The Classical Problem Statement

At the heart of the Riemann Hypothesis lies the Riemann zeta function, which we can define as follows:

The Riemann zeta function is a complex holomorphic function defined for complex numbers where the real part is greater than 1. This condition ensures that the series converges. However, when referring to the Riemann zeta function, we typically imply its analytically continued version, which is defined for all complex numbers except for 1, where it has a simple pole.

This leads us to understand that the earlier definition provides an expression for the zeta function limited to the half-plane where Re(s) > 1.

Leonhard Euler demonstrated that this function can be expressed as an infinite product over the prime numbers:

zeta(s) = prod_{p in mathbb{P}} frac{1}{1 - p^{-s}}

where ℙ signifies the set of prime numbers. This connection is profound, linking the analytic characteristics of the zeta function to the distribution of prime numbers within the natural numbers.

The theory of the Riemann zeta function serves as a crossroads between number theory and complex analysis. For example, the most straightforward proof of the prime number theorem, which indicates that the number of primes approximates x/ln(x), utilizes the zeta function.

In fact, it can be shown that the prime number theorem is closely related to the absence of zeros of the zeta function on the line Re(s) = 1. Despite this, the analytically continued zeta function possesses infinitely many zeros, which correspond to solutions of the equation ζ(s) = 0.

These zeros are pivotal as they reveal the distribution of prime numbers. Thus, understanding where these zeros are located within the complex plane could provide the best possible bounds on the growth of primes.

We know that the zeros can be categorized into two families: the trivial zeros, which are the negative even integers, and the non-trivial zeros, which must lie within the range of 0 and 1 in terms of their real part. This has been proven, primarily due to the lack of zeros on the lines Re(s) = 1 and Re(s) = 0.

The Euler product confirms there are no zeros in the half-plane Re(s) > 1, and a functional equation established by Bernhard Riemann in a groundbreaking paper in 1859 demonstrates the trivial zeros at negative even integers, while precluding other zeros from having negative real parts.

Riemann’s paper also computed the first few non-trivial zeros, all of which lay on a straight line: Re(s) = 1/2. This observation led to the famous Riemann Hypothesis:

The Riemann Hypothesis: All non-trivial zeros of ζ have a real part of 1/2.

This question has captivated mathematicians, including myself, since 1859, and no one has yet succeeded in proving or disproving this statement.

Connecting with Reality

There exist multiple equivalent formulations of the Riemann Hypothesis. Most are deeply rooted in complex analysis or complex insights into prime number theory, which remain elusive. I do not assert that my approach is superior; rather, I aim to present an equivalent problem using solely real analysis, devoid of apparent number theory or holomorphic functions, in hopes of engaging more individuals in this monumental quest. At the very least, they may enjoy the thrill of the pursuit!

To initiate this approach, we will examine a related function, decompose it into its real and complex parts, and perform a few manipulations to generate a problem equivalent to the Riemann Hypothesis.

The Dirichlet Eta Function

As I previously noted, the series definition of the zeta function converges only in the half-plane Re(s) > 1, which is not particularly useful for analyzing zeros since the zeros of interest reside in the critical strip where 0 < Re(s) < 1. Fortunately, a fascinating related function exists: the Dirichlet eta function, defined as follows:

eta(s) = sum_{n=1}^{infty} frac{(-1)^{n-1}}{n^s}

The above series converges for Re(s) > 0. While this may not be immediately obvious, it becomes a rewarding exercise to ponder while grappling with sleepless nights. Essentially, such series (like the zeta and eta functions) are categorized as Dirichlet series, each possessing an abscissa of convergence α, which delineates the boundary between convergence and divergence.

For the Dirichlet series, the abscissa of convergence α is the real number that marks this boundary. Formally, if Re(s) > α, then the series converges. Informally, since the eta function series converges for any real number s > 0 and diverges for s ≤ 0, its abscissa of convergence must be 0.

Moreover, it can be verified that the eta function and the zeta function satisfy the following functional equation:

eta(s) = (1 - 2^{1-s}) zeta(s)

This relationship is noteworthy as it reveals information about the zeros of the eta function. Firstly, η shares all the zeros with ζ. Furthermore, η possesses infinitely many zeros on the line Re(s) = 1, deriving from the first factor above. Interestingly, within the critical strip, where all non-trivial zeros of the zeta function lie, the eta function also has identical zeros.

In other words, a Riemann hypothesis exists for the eta function, stating that all non-trivial zeros of η (located within the critical strip) have a real part of 1/2. This claim is equivalent to the traditional Riemann hypothesis, yet the series definition of the eta function holds validity within the critical strip, while that of the zeta function does not.

Let us now separate the eta function into its real and imaginary components. Since s is a complex number, we must clarify what it means to raise a real number to a complex power. This is where Euler's identity comes into play:

e^{ix} = cos(x) + isin(x)

Utilizing this elegant fact, we recognize that the exponential function exhibits periodicity with an imaginary period.

Now, let’s express the Dirichlet eta function as follows:

eta(s) = sum_{n=1}^{infty} frac{(-1)^{n-1}}{n^{sigma + it}} = sum_{n=1}^{infty} frac{(-1)^{n-1}}{n^{sigma}} e^{-it log(n)}

As we delve deeper, we can define two functions, where σ > 0 and σ, t ∈ ℝ are real numbers. We could conclude here and assert that the Riemann hypothesis is equivalent to the statement that if both α and β vanish and 0 < σ < 1, then σ = 1/2. However, it becomes somewhat cumbersome to exclude zeros along the line σ = 1, necessitating a restriction of the functions.

Let’s add one more twist. We can substitute σ with a logistic function, sometimes referred to as the sigmoid function:

sigma(r) = frac{1}{1 + e^{-r}}

This function ensures that for any real number r, 0 < σ(r) < 1.

Thus, the Riemann hypothesis can be expressed as follows:

f(r, t) = g(r, t) = 0 implies r = 0.

Alternatively, we can construct a mapping T defined as:

T: mathbb{R}^2 rightarrow mathbb{R}^2 text{ given by } (r, t) mapsto (f(r, t), g(r, t))

In this context, the Riemann Hypothesis asserts that if T((r, t)) = (0, 0), then r must equal 0.

The Riemann Hypothesis remains an intriguing problem, and approaching it through the lens of real analysis does not necessarily yield a solution. In fact, complex analysis possesses a richer theoretical framework and more potent tools. However, it is beneficial to view a problem from multiple perspectives.

I hope this explanation has rendered the topic more comprehensible and inviting for those unfamiliar with complex function theory, while providing a fresh perspective for those who are well-acquainted with it. Ultimately, pondering this subject is always worthwhile, as the Riemann Hypothesis resembles a captivating book: no matter how many times one revisits it, the experience remains delightful.

The Riemann Hypothesis, Explained - A comprehensive overview of this significant mathematical problem.

What is the Riemann Hypothesis, and why does it matter? by Prof. Ken Ono - An insightful exploration of the implications of the Riemann Hypothesis.