The Classical Problem Statement

At the core of the Riemann hypothesis lies the Riemann zeta function, thus it is prudent to begin with its definition. The Riemann zeta function is defined as a complex holomorphic function:

It is crucial to note that this definition applies only to complex numbers where the real part exceeds 1, which is necessary to ensure the convergence of the series.

Typically, when discussing the Riemann zeta function, we refer to its analytically continued form, which encompasses all complex numbers except for 1, where a simple pole exists. Therefore, the aforementioned definition can be viewed as a representation of the zeta function limited to the half-plane where Re(s) > 1.

The eminent mathematician Leonhard Euler demonstrated that this function can be expressed through an infinite product of prime numbers:

In this context, ? denotes the set of prime numbers. This relationship is foundational to the theory, linking the analytical properties of the zeta function to the distribution of prime numbers, which are considered as an ordered subset of natural numbers.

This intersection of number theory and complex analysis provides a profound insight into the nature of the Riemann zeta function. For example, one of the simplest and most elegant proofs of the prime number theorem, which asserts that the number of prime numbers increases roughly as x/ln(x), utilizes the zeta function.

Moreover, it can be shown that the prime number theorem is closely tied to the absence of zeros of the zeta function along the line Re(s) = 1. Although there are no zeros along this line, the analytically continued zeta function has infinitely many zeros, which are the solutions to the equation ?(s) = 0.

These zeros play a crucial role in understanding the distribution of prime numbers, and we are eager to locate them within the complex plane. Such knowledge would provide the best possible bounds on the growth of primes.

We know that zeros occur in two distinct families. The first are referred to as the trivial zeros, which are precisely the negative even integers:

The second set consists of the non-trivial zeros. All non-trivial zeros are proven to have a real part that lies between 0 and 1. This can be easily demonstrated, particularly when we establish that there are no zeros on the lines Re(s) = 1 and Re(s) = 0.

The Euler product confirms that there are no zeros in the half-plane where Re(s) > 1, and a functional equation:

indicates that the trivial zeros occur at negative even integers, while symmetry ensures there are no other zeros with negative real parts.

The functional equation was established by Bernhard Riemann in a concise yet groundbreaking paper published in 1859. In that same paper, Riemann calculated the initial few non-trivial zeros and observed that all were situated on a vertical line, specifically Re(s) = 1/2, leading to the famous hypothesis.

The Riemann Hypothesis:

> All non-trivial zeros of ? have a real part of 1/2.

This query has perplexed mathematicians, including myself, since 1859. The proof (or disproof) of this assertion remains elusive.

Let’s Make it Real

Numerous equivalent expressions of the Riemann hypothesis exist, most involving complex analysis or intricate insights into prime number theory that we are far from solving, such as precise bounds on the divisor function or the exact distribution of prime numbers.

I do not suggest that the approach I will present here is superior; in fact, it may be less effective. However, my goal is to foster interest in this monumental quest by framing it through real analysis, devoid of complex jargon or theorems, thus inviting more individuals to engage with this challenge.

> At the very least, they may find joy in the pursuit!

The foundation of this approach will be to consider a related function and subsequently break it down into its real and complex components. Ultimately, a few manipulations will yield a problem equivalent to the Riemann Hypothesis.

The Dirichlet Eta Function

As mentioned earlier, the series definition of the zeta function converges solely in the half-plane where Re(s) > 1, which is not particularly useful for examining zeros since our interest lies within the critical strip 0 < Re(s) < 1.

Fortunately, we have a fascinating related function known as the Dirichlet eta function, defined as follows:

The series for the eta function converges when Re(s) > 0. This isn’t immediately obvious, but it serves as an interesting exercise for those sleepless nights fueled by too much caffeine.

Functions of this nature, like the zeta and eta functions, are classified as Dirichlet series, each possessing an abscissa of convergence ?.

For any given Dirichlet series, the abscissa of convergence is a real number that delineates the boundary between the half-plane of convergence and that of divergence. Formally, if Re(s) > ?, then the series converges.

Informally, since the series for the eta function converges for any real number s > 0 and diverges for s ? 0, the abscissa of convergence must be 0.

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

This is significant as it reveals important information regarding the zeros of the eta function.

Firstly, ? shares all the zeros of ?. Additionally, ? possesses infinitely many zeros along the line Re(s) = 1, resulting from the first factor above.

Interestingly, within the critical strip, where we know all the non-trivial zeros of the zeta function reside, the eta function has precisely the same zeros.

In other words, a version of the Riemann Hypothesis applies to the eta function as well, stating that all non-trivial zeros of the eta function (those located within the critical strip) have a real part of 1/2.

This is equivalent to the standard Riemann hypothesis, yet the series definition of the eta function is valid within the critical strip, unlike that of the zeta function.

Let us now separate the eta function into its real and imaginary components.

Given that s is a complex number, we must first comprehend what it means for a real number to be raised to the power of a complex number.

We require Euler’s identity!

Recall from Euler that

This remarkable fact indicates that the exponential function is periodic with an imaginary period.

Let’s utilize this to decompose the eta function. First, observe that if we express s as ? + it, we have

Thus, we can express the Dirichlet eta function as follows:

It is not immediately clear why we can separate the series into real and imaginary parts and assume convergence. I encourage the reader to contemplate this.

Next, we will define two functions:

Where ? > 0 and ?, t ? ? are real numbers.

We could conclude here, stating that the Riemann hypothesis is equivalent to the assertion that if both ? and ? equal zero and 0 < ? < 1, then ? = 1/2.

However, it is somewhat inconvenient to exclude zeros on the line ? = 1, necessitating restrictions on the functions.

Let’s introduce one more twist. We will substitute ? with a logistic function, often referred to as the sigmoid function.

Define f, g: ?² ? ? as follows:

Note that the exponent of n, when viewed as a function of r, is the logistic function ?(r) = 1/(1 + exp(-r)). The function ? takes any real number and satisfies 0 < ?(r) < 1 for all r ? ?.

The Riemann hypothesis can be articulated as follows:

> f(r, t) = g(r, t) = 0 ? r = 0.

Alternatively, we can construct a mapping T as follows:

> T: ?² ? ?² defined by (r, t) ? (f(r, t), g(r, t)) as two-dimensional real vectors. Consequently, the Riemann Hypothesis posits that if T((r, t)) = (0, 0), then r must equal 0.

The Riemann Hypothesis is a captivating challenge, and viewing it through the lens of real analysis may not provide all the answers. Indeed, complex analysis offers a wealth of theory and more potent tools. However, it is beneficial to approach a problem from multiple perspectives.

I hope this approach renders the problem more comprehensible and accessible to those who may not have encountered complex function theory, while also providing a fresh perspective for those who have.

Ultimately, it is always intriguing to revisit this subject, as the Riemann Hypothesis resembles a beloved book; no matter how many times it has been read, it is always a delight to explore again.

I am left wondering how this story will conclude.