The Riemann Hypothesis Unveiled

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You’re likely familiar with prime numbers, those unique integers that can only be divided by 1 and themselves. Here’s a question that has puzzled mathematicians for over three millennia:

• 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, p. What is p? 31. What follows p? It’s 37. And after that? 41. Then? 43. But how can one predict the next prime number?

Present a theorem or equation that even vaguely anticipates the next prime in any series of numbers, and your name will forever be associated with one of humanity's most significant intellectual feats, akin to Newton, Einstein, and Gödel. Understand the behavior of prime numbers, and you might never have to engage in any other pursuit.

Introduction

Prime number properties have captivated many of history's mathematical luminaries. From Euclid's initial proof of the infinitude of primes to Euler's product formula linking primes to the zeta function, the quest to understand primes continued. Gauss and Legendre formulated the prime number theorem, which was later validated by Hadamard and de la Vallée Poussin. Among all, Bernhard Riemann stands out as the mathematician who made a groundbreaking contribution to prime number theory. His seminal 1859 paper, comprising just eight pages, unveiled new insights about prime distribution and remains a cornerstone of number theory.

Since its release, Riemann's paper has been the focal point of prime number research and was instrumental in proving what is known as the prime number theorem in 1896. Since then, various new proofs have emerged, including elementary ones by Selberg and Erd?s. However, Riemann's hypothesis regarding the roots of the zeta function continues to baffle mathematicians.

How Many Primes Exist?

Let’s begin with a straightforward premise. Every number is either prime or composite. Composite numbers can be decomposed into products of prime numbers, making primes the essential components of all integers. Euclid demonstrated their infinite nature around 300 BCE, with an elegant proof that unfolds as follows:

Euclid’s Theorem

Assume the set of prime numbers is finite. List all known primes. Let P be the product of this list (the multiplication of all primes). Add 1 to this number, resulting in Q = P + 1. Every integer must either be prime or composite:

• If Q is prime, it’s a prime not included in your original list.
• If Q is composite, it can be expressed as a product of primes, one of which, p, must divide Q. Since every prime p in P divides P, it must also divide the difference between P and Q, which is 1. No prime divides 1, leading to a contradiction; thus, p cannot be on your list.

Consequently, there will always be another prime p not accounted for, confirming the infinitude of primes.

Why Are Primes So Challenging?

The very fact that anyone can comprehend the preceding problem speaks volumes about its complexity. Despite extensive research, the arithmetic properties of primes remain elusive. The scientific community is so aware of our limited understanding of prime behavior that the factorization of large numbers (determining which two primes yield a specific number) forms the basis of encryption theories. Here's one perspective:

We have a clear grasp of composite numbers—those that aren't prime. They consist of primes, and one can easily devise a formula to predict or generate composite numbers. Such a mechanism is termed a sieve. The most renowned example is the “Sieve of Eratosthenes,” dating back to around 200 BCE. It simply marks multiples of each prime up to a specified limit. For instance, for the prime number 2, you'd mark 4, 6, 8, 10, etc. Then, for 3, you’d mark 6, 9, 12, 15, and so forth. What remains will be solely prime numbers. Although easy to grasp, the Sieve of Eratosthenes proves to be inefficient.

A simplified function for your task is 6n ± 1. This function generates all primes except 2 and 3, filtering out all multiples of 3 and all even numbers. Inputting values for n = 1, 2, 3, 4, 5, 6, 7 yields: 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43. The only non-prime results from this function are 25 and 35, which can be expressed as 5 x 5 and 5 x 7, respectively. The next non-primes are 49 = 7 x 7, 55 = 5 x 11, and so forth. Simple, right?

To illustrate this visually, I’ve created what I refer to as “composite ladders,” a straightforward method to visualize how the composite numbers generated by the function relate to each prime, combined. In the first three columns of the image below, you can see the prime numbers 5, 7, and 11 along with their respective composite ladders up to and including 91. The disorder in the fourth column, which shows how the sieve has eliminated all but the prime numbers, aptly illustrates the perplexity surrounding prime numbers.

Fundamental Resources

What relevance does all this have to the well-known "Riemann hypothesis"? In essence, to deepen our understanding of primes, mathematicians in the 1800s shifted their focus from pinpointing individual prime locations to examining the phenomenon of primes collectively. This analytic approach was Riemann's expertise and the foundation of his renowned hypothesis. Before delving into it, we must familiarize ourselves with some essential resources.

The Harmonic Series

The harmonic series is an infinite sequence of numbers first explored by Nicholas Oresme in the 14th century. Its name derives from the concept of musical harmonics, which are overtones beyond the fundamental frequency. The series is expressed as follows:

Oresme proved that this sum diverges, meaning it doesn’t approach any finite limit but extends infinitely.

Zeta Functions

The harmonic series represents a specific case of a broader function known as a zeta function ?(s). The real-valued zeta function is defined for two real numbers, r and n:

For n = 1, it yields the harmonic series, which diverges. However, for all n > 1, the series converges, indicating that the sum approaches a specific number as r increases, rather than extending to infinity.

The Euler Product Formula

Euler was the first to connect zeta functions with prime numbers, demonstrating that for two natural numbers, n and p, where p is prime:

This expression first appeared in a 1737 paper titled Variae observationes circa series infinitas. The equation asserts that the sum of the zeta function equals the product of the reciprocals of one minus the reciprocals of primes raised to the power of s. This remarkable connection laid the groundwork for contemporary prime number theory, which subsequently utilized the zeta function ?(s) to study primes.

I find the proof of this formula particularly captivating, so I will include it here, even though it's not strictly necessary for our discussion (it's just too beautiful!):

Proof of Euler’s Product Formula

Euler starts with the general zeta function:

Initially, he multiplies both sides by the second term:

Next, he subtracts the resulting expression from the zeta function:

He continues this process, multiplying both sides by the third term:

Then, he subtracts the new expression from the zeta function:

Continuing this process infinitely leads to the expression:

If this process seems familiar, it’s because Euler effectively constructed a sieve, similar to the Sieve of Eratosthenes. He filtered out non-prime numbers from the zeta function.

Next, by dividing the expression by all prime reciprocal terms, he arrives at:

In summary, we have demonstrated that:

Wasn't that elegantly done? When substituting s = 1, one retrieves the infinite harmonic series, thereby reaffirming the infinite nature of primes.

The Möbius Function

August Ferdinand Möbius later reformulated the Euler product formula to generate a new sum. In addition to incorporating prime reciprocals, Möbius’ function also encompasses every natural number that results from an odd or even number of prime factors. Numbers divisible by any prime squared are excluded from his series. His sum, represented by ?(n), is as follows:

This sum comprises reciprocals of:

1. Every prime;
2. Every natural number that is the product of an odd count of distinct primes, prefixed by a negative sign;
3. Every natural number that is the product of an even count of distinct primes, prefixed by a positive sign;

Below are the initial terms:

The sum excludes reciprocals of numbers that are multiples of any prime squared, such as 4, 8, 9, and so on.

The Möbius function ?(n) takes on three possible values which either include (1 or -1) or exclude (0) terms from the sum:

Though formally defined by Möbius, this peculiar sum was notably considered by Gauss in a note over 30 years earlier, where he remarked:

"The sum of all primitive roots (of a prime number p) is either ?(0) when p-1 is divisible by a square, or ?(±1) (mod p) when p-1 is the product of distinct prime numbers; if the count of these is even, the sign is positive, while if odd, the sign is negative."

The Prime Counting Function

To grasp how primes are distributed as one ascends the number line, without pinpointing their locations, it’s useful to count how many exist up to a specified number.

The prime counting function ?(x), introduced by Gauss, performs this task, providing the quantity of primes less than or equal to a given real number. Given the absence of a known formula to determine primes, the prime counting function is recognized solely as a plot or step function that increases by 1 whenever x is prime. The graph below depicts the function up to x = 200.

The Prime Number Theorem

The prime number theorem, also conceptualized by Gauss (and independently by Legendre), asserts:

In simpler terms: "As x approaches infinity, the prime counting function ?(x) will approximate the function x/ln(x)." In other words, if one counts sufficiently high and plots the number of primes up to a significantly large number x, while also plotting x divided by the natural logarithm of x, the ratio of the two will converge to 1. The two functions are illustrated below for x = 1000.

In probabilistic terms, the prime number theorem indicates that if one randomly selects a natural number x, the probability P(x) that this number is prime is approximately 1/ln(x). This implies that the average interval between consecutive prime numbers within the first x integers is roughly ln(x).

The Logarithmic Integral Function

The function Li(x) is defined for all positive real numbers except x = 1. It is represented by an integral from 2 to x:

When plotted alongside the prime counting function and the formula from the prime number theorem, it becomes evident that Li(x) offers a superior approximation compared to x/ln(x):

The enhanced accuracy of Li(x) can be observed in a table featuring large values of x, the corresponding number of primes up to x, and the discrepancies of the older (prime number theorem) and newer (logarithmic integral) functions:

As demonstrated here, the logarithmic integral function significantly outperforms the prime number theorem's function, "overshooting" by only 314,890 primes for x = 10^14. Nevertheless, both functions converge towards the prime counting function ?(x). The function Li(x) converges much quicker, yet as x approaches infinity, the ratio of the prime counting function to both Li(x) and x/ln(x) approaches 1. This convergence is visualized below:

The Gamma Function

The gamma function ?(z) has been a significant area of study since Daniel Bernoulli and Christian Goldbach explored the extension of the factorial function to non-integer arguments in the 1720s. This function extends the factorial n! (1 x 2 x 3 x 4 x 5 x … n), shifted down by 1:

Its representation is intriguing:

The gamma function ?(z) is defined for all complex values of z greater than zero. Complex numbers, as you may know, encompass numbers with an imaginary part, expressed as Re(z) + Im(z), where Re(z) denotes the real part (ordinary real number) and Im(z) indicates the imaginary part, represented by the letter i. A complex number is typically expressed in the form z = ? + it, where ? is the real portion and it is the imaginary part. Complex numbers are invaluable, enabling mathematicians and engineers to address problems that ordinary real numbers cannot resolve. Visually, complex numbers extend the traditional one-dimensional "number line" into a two-dimensional "number plane," termed the complex plane, where the real part of a complex number is plotted on the x-axis and the imaginary part on the y-axis.

To utilize the gamma function ?(z), it is commonly rewritten in the form:

Using this identity enables one to derive values for z below zero. However, it does not yield values for negative integers, as they remain undefined (technically termed singularities, or simple poles).

Zeta and Gamma

The relationship between the zeta function and the gamma function is articulated through the following integral:

Bernhard Riemann

Now that we’ve examined the essential resources, we can begin to connect prime numbers with the Riemann Hypothesis.

German mathematician Bernhard Riemann was born in Breselenz in 1826. A student of Gauss, Riemann made significant contributions in analysis and geometry. His most notable achievement was likely in differential geometry, where he laid the foundation for the geometric language later adopted in Einstein’s General Theory of Relativity.

Riemann’s sole foray into number theory, his 1859 paper Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse (On prime numbers less than a given magnitude), is regarded as the most pivotal work in the field. In its four concise pages, he detailed:

1. A definition of the Riemann zeta function ?(s), a complex-valued zeta function;
2. The analytic continuation of the zeta function to all complex numbers s ? 1;
3. A definition for the Riemann xi function ?(s), an entire function related to the Riemann zeta function through the gamma function;
4. Two proofs of the functional equation of the Riemann zeta function;
5. A definition of the Riemann prime counting function J(x) employing the prime counting function and the Möbius function;
6. An explicit formula for the number of primes less than a specified number using the Riemann prime counting function, based on the non-trivial zeros of the Riemann zeta function.

This remarkable feat of intellectual engineering and creativity is unparalleled in the annals of mathematics.

The Riemann Zeta Function

We have already seen the deep connection between prime numbers and the zeta function as demonstrated by Euler in his product formula. Beyond this link, however, little was known about the relationship until the advent of complex numbers revealed just how interwoven the two are.

Riemann was the first to examine the zeta function ?(s) for a complex variable s, where s = ? + it.

Known as the Riemann zeta function ?(s), it is an infinite series that is analytic (having definable values) for all complex numbers with a real part greater than 1 (Re(s) > 1). In this domain, it converges absolutely.

To analyze the function in areas beyond its regular convergence (when the real part of the complex variable s exceeds 1), the function requires redefinition. Riemann accomplished this through analytic continuation to an absolutely convergent function in the half-plane Re(s) > 0.

This new zeta function definition is analytic across the half-plane Re(s) > 0, except at s = 1, where a singularity/simple pole exists. This is termed a meromorphic function in this domain since it is holomorphic (complex differentiable in a neighborhood of every point within its domain) except at the simple pole s = 1. It also exemplifies a Dirichlet L-function.

In his paper, Riemann did not stop there. He extended his zeta function ?(s) to the entire complex plane using the gamma function ?(z). To keep this article straightforward, I won’t delve into that calculation here, but I highly encourage you to read it yourself as it showcases Riemann’s exceptional intuition and technique (edit 03.13.20: the calculation is available in Veisdal (2013) pp. 28).

Riemann’s method employs the integral representation of gamma ?(z) for complex variables and a function known as the Jacobi theta function ?(x), which can be rewritten to incorporate the zeta function. To express zeta,

In this representation, one can observe that the term ?(s) decreases more rapidly than any power of x, ensuring convergence for all values of s.

Furthermore, Riemann noted that the first term in the braces (-1 / s(1 - s)) remains unchanged if one substitutes s with 1 - s. By doing this, Riemann expanded the utility of the equation by eliminating the two poles at s=0 and s=1, thus defining the Riemann xi function ?(s) with no singularities:

Zeros of the Riemann Zeta Function

The roots or zeros of the zeta function, occurring when ?(s)=0, can be categorized into two types: "trivial" and "non-trivial" zeros of the Riemann zeta function.

Existence of Zeros with Re(s) < 0

The trivial zeros are easily identifiable. They manifest clearly in the following functional representation of the zeta function:

This product equals zero when the sine term equals zero, occurring at k?. For instance, for negative even integers s = -2n, the zeta function becomes zero. However, for positive even integers s = 2n, the zeros cancel out due to the poles of the gamma function ?(z). This is more evident in the original functional format, where substituting s = 2n renders the first part of the term undefined.

Thus, the Riemann zeta function has zeros at every negative even integer s = -2n. These are the trivial zeros, illustrated in the function plot below:

Existence of Zeros with Re(s) > 1

From Euler’s product formulation of zeta, it is evident that ?(s) cannot be zero in the region where the real part of s is greater than 1, as a convergent infinite product can only be zero if one of its factors equals zero. The proof of the infinitude of primes corroborates this assertion.

Existence of Zeros with 0 ? Re(s) ? 1

We have now identified the trivial zeros of zeta in the negative half-plane where Re(s) < 0 and established that no zeros exist for Re(s) > 1.

The region between these two areas, known as the critical strip, has been the focal point of analytic number theory for centuries.

In the above plot, I have graphed the real parts of ?(s) in red and the imaginary parts in blue. The first two trivial zeros are evident in the lower left when the real part of s is -2 and -4. Within the range of 0 to 1, I have marked the critical strip and highlighted where the real and imaginary parts of ?(s) intersect. These intersections represent the non-trivial zeros of the Riemann zeta function. As we progress to higher values, more zeros appear, alongside two seemingly random functions that appear to become denser as the imaginary part of s increases.

The Riemann Xi Function

We have defined the Riemann Xi function ?(s) (the variant of the functional equation devoid of singularities, hence defined for all values of s) as:

This function exhibits the relationship:

This symmetry around the vertical line Re(s) = 1/2 implies that ?(1) = ?(0), ?(2) = ?(-1), and so forth. This functional relationship (the symmetry of s and 1-s) combined with the Euler product formula indicates that the Riemann xi function ?(s) can only possess zeros within the range 0 ? Re(s) ? 1. Thus, the zeros of the Riemann xi function correspond to the non-trivial zeros of the Riemann zeta function. Essentially, the critical line R(s) = 1/2 for the Riemann zeta function ?(s) parallels the real line (Im(s) = 0) for the Riemann xi function ?(s).

Upon inspecting the two charts above, one should immediately observe that all non-trivial zeros of the Riemann zeta function ?(s) (the zeros of the Riemann xi function) possess real parts Re(s) equal to 1/2. Riemann briefly alluded to this occurrence in his paper, a passing remark that would evolve into one of his most significant legacies.

The Riemann Hypothesis

The non-trivial zeros of the Riemann zeta function ?(s) possess real parts Re(s) = 1/2.

This constitutes the contemporary formulation of the unproven conjecture posited by Riemann in his renowned paper. In essence, it asserts that the points at which zeta equals zero, ?(s) = 0, within the critical strip 0 ? Re(s) ? 1, all have real parts Re(s) = 1/2. If validated, all non-trivial zeros of Zeta will take the form ?(1/2 + it).

An equivalent assertion (Riemann’s original declaration) is that all roots of the Riemann xi function ?(s) are real.

In the plot below, the line Re(s) = 1/2 serves as the horizontal axis. The real part Re(s) of zeta ?(s) is represented by the red graph, while the imaginary part Im(s) is depicted by the blue graph. The non-trivial zeros are the points of intersection between the red and blue graphs along the horizontal line.

Should the Riemann hypothesis prove to be accurate, all non-trivial zeros of the function will manifest on this line as intersections between the two graphs.

Reasons to Believe the Hypothesis

Numerous reasons bolster the belief in the validity of Riemann’s hypothesis regarding the zeros of the zeta function. Perhaps the most persuasive reasoning for mathematicians lies in the implications it would have for the distribution of prime numbers. The numerical validation of the hypothesis to exceptionally high values suggests its truth. In fact, the empirical evidence supporting the hypothesis is robust enough to be considered experimentally verified in other disciplines, such as physics and chemistry. However, the history of mathematics reveals several conjectures that were numerically validated to extreme extents yet ultimately disproven. Derbyshire (2004) recounts the story of the Skewes number, an extraordinarily large value that provided an upper bound, disproving one of Gauss’ conjectures—that the logarithmic integral Li(x) is always greater than the prime counting function. It was refuted by Littlewood without a counterexample, then subsequently shown to fail beyond Skewes’ immensely large number, 10^(10^(34)), illustrating that while Gauss’ assertion was disproven, an example of exactly where remains beyond the reach of numerical computation even today. This might also hold true for Riemann’s hypothesis, which has been "verified" only up to 10^12 non-trivial zeros.

The Riemann Zeta Function and Prime Numbers

Using the assumption of the Riemann hypothesis as a starting point, Riemann began to explore its implications. In his paper, he stated: “…it is very probable that all roots are real. Of course, one would wish for a rigorous proof here; I have, for the time being, after some fleeting vain attempts, provisionally set aside the search for this, as it appears dispensable for the next objective of my investigation.” His subsequent goal was to connect the zeros of the zeta function to prime numbers.

Recall the prime counting function ?(x), which denotes the number of primes up to and including a real number x. Riemann utilized ?(x) to define his own prime counting function, the Riemann prime counting function J(x). It is articulated as follows:

The first notable aspect of this function is that it is not infinite. At some point, the counting function will yield zero because no primes exist for x < 2. Taking J(100) as an example, the function will comprise seven terms since the eighth term includes an eighth root of 100, approximately 1.778279..., which renders this prime counting term zero, resulting in J(100) = 28.5333… .

Similar to the prime counting function, the Riemann prime counting function J(x) acts as a step function that increases in value when:

To relate the value of J(x) to the count of primes up to and including x, we can retrieve the prime counting function ?(x) through a process known as Möbius inversion (which I will not elaborate on here). The resulting expression is:

Remember that the possible values of the Möbius function are:

This implies we can now express the prime counting function in terms of the Riemann prime counting function, leading us to:

This new expression remains a finite sum because J(x) equals zero when x < 2 since no primes exist below 2.

Now, examining our example of J(100), we arrive at the sum:

This indicates the total number of primes below 100.

Translating the Euler Product Formula

Riemann subsequently employed the Euler product formula as a basis to analytically evaluate prime numbers using the infinitesimal language of calculus. Beginning with Euler:

By initially taking the logarithm of both sides and then reconfiguring the denominators within the parentheses, he derived the relationship:

Next, applying the well-known Maclaurin Taylor series, he expanded each log term on the right side, creating an infinite sum of infinite sums—one for each term in the prime number series.

Examining a specific term, for instance:

This term, along with all others in the calculation, signifies a segment of the area beneath the J(x) function. Expressed as an integral:

In essence, employing the Euler product formula, Riemann demonstrated that one can represent the discrete prime counting step function as a continuous sum of integrals. The example term below illustrates its relation to the area beneath the Riemann prime counting function graph.

Thus, each expression in the finite sum forming the prime reciprocal series of the Euler product formula can be articulated as integrals, culminating in an infinite sum of integrals that correspond to the area beneath the Riemann prime counting function. For the prime number 3, this infinite product of integrals is expressed as:

Collectively, all these infinite sums can be consolidated into a single integral, the integral beneath the Riemann prime counting function J(x), expressed as:

Or, in the more widely recognized format:

Through this, Riemann has connected his zeta function ?(s) with his Riemann prime counting function J(x) in an identity statement equivalent to the Euler product formula, articulated in the language of calculus.

The Error Term

After deriving his analytic version of the Euler product formula, Riemann proceeded to formulate his own prime number theorem. The explicit form he proposed was:

This represents Riemann’s explicit formula. It serves as an enhancement of the prime number theorem, offering a more precise estimation of the number of primes up to and including a number x. The formula comprises four terms:

1. The first term, or "principal term," is the logarithmic integral Li(x), which serves as a superior estimate of the prime counting function ?(x) as compared to the prime number theorem. It is by far the dominant term and, as previously discussed, tends to overestimate the number of primes present up to a certain value x.
2. The second term, or "periodic term," is the sum of the logarithmic integral of x raised to the power of each of the non-trivial zeros of the Riemann zeta function. This term adjusts for the overestimation created by the principal term.
3. The third term is the constant -log(2) = -0.6993147… .
4. The fourth and final term is an integral that equals zero for x < 2, as there are no primes less than 2. It reaches its maximum value at 2, with an integral of approximately 0.1400101… .

The last two terms contribute infinitesimally to the function's value as x increases. For large numbers, the primary "contributors" are the logarithmic integral function and the periodic sum. Observe their effects in the following chart:

In this graph, I have approximated the prime counting function ?(x) using Riemann's explicit formula for the Riemann prime counting function J(x) and summed over the first 35 non-trivial zeros of the Riemann zeta function ?(s). The periodic term induces a "resonance" effect, causing the function to closely resemble the shape of the prime counting function ?(x).

The subsequent chart illustrates the same concept, now utilizing a greater number of non-trivial zeros.

With Riemann's explicit function, one can estimate the number of primes up to and including a specific number x with remarkable accuracy. In fact, Von Koch proved in 1901 that employing the non-trivial zeros of the Riemann zeta function for error correction in the logarithmic integral function yields the "best possible" bound for the error term in the prime number theorem.

"These zeros act like telephone poles, and the special nature of Riemann’s zeta function dictates precisely how the wire—its graph—must be strung between them." — Dan Rockmore

Epilogue

Since Riemann’s passing in 1866 at the young age of 39, his groundbreaking paper has remained a landmark in prime and analytic number theory. To this day, Riemann’s hypothesis regarding the non-trivial zeros of the Riemann zeta function remains unresolved, despite extensive research conducted by numerous eminent mathematicians over centuries. Each year, new results and conjectures associated with the hypothesis are presented, with the hope that a proof will eventually materialize.

This essay is an abridged version of my undergraduate thesis, “Prime Numbers and the Riemann Zeta Function.” The full thesis is available here. For those interested in further exploring the topic, I particularly recommend John Derbyshire’s book Prime Obsession (Amazon Affiliate Link).

This essay is part of a series on math-related topics published in Cantor’s Paradise, a weekly Medium publication. Thank you for reading!