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I’ve only come across one thing in pure mathematics that rivals or even surpasses Euler’s identity. Unlike the previous math section, this one takes more of the form of a narrative.
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Recap of Convergent and Divergent Infinite Series
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“A convergent infinite series has a finite limit. The following are examples of convergent infinite series:
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A divergent infinite series either doesn’t have a limit or the limit goes to ±∞. The following are examples of divergent infinite series:
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Now this is where the story begins. The following is a famous infinite series known as the harmonic series:
Here’s an interesting question: Does the harmonic series converge or diverge?
It’s surprisingly hard to tell, even though it too is one of the simplest and most natural infinite series.
The answer (along with its proof) was discovered around 1350 by a French monk named Oresme. It turns out that the harmonic series diverges, albeit very slowly.” [1]
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The Basel Problem
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“The harmonic series can be described as the sum of the reciprocals of the natural numbers. Another series that presents itself as being similar is the sum of the reciprocals of the squares of the natural numbers:
A problem, first posed by Pietro Mengoli in 1650, asks for the exact sum of this series (in closed form), as well as a proof that this sum is correct. It was known that, unlike the harmonic series, this series does converge, but the exact value of the sum proved hard to find. Mengoli failed to find it, and so did Bernoulli and Leibniz. This became known as the Basel problem.
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It wasn’t until 1731 that the problem was finally solved by Euler. He arrived at the following exact value:
This was a truly remarkable result, for no one had expected π to appear.” [2]
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I’ve come to think of the solved Basel problem as the second most beautiful equation in mathematics. I always love when the fundamental constants turn up serendipitously.
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When I first read about the Basel problem, I thought back to middle school. I remembered discussing the fact that the harmonic series was divergent. But I could’ve sworn we learned an easy way to calculate the sums of infinite series, at least for simple ones like this. Were we really capable as middle-schoolers of solving something that stumped everyone until Euler?
Looking into it, it turns out we’d only learned how to calculate the sums of geometric series. “A geometric series has a constant ratio between successive terms.”i[3]
There’s a sense in which the series mentioned above and the Basel problem are opposites.
In the former, the denominator is given while the exponent ‘travels’ through the natural numbers. In the latter, the exponent is given while the denominator ‘travels’ through the natural numbers.
Despite appearances, the latter turns out to be much stranger and more interesting. It’s an example of what’s known as a p-series as opposed to a geometric series. And with these, there’s no analogous simple way to calculate the sums.
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Prime Numbers
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“More than 2,000 years ago, the Greeks proved that there are infinitely many prime numbers. In my mind, this discovery is the beginning of mathematics ― when humankind realized that, by pure thought alone, it could prove eternal truths.
The primes are the ‘indivisible’ numbers, numbers that can be divided only by themselves and one. They’re fundamental in mathematics, because every natural number (2 or greater) is built by multiplying prime numbers together. They’re like the atoms, the hydrogen and oxygen of arithmetic.” [4]
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“No formula predicts the occurrence of primes, and mathematicians tend to treat them as if they occur randomly.” [5]
“One of the reasons we’re fascinated by prime numbers is that they’re quite weird in the way they behave. On one hand, they kind of feel random. Sometimes you get these long gaps between primes, and then suddenly, like buses, a couple of primes turn up all at once. On the other hand, there are things that we can predict about primes and when they’ll turn up, which is slightly unexpected. They’re not completely random.” [6]
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“The Ulam spiral is constructed by writing the positive integers in a square spiral and specially marking the prime numbers.
Below is a zoomed-out Ulam spiral compared to random odd numbers at the same density.
In both cases, because the numbers are all odd (with the exception of 2), they line up on diagonals. In the Ulam spiral, however, some diagonals have lots of primes and others don’t. It’s something more than just being random.”i[7]
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I never found the prime numbers interesting until I heard them described in these ways. It’s reminiscent of, “Monitoring an awake brain shows activity that’s neither totally predictable nor completely random ― and is emblematic of what’s meant by ‘complex’.”i[8]
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“The Sachs spiral is constructed by lining up the square numbers and then evenly spacing the numbers between them.” [6]
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“Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it’s a mystery into which the human mind will never penetrate.” [Euler]
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“The harmonic series and the Basel problem can be generalized to the following function:
We know that the series diverges when s.=.1, and that it converges when s.=.2. So a natural question is, where between 1 and 2 does this change occur?
It turns out that if s.>.1 (even by just the tiniest bit, such as 1.000001), then the series converges. In other words, 1 is the largest integer value of s for which ζ(s) doesn’t converge, and 2 is smallest integer value of s for which ζ(s) does converge.
What about the integers after 2?
The odd values of s don’t have solutions in closed form.
This pattern continues throughout the natural numbers.” [9]
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The Riemann Zeta Function
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“It turns out that ζ(s) converges not only for all real numbers greater than 1, but for all complex numbers with a real part greater than 1.
The following is a visualization of the function from s on the complex numbers to ζ(s) on the complex numbers, where s has a real part greater than 1:
ζ(s) abruptly stops on the left side. However, there’s a single natural way to extend it, through what’s called ‘analytic continuation’.
Working backwards, an amended version of the function can be created, in which all complex values of s can be assigned a finite value, with the exception of a singularity at 1.” [10]
“It’s defined in the following way:
It’s known as the Riemann zeta function.
It was first published in 1859 by Bernhard Riemann. In a sense, it was discovered piece by piece over centuries.” [11]
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It’s interesting that even after all of these advanced discoveries and strategies, s.=.1 (the one that started it all) still remains undefined.
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The Riemann zeta function unlocks two interesting, important, unexpected, new values:
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“The infinite series whose terms are the natural numbers 1.+.2.+.3.+.4.+.… is a divergent series whose limit goes to infinity. Because the sequence of partial sums fails to converge to a finite limit, the series doesn’t have a sum.
Although the series seems at first sight not to have any meaningful value, it can be manipulated to yield a number of mathematically interesting insights. For example, many summation methods are used to assign numerical values even to a divergent series. In particular, the methods of zeta function regularization (what we’ve been discussing) and Ramanujan summation assign the series a value of -1/12,
where the left-hand side is interpreted as being one of the aforementioned summation methods and not as the sum of an infinite series in its usual meaning. These methods have applications in other fields such quantum field theory and string theory.
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The following is one example of how -1/12 pops out of the series of natural numbers.
The graph below shows the (first few) partial sums of the series 1 + 2 + 3 + 4 + …
The parabola is the smoothed asymptote.
Astonishingly, its y-intercept isn’t 0.
It’s -1/12.” [12]
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The Riemann Hypothesis
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“A new question that arises is, for which values of s does ζ(s).=.0?
It turns out that for all negative even integer values of s, ζ(s).=.0. These have been named the ‘trivial zeros’.
But there are other values of s for which ζ(s).=.0 as well, and these have been named the ‘non-trivial zeros’.
All non-trivial zeros necessarily have a real part between 0 and 1 and are mirrored over the real axis.
However, every non-trivial zero discovered so far has a real part of exactly ½.” [13]
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“The Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part ½.
Although it was first posed in 1859, it has yet to be proved or disproved.” [13]
“Any non-trivial zeros would be mirrored over the critical line (in addition to being mirrored over the real axis). So in a sense, the Riemann hypothesis being true would mean that the Riemann zeta function has the ‘minimum possible’ zeros.” [15]
“The first 10 trillion non-trivial zeros have all been found to have a real part of ½.
It’s consider by many to be the most important unsolved problem in pure mathematics.”i[13]
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“Perhaps most surprisingly of all, it turns out that there’s a deep connection between the Riemann zeta function and the prime numbers ― the behavior and location of the zeros has a direct bearing on the distribution of primes. Riemann’s explicit formula for the number of primes less than a given number says that the magnitude of the oscillations of primes around their expected position is controlled by the real parts of the zeros of the zeta function.
The result has caught the imagination of countless mathematicians because it’s so unexpected, connecting two seemingly unrelated areas in mathematics: number theory, which is the study of the discrete, and complex analysis, which deals with continuous processes.
One of the most compelling reasons for mathematicians to believe that the Riemann hypothesis is true is the hope that the prime numbers are distributed as regularly as possible.” [13]
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“The prime numbers are unpredictable, unconstrained, tantalizingly regular, but never quite the same ― and the Riemann zeta function is a rippled curtain of the imagined and real, deeply tied to them in incomprehensible ways.” [16]
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This is one of the few things that fills me with awe ― like we’re staring into this timeless, higher-order pattern, beyond our comprehension.
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Citations
- C. J. Sangwin & Wikipedia’s Convergent Series article
- C. J. Sangwin & William Dunham
- Wikipedia’s Geometric Series article
- Marcus du Sautoy
- Yitang Zhang
- James Grime & Numberphile
- James Grime, Numberphile, & Wikipedia’s Ulam Spiral article
- Christof Coch
- Wikipedia’s Particular Values of the Riemann Zeta Function article
- Grant Sanderson (3 Blue 1 Brown)
- Numberphile, Wikipedia’s Riemann Zeta Function article, & mathworld.wolfram.com/RiemannZetaFunction
- Wikipedia’s 1 + 2 + 3 + 4 + ⋯ article
- Wikipedia’s Riemann Hypothesis article
- Lukas Golino
- (source lost)
- Randall Munroe
philosophicaltreasures.com 