.e. $0<{ϵ}_1<e-1$). Then,

$$log\left(1-e^{-1}·({ϵ}_1+1)\right)\ge -\frac{e^{-1}·({ϵ}_1+1)}{1-e^{-1}·({ϵ}_1+1)}.$$

Proof. 

We can apply the Proposition 2 since $-e^{-1}·({ϵ}_1+1)>-1$. Therefore, we only need to replace x by $-e^{-1}·({ϵ}_1+1)$ in the following expression

$$\frac{x}{1+x}\le log(1+x).$$

□

Several analogues of the Riemann hypothesis have already been proved. Many authors expect (or at least hope) that it is true. Nevertheless, there exist some implications in case the Riemann hypothesis could be false. The following is a key Lemma.

Lemma 2. 

If the Riemann hypothesis is false, then there exist infinitely many prime numbers $q_n$ such that $\mathsf{Dedekind}\left(q_n\right)$ fails (i.e. $\mathsf{Dedekind}\left(q_n\right)$ does not hold).

Proof. 

Let’s define a function called $g\left(x\right)$:

$$g\left(x\right)=\frac{e^{\gamma }}{\zeta \left(2\right)}·log\theta \left(x\right)·\prod _{q\le x}{\left(1+\frac{1}{q}\right)}^{-1}.$$

This function is based on some previously proven results (reference: [5] (Theorem 4.2 pp. 5)). It involves several things: the constants $\gamma$ and $\zeta \left(2\right)$, the Chebyshev function $\theta \left(x\right)$, and a product considering all prime numbers less than or equal to x.

We’re interested in a specific condition, called $\mathsf{Dedekind}\left(q_n\right)$ (see Definition 1). This proof argues that $\mathsf{Dedekind}\left(q_n\right)$ could fail under the possibility that the Riemann hypothesis is false. That circumstance involves an infinitely many real numbers $x_0$ greater than or equal to 5. We claim that $\mathsf{Dedekind}\left(q_n\right)$ fails for infinitely many prime numbers $q_n$ such that $q_n$ refers to the largest prime number less than or equal to $x_0$. For this $x_0$, the value of $g\left(x_0\right)$ must be greater than 1 (or equivalently, $logg\left(x_0\right)>0$).

There’s a previously established relationship between $g\left(x\right)$ and $f\left(x\right)$ [5] (Theorem 4.2 pp. 5):

$$logg\left(x\right)\ge logf\left(x\right)-\frac{2}{x}.$$

If the Riemann hypothesis (RH) is false, then there must be infinitely many numbers x for which $logf\left(x\right)={\Omega }_{+}\left(x^{-b}\right)$ by Proposition 5. This result depends on another number b between 0 and $\frac{1}{2}$ (i.e. $0<b<\frac{1}{2}$). Nicolas proved the general case $logf\left(x\right)={\Omega }_{\pm }\left(x^{-b}\right)$, but we only need to use the notation ${\Omega }_{+}$ under the domain of the real numbers. According to the Hardy and Littlewood definition, this would mean

$$\exists k>0,\forall y_0\in \mathbb{R},\exists y\in \mathbb{R}(y>y_0):logf\left(y\right)\ge k·y^{-b}.$$

The previous inequality is $logf\left(y\right)\ge \left(k·y^{-b}·\sqrt{y}\right)·\frac{1}{\sqrt{y}}$, where we notice that

$$\underset{y\to \infty }{lim}\left(k·y^{-b}·\sqrt{y}\right)=\infty$$

for $k>0$ and $0<b<\frac{1}{2}$. Now, this implies

$$\forall y_0\in \mathbb{R},\exists y\in \mathbb{R}(y>y_0):logf\left(y\right)\ge \frac{1}{\sqrt{y}}.$$

This inequality would mean that under a false RH, there are infinitely many widely spaced real numbers x where $logf\left(x\right)\ge \frac{1}{\sqrt{x}}$. Here’s how this connects back to our original function $g\left(x\right)$. Because of $\frac{1}{\sqrt{x_0}}>\frac{2}{x_0}$ for $x_0\ge 5$, hence if the false RH scenario holds, then there must be infinitely many such $x_0$ where $logg\left(x_0\right)>0$.

Finally, the proof establishes a link between these positive $logg\left(x_0\right)$ values and the prime numbers. It shows that if the logarithm of $g\left(x_0\right)$ is positive for a specific $x_0\ge 5$, then it must also be positive for the largest prime number $q_n$ less than or equal to $x_0$. This connection arises from the properties of the terms used in the definition of $g\left(x\right)$ and the Chebyshev function. □

Lemma 3. 

If $R\left(N_n\right)$ is strictly decreasing (i.e. $R\left(N_n\right)>R\left(N_{n+1}\right)$) for n big enough then $\mathsf{Dedekind}\left(q_n\right)$ holds for n big enough.

Proof. 

Assume $R\left(N_n\right)>R\left(N_{n+1}\right)$ for $n>n_0$ and that $\mathsf{Dedekind}\left(q_m\right)$ fails for $m>n_0$ that is

$$R\left(N_m\right)<\frac{e^{\gamma }}{\zeta \left(2\right)},$$

then for $n\ge m+1$ we have $R\left(N_{n+1}\right)<R\left(N_n\right)<\frac{e^{\gamma }}{\zeta \left(2\right)}$. This implies

$$\underset{n\to \infty }{lim\; sup}R\left(N_n\right)<\frac{e^{\gamma }}{\zeta \left(2\right)}$$

contradicting Proposition 4. □

This is the main insight.

Theorem 1. 

The inequality $R\left(N_n\right)>R\left(N_{n+1}\right)$ holds for n big enough.

Proof. 

By Lemma 3, $\mathsf{Dedekind}\left(q_n\right)$ holds for n big enough if the following inequality is satisfied for a sufficiently large value of n:

$$R\left(N_{n+1}\right)<R\left(N_n\right).$$

This translates to:

$$\frac{{\prod }_{q\le q_{n+1}}\left(1+\frac{1}{q}\right)}{log\theta \left(q_{n+1}\right)}<\frac{{\prod }_{q\le q_n}\left(1+\frac{1}{q}\right)}{log\theta \left(q_n\right)}.$$

Applying logarithms to both sides and expanding the terms, we get:

$$loglog\theta \left(q_{n+1}\right)>loglog\theta \left(q_n\right)+\sum _{q_n<q\le q_{n+1}}log\left(1+\frac{1}{q}\right).$$

Dividing both sides by $loglog\theta \left(q_{n+1}\right)$ (since $q_{n+1}$ is large enough to ensure $loglog\theta \left(q_{n+1}\right)>0$), we have:

$$1>\frac{loglog\theta \left(q_n\right)}{loglog\theta \left(q_{n+1}\right)}+\frac{{\sum }_{q_n<q\le q_{n+1}}log\left(1+\frac{1}{q}\right)}{loglog\theta \left(q_{n+1}\right)}.$$

Taking exponentials of both sides yields:

$$e>exp\left(\frac{loglog\theta \left(q_n\right)}{loglog\theta \left(q_{n+1}\right)}\right)·{\left(\prod _{q_n<q\le q_{n+1}}\left(1+\frac{1}{q}\right)\right)}^{\frac{1}{loglog\theta \left(q_{n+1}\right)}}.$$

For a sufficiently large prime $q_{n+1}$, we can leverage the property $e=x^{\frac{1}{logx}}$ for $x>0$ to obtain:

$$e={\left(log\theta \left(q_{n+1}\right)\right)}^{\frac{1}{loglog\theta \left(q_{n+1}\right)}}.$$

Therefore, it suffices to show that:

$$log\theta \left(q_{n+1}\right)>\prod _{q_n<q\le q_{n+1}}\left(1+\frac{1}{q}\right).$$

This simplifies to:

$$log\theta \left(q_{n+1}\right)>1+\frac{1}{q_{n+1}}$$

which is trivially true for n big enough. That would mean

$$e·(1-{ϵ}_2)={\left(\prod _{q_n<q\le q_{n+1}}\left(1+\frac{1}{q}\right)\right)}^{\frac{1}{loglog\theta \left(q_{n+1}\right)}}$$

for some positive integer ${ϵ}_2$ between 0 and 1 (i.e. $0<{ϵ}_2<1$). Besides, we have:

$$1+{ϵ}_1=exp\left(\frac{loglog\theta \left(q_n\right)}{loglog\theta \left(q_{n+1}\right)}\right)$$

where ${ϵ}_1$ is a positive integer between 0 and $e-1$ (i.e. $0<{ϵ}_1<e-1$). Our goal is to prove:

$$e>(1+{ϵ}_1)·e·(1-{ϵ}_2),$$

which simplifies to:

$${ϵ}_2>\frac{{ϵ}_1}{{ϵ}_1+1}.$$

We can also see that:

$$1-e^{-1}·{\left(\prod _{q_n<q\le q_{n+1}}\left(1+\frac{1}{q}\right)\right)}^{\frac{1}{loglog\theta \left(q_{n+1}\right)}}={ϵ}_2.$$

Using Proposition 1 and the fact that $1\le loglog\theta \left(q_{n+1}\right)$ (due to a sufficiently large $q_{n+1}$), we obtain

$$\begin{array}{cc}\hfill {\left(\prod _{q_n<q\le q_{n+1}}\left(1+\frac{1}{q}\right)\right)}^{\frac{1}{loglog\theta \left(q_{n+1}\right)}}& ={\left(1+\frac{1}{q_{n+1}}\right)}^{\frac{1}{loglog\theta \left(q_{n+1}\right)}}\hfill \\ \hfill & \le 1+\frac{1}{q_{n+1}·loglog\theta \left(q_{n+1}\right)}\hfill \\ \hfill & <1+\frac{1}{q_{n+1}·loglog\theta \left(q_{n+1}\right)-1}\hfill \\ \hfill & =\frac{loglog\theta \left(q_{n+1}\right)}{loglog\theta \left(q_{n+1}\right)-\frac{1}{q_{n+1}}}.\hfill \end{array}$$

So, we arrive at:

$$1-\frac{e^{-1}·loglog\theta \left(q_{n+1}\right)}{loglog\theta \left(q_{n+1}\right)-\frac{1}{q_{n+1}}}<{ϵ}_2.$$

Combining steps, this follow as

$$1-\frac{e^{-1}·loglog\theta \left(q_{n+1}\right)}{loglog\theta \left(q_{n+1}\right)-\frac{1}{q_{n+1}}}>\frac{{ϵ}_1}{{ϵ}_1+1}.$$

After simple distribution, we make

$$\frac{{ϵ}_1+1}{{ϵ}_1}-\frac{e^{-1}·\frac{{ϵ}_1+1}{{ϵ}_1}·loglog\theta \left(q_{n+1}\right)}{loglog\theta \left(q_{n+1}\right)-\frac{1}{q_{n+1}}}>1$$

and

$$1>\frac{e^{-1}·({ϵ}_1+1)·loglog\theta \left(q_{n+1}\right)}{loglog\theta \left(q_{n+1}\right)-\frac{1}{q_{n+1}}}$$

where

$$loglog\theta \left(q_{n+1}\right)-\frac{1}{q_{n+1}}>e^{-1}·({ϵ}_1+1)·loglog\theta \left(q_{n+1}\right).$$

Using further manipulations, we arrive at:

$$-\frac{1}{q_{n+1}}>\left(e^{-1}·({ϵ}_1+1)-1\right)·loglog\theta \left(q_{n+1}\right).$$

and

$$1<q_{n+1}·\left(1-e^{-1}·({ϵ}_1+1)\right)·loglog\theta \left(q_{n+1}\right)$$

which is

$$0<log{q}_{n+1}+log\left(1-e^{-1}·({ϵ}_1+1)\right)+logloglog\theta \left(q_{n+1}\right)$$

after of applying the logarithm to both sides. That could be rewritten as

$$0<-\frac{e^{-1}·({ϵ}_1+1)}{1-e^{-1}·({ϵ}_1+1)}+log{q}_{n+1}+logloglog\theta \left(q_{n+1}\right)$$

by Lemma 1. That is equivalent to

$$\frac{1}{e·{({ϵ}_1+1)}^{-1}-1}<log{q}_{n+1}+logloglog\theta \left(q_{n+1}\right)$$

since

$$\frac{e^{-1}·({ϵ}_1+1)}{1-e^{-1}·({ϵ}_1+1)}=\frac{1}{e·{({ϵ}_1+1)}^{-1}-1}$$

after multiplying the fraction (so above as below) by $e·{({ϵ}_1+1)}^{-1}$. The inequality

$$\frac{1}{e·{({ϵ}_1+1)}^{-1}-1}<log{q}_{n+1}+logloglog\theta \left(q_{n+1}\right)$$

is the same as

$$\frac{1}{exp\left(1-\frac{loglog\theta \left(q_n\right)}{loglog\theta \left(q_{n+1}\right)}\right)-1}<log{q}_{n+1}+logloglog\theta \left(q_{n+1}\right)$$

because of

$${ϵ}_1=exp\left(\frac{loglog\theta \left(q_n\right)}{loglog\theta \left(q_{n+1}\right)}\right)-1.$$

We can further deduce that

$$\frac{1}{exp\left(1-\frac{loglog\theta \left(q_n\right)}{loglog\theta \left(q_{n+1}\right)}\right)-1}<log{q}_{n+1}+logloglog\theta \left(q_{n+1}\right)$$

holds whenever

$$log{q}_{n+1}+logloglog\theta \left(q_{n+1}\right)<exp\left(1-\frac{loglog\theta \left(q_n\right)}{loglog\theta \left(q_{n+1}\right)}\right)·\left(log{q}_{n+1}+logloglog\theta \left(q_{n+1}\right)\right)$$

also holds. Finally, we can infer that

$$log{q}_{n+1}+logloglog\theta \left(q_{n+1}\right)<exp\left(1-\frac{loglog\theta \left(q_n\right)}{loglog\theta \left(q_{n+1}\right)}\right)·\left(log{q}_{n+1}+logloglog\theta \left(q_{n+1}\right)\right)$$

trivially holds by the fact that

$$exp\left(1-\frac{loglog\theta \left(q_n\right)}{loglog\theta \left(q_{n+1}\right)}\right)>1$$

under the supposition that n is big enough. □

This is the main theorem.

Theorem 2. 

The Riemann hypothesis is true.

Proof. 

In virtue of Lemmas 2 and 3, the Riemann hypothesis is true if the inequality

$$R\left(N_{n+1}\right)<R\left(N_n\right)$$

holds for n big enough. Consequently, the Riemann hypothesis is true by Theorem 1. □

4. Discussion

In number theory, the difference between consecutive prime numbers is called a prime gap. The $n^{\mathrm{th}}$ prime gap refers specifically to the difference between the prime numbers at positions n and $(n+1)$ in the sequence of primes.

In 1936, Harald Cramér, a Swedish mathematician, proposed a conjecture about the size of prime gaps. Cramér conjecture states that the difference between consecutive prime gaps grows no faster than the square of the logarithm of the larger prime gap (i.e. $q_{n+1}-q_n=O\left({(log{q}_n)}^2\right)$). Here, the big O notation represents an upper bound on the order of magnitude of a function.

However, there’s growing evidence that Cramér conjecture might be incorrect [10]. Recent research suggests the conjecture may be violated for infinitely many prime gaps. This conclusion is based on results from a yet-to-be-peer-reviewed paper [11] (Proposition 4 pp. 5; Proposition 7 pp. 7). While Theorem 1 in this work seems to disprove the conjecture, its validity depends on the full publication of [11] through peer review.

5. Conclusion

The Riemann hypothesis holds immense significance not only for number theory, but also for fields as diverse as cryptography and particle physics. A proof wouldn’t just offer deep insights into the nature and distribution of prime numbers, the fundamental building blocks of integers. It would fundamentally reshape various mathematical landscapes, sparking entirely new lines of inquiry. For example, a proven Riemann hypothesis could lead to more efficient methods of prime number generation, which are crucial for securing online communication in cryptography. Furthermore, its implications might extend beyond pure mathematics, potentially influencing our understanding of the distribution of energy levels in complex systems studied in particle physics. In essence, a resolution to the Riemann hypothesis could be a catalyst for groundbreaking discoveries across a wide range of scientific disciplines.