>[!abstract]
>Some numbers have the special property that they cannot be expressed as the product of two smaller numbers, e.g., 2, 3, 5, 7, etc. Such numbers are called *prime numbers*, and they play an important role, both in pure mathematics and its applications. The distribution of such prime numbers among all natural numbers does not follow any regular pattern. However, the German mathematician G.F.B. Riemann (1826–1866) observed that the frequency of prime numbers is very closely related to the behavior of an elaborate function $\zeta(s) = 1 + 1/2^s + 1/3^s + 1/4^s + \ldots$ called the Riemann Zeta function. The Riemann hypothesis asserts that all interesting [*non-trivial*] solutions of the equation $\zeta(s) = 0$ lie on a certain vertical straight line [*called the critical line*] (Clay Mathematics Institute, 2025).
^c32e86
<small><i>Note: I added the two italicized brackets to this abstract because they are important concepts for later.</i></small>
## About zeta functions in general
1. **There is a family of mathematical functions called *zeta functions*.** They are represented using the greek letter $\zeta$ (zeta) and the input variable $s$ (the use of $s$ instead of the more common $x$ is just mere convention). So, those functions are expressed as $\zeta(s)$, just like some other functions are written $f(x)$.
<div style="font-size: x-large; text-align:center;">∴</div>
2. **These zeta functions encode certain properties about numbers, geometric shapes, or algebraic objects.** They serve to bridge those *discrete* structures (such as the distribution of primes) and *continuous* analysis. They are versatile and appear in number theory (as is the case of the Riemann zeta function related to primes), physics (quantum mechanics, statistical mechanics, string theory), dynamical systems, and algebraic geometry.
<div style="font-size: x-large; text-align:center;">∴</div>
3. **There are *infinitely many* zeta functions, but they all share some similarities.** For example, many of them are infinite sums of some sequence, such as the reciprocals of powers of numbers (e.g., $1/n^s$). Others involve products, integrals, or even solutions to differential equations.
## About the Riemann zeta function
4. **One particular such zeta function is the *Riemann zeta function***, which takes its name from German mathematician Bernhard Riemann who published his eponymous *hypothesis* in 1856, although this function had been known long before him. In this note, I will occasionally refer to the Riemann zeta function simply as *the zeta function* for brevity. In its classical *Dirichlet series* form, it is defined as the sum from $n=1$ to $\infty$ of the reciprocal of all natural numbers $n$ raised to the power of a real number $s$, written as:
$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$ ^ea46f6
5. A key property of **the Riemann zeta function is that it is well defined** (in the sense that it *converges* to a finite result) for all real numbers $s \gt 1$. Let's test that on a few examples; for simplicity, we will only focus on integer values for $s$, although the function is defined for all real numbers $\mathbb{R}$ (greater than $1$) and not just integers $\mathbb{Z}$. ^ce0ccc
## Zeta values of even natural numbers
6. Let's evaluate the Riemann zeta function when $s = 2$. The expansion below shows that $\zeta(2)$ is the sum of the reciprocal of the *squares* of all natural numbers.
$\zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \cdots$
7. The first term of the expansion tells us that $\zeta(2) \ge 1$; the second term, that $\zeta(2) \ge 1.25$; the third term, that $\zeta(2) \ge 1.36$, etc. Each additional term *decreases*, and the rate of decrease is such that $\zeta(2)$ is *converging* to some result. (Note: other infinite series whose terms decrease, but not fast enough, can actually diverge; for example, the *harmonic series* which we will discuss in [[Riemann hypothesis#^29620c|§20]]). Let's run R code to approximate that result:
```
# Calculate first billion terms of zeta(2)
sum(1/(1:1e9)^2)
# Result: 1.644934
```
^21091d
8. In 1650, Italian mathematician Pietro Mengoli asked whether a *closed form* of $\zeta(2)$ could be found. This became known as the *Basel problem*, named this way because one of the early mathematicians to study it was Basel resident Jacob Bernoulli (for more history about the Basel problem, see [[Gamma — Exploring Euler's Constant#^4308e2|Havil (2003) p. 37]]). ^7b2067
<div style="font-size: x-large; text-align:center;">∴</div>
9. In 1734, Leonhard Euler (coincidentally, also from Basel) empirically derived such a closed form using the Taylor expansion of the *sine* function (for details, see [[Gamma — Exploring Euler's Constant|Havil (2003) p. 39]]), and arrived at the stunning result:$\zeta(2) = \frac{\pi^2}{6}$ ^ab9768
10. Euler's result is astonishing not just because a solution to the Basel problem had eluded brilliant mathematicians for decades, but also because of the unexpected appearance of *pi*, a constant associated with geometry rather than number theory (more to the point — it is Euler's brilliant intuition to start from the Taylor expansion of $\sin(\pi x)$, a trigonometric construct, which led to surfacing $\pi$ in $\zeta(2)$).
<div style="font-size: x-large; text-align:center;">∴</div>
11. We can now compare our approximated result from [[Riemann hypothesis#^21091d|§7]] above to the closed form result using R:
```
# Calculate the analytical form π²/6
pi^2/6
# Result: 1.644934
```
12. While both results are identical to the sixth decimal of precision shown here, they are not equivalent, as the former is only an approximation while Euler's analytical form is exact; they will eventually differ when more decimals are computed.
```
# Compare the two forms
sum(1/(1:1e9)^2) == pi^2/6
# FALSE
```
13. Euler did not stop there. He painstakingly reused that same approach to calculate the zeta values of all even natural numbers up to 26, which converge to $1$ as $s$ gets larger. Imagine how much more science Euler could have given us if he had had access to a computer!
$\zeta(4) = 1/1^4 + 1/2^4 + 1/3^4 + 1/4^4 + \cdots = \pi^4/90 \approx 1.0823$
$\zeta(6) = 1/1^6 + 1/2^6 + 1/3^6 + 1/4^6 + \cdots = \pi^6/945 \approx 1.0173$
$\zeta(8) = 1/1^8 + 1/2^8 + 1/3^8 + 1/4^8 + \cdots = \pi^8/9450 \approx 1.0040$
$\cdots$
$\zeta(26) = 1/1^{26} + 1/2^{26} + 1/3^{26} + \cdots = 8553103\pi^{26}/71064372752050625000 \approx 1.0000$
<div style="font-size: x-large; text-align:center;">∴</div>
14. Finally, in 1750, Euler devised a more general analytical form ([[Gamma — Exploring Euler's Constant|Havil (2003) p. 51]]) also involving $\pi$ for the zeta value of *any* even natural number $n$. This general form is provided below but explaining it (and the Bernoulli numbers $B_{2n}$ it contains) is beyond the scope of this note.$\zeta(2n)=\sum_{r=1}^{\infty} \frac{1}{r^{2n}}=(-1)^{n-1} \frac{(2\pi)^{2n}}{2(2n)!}B_{2n}$ ^26dab0
## Zeta values of odd natural numbers
15. Let's now see how zeta behaves when $s$ is odd. We already know from [[Riemann hypothesis#^ce0ccc|§5]] that $\zeta(1)$ is undefined, so let's skip it for now and go to the next odd integer. The expansion below shows that $\zeta(3)$ is the sum of the reciprocal of the *cubes* of all natural numbers.$\zeta(3) = \sum_{n=1}^{\infty} \frac{1}{n^3} = \frac{1}{1^3} + \frac{1}{2^3} + \frac{1}{3^3} + \frac{1}{4^3} + \cdots = 1 + \frac{1}{8} + \frac{1}{27} + \frac{1}{64} + \cdots$
16. The first term of this expansion tells us that the value of $\zeta(3)$ must be at least $1$; adding the second term, we get to $1.125$, and adding the third term, $1.162$, etc. Just like $\zeta(2)$, and as expected from [[Riemann hypothesis#^ce0ccc|§5]], $\zeta(3)$ is *converging* to some result. Let's run R code to approximate that result:
```
# Calculate first billion terms of zeta(3)
sum(1/(1:1e9)^3)
# Result: 1.202057
```
17. This number toward which $\zeta(3)$ converges is known as *Apéry's constant* because Greek-French mathematician Roger Apéry proved it to be *irrational* in 1978. Unlike their even neighbors, odd zeta values are not believed to have a closed form, given the extensive and unfruitful numerical and theoretical attempts at finding one.
<div style="font-size: x-large; text-align:center;">∴</div>
18. Of note, $\zeta(3)$ is a deeply mysterious constant that appears in physics (quantum electrodynamics, quantum field theory, statistical mechanics, and string theory).
## Undefined zeta values
19. I wrote earlier in [[Riemann hypothesis#^ce0ccc|§5]] that the Riemann zeta function is only well defined for $s \gt 1$. What happens at $s = 1$? Let's expand $\zeta(1)$:
$\zeta(1) = \sum_{n=1}^{\infty} \frac{1}{n^1} = \frac{1}{1^1} + \frac{1}{2^1} + \frac{1}{3^1} + \frac{1}{4^1} + \cdots = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots \rightarrow \infty$
20. This is simply the *harmonic series* of all positive unit fractions. This series was first proven to diverge to positive infinity by French philosopher Nicolas d'Oresme in 1360; though his proof was lost and rediscovered independently in the 17<sup>th</sup> century by the same Pietro Mengoli mentioned in [[Riemann hypothesis#^7b2067|§8]]. In other words, $\zeta(1)$ is undefined because it *blows up to infinity*. In mathematical terms, the Riemann zeta function has a *simple pole* at $s=1$. ^29620c
<div style="font-size: x-large; text-align:center;">∴</div>
21. What, then, happens at $s \le 0$? The zeta function again *appears* to diverge to positive infinity (emphasis on "appears", because there will be a twist later):$\zeta(0) = \sum_{n=1}^{\infty} \frac{1}{n^0} = \frac{1}{1^0} + \frac{1}{2^0} + \frac{1}{3^0} + \frac{1}{4^0} + \cdots = 1 + 1 + 1 + 1 + \cdots \rightarrow \infty$$\zeta(-1) = \sum_{n=1}^{\infty} \frac{1}{n^{-1}} = \frac{1}{1^{-1}} + \frac{1}{2^{-1}} + \frac{1}{3^{-1}} + \frac{1}{4^{-1}} + \cdots = 1 + 2 + 3 + 4 + \cdots \rightarrow \infty$$\zeta(-2) = \sum_{n=1}^{\infty} \frac{1}{n^{-2}} = \frac{1}{1^{-2}} + \frac{1}{2^{-2}} + \frac{1}{3^{-2}} + \frac{1}{4^{-2}} + \cdots = 1 + 4 + 9 + 16 + \cdots \rightarrow \infty$$\zeta(-3) = \sum_{n=1}^{\infty} \frac{1}{n^{-3}} = \frac{1}{1^{-3}} + \frac{1}{2^{-3}} + \frac{1}{3^{-3}} + \frac{1}{4^{-3}} + \cdots = 1 + 8 + 27 + 64 + \cdots \rightarrow \infty$^6fc7cc
## Analytical continuation
22. So far, our testing of several values of $s$ has allowed us to confirm the defining property given in [[Riemann hypothesis#^ce0ccc|§5]]: that the Riemann zeta function is only well defined within the domain $s \in \mathbb{R}, s \gt 1$. Outside of that domain, when $s \le 1$, it diverges to infinity.
<div style="font-size: x-large; text-align:center;">∴</div>
23. The genius of Bernard Riemann in 1859 was to find a way to *analytically continue* the zeta function outside of its domain of definition in an attempt to give it meaning and utility for $s \leq 1$. To do so, he came up with a functional equation, i.e., he showed that the zeta function as defined in [[Riemann hypothesis#^ea46f6|§4]] is equivalent to *another* function. Crucially, that other function *must* match the properties of the original wherever the original is defined; *and*, to be useful, it should also *extend* the domain of definition of the original.
<div style="font-size: x-large; text-align:center;">∴</div>
24. Remember from [[Riemann hypothesis#^26dab0|§14]] that Euler had found a closed form of the zeta function involving $\pi$ for any zeta value of $2k$ (i.e., any positive even integer). Riemann generalized Euler's contribution and was able to extend the zeta function to the *complex plane*, i.e., treating $s$ as a complex number $\mathbb{C}$ rather than a real number $\mathbb{R}$. This is the functional equation that Riemann came up with:$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} = 2^s \cdot \pi^{s-1} \cdot \sin({\frac{\pi s}{2}}) \cdot \Gamma(1-s) \cdot \zeta(1-s)$ ^8b2a36
25. Note the introduction of a completely new term, $\Gamma (1-s)$. This is the *gamma function*, which is used in mathematics to extend the factorial function ($f(n)=n!$) to the complex plane. This should not be a surprise because Euler's less general analytical form of the zeta function described in [[Riemann hypothesis#^26dab0|§14]] does contain a factorial.
<div style="font-size: x-large; text-align:center;">∴</div>
26. Crucially, when $s=1$, the term $\Gamma (1-s)$ becomes $\Gamma (0)$, which diverges to infinity (without going into the details here, $0$ is a well-known pole of the gamma function; if you want to learn more about the gamma function, consider [[Gamma — Exploring Euler's Constant|Havil, 2003]]). This means that once again, the functional equation in [[Riemann hypothesis#^8b2a36|§24]] blows up to infinity when $s=1$, just like the original definition of the zeta function ([[Riemann hypothesis#^29620c|§20]]). Despite Riemann's generalization of Euler's earlier analytical form, it appears that the singularity at $s=1$ is an unavoidable and fundamental property of the Riemann zeta function. ^ba750f
<div style="font-size: x-large; text-align:center;">∴</div>
27. Also note the term $\sin(\pi s /2)$, in which the presence of $\pi$ is also inherited from Euler's analytical form of the zeta function (see [[Riemann hypothesis#^26dab0|§14]]). This term tells us something very important: that every negative even integer ($s=-2$, $-4$, $-6$, etc.) causes the zeta function to return $0$. Let me explain why.
## Trivial zeros
28. In trigonometry, it is a basic property of the sine function that $\sin(\pi)$ evaluates to $0$. And because the sine function is periodic, and its periodicity is $2\pi$, every multiple $k$ of $\pi$ is a *root* of the sine function, such that $\sin(\pi k)=0$. Replacing $k$ with $s$, the term $sin(\pi s)$ in the functional equation in [[Riemann hypothesis#^8b2a36|§24]] must therefore return $0$ for every integer value of $s$ (which acts as a multiplier to $\pi$). ^7c3943
<div style="font-size: x-large; text-align:center;">∴</div>
29. However, the term is actually $\sin(\pi s/2)$ — notice the extra division by $2$. This means that for *odd* integer values of $s$, $s/2$ is *not* an integer, and the sine function *does not* return $0$. For example, at $s=-1$, the term becomes $\sin(-\pi/2)$, and for $s=-3$, it becomes $\sin(-3\pi/2)$, neither of which return $0$. As a consequence, *only even integer* values of $s$ will cause the entire zeta function in [[Riemann hypothesis#^8b2a36|§24]] to evaluate to $0$. ^2341f2
<div style="font-size: x-large; text-align:center;">∴</div>
30. Furthermore, in the case of *positive even integer* values of $s$, the term $\sin(\pi s/2)$ evaluates to $0$ (as explained in [[Riemann hypothesis#^2341f2|§29]]), *but* the term $\Gamma (1-s)$ diverges to infinity. This is because the gamma function has poles not only at $0$ (as explained in [[Riemann hypothesis#^ba750f|§26]]) but also at *every negative integer* ($-1$, $-2$, etc.). For example, if $s=2$, $\Gamma (1-s)$ evaluates to $\Gamma (1-2) = \Gamma (-1)$, which is a pole. In this case, rather than saying that $\zeta(2)$ is undefined, we retain the known value of $\zeta(2)$ from the *original* formulation of the zeta function as expressed in [[Riemann hypothesis#^ab9768|§9]], which is $\pi^2/6$. Keep in mind that Riemann's functional equation in [[Riemann hypothesis#^8b2a36|§24]] is meant to *extend* the domain of definition of the zeta function to the left of $s \gt 1$, *not* to create singularities in the domain where it *is* defined! ^1299a1
<div style="font-size: x-large; text-align:center;">∴</div>
31. So, we have just seen *why* the inclusion of the term $\sin(\pi s/2)$ to Riemann's functional equation in [[Riemann hypothesis#^8b2a36|§24]] creates *infinitely many zeros* of the zeta function at negative even integers, and *not* at odd integers ([[Riemann hypothesis#^2341f2|§29]]) nor at positive even integers ([[Riemann hypothesis#^1299a1|§30]]). The proof for that is simple (basic trigonometry), which is why the zeros at every negative even integer value of $s$ are called *trivial zeros*. ^d3c5c7
## The extended domain
32. Let's recapitulate. We know from the original formulation of the zeta function that it returns well-defined values for all $s \gt 1$, and Riemann's analytical reformulation from [[Riemann hypothesis#^8b2a36|§24]] does not change those. We also know that as $s=1$, a pole remains, regardless of which of the two formulations of the zeta function we use. We also know that it is possible to extend the domain of the zeta function to include $s \lt 1$ using the functional equation in [[Riemann hypothesis#^8b2a36|§24]], and that it leads to infinitely many trivial zeros for every negative even integer value of $s$.
<div style="font-size: x-large; text-align:center;">∴</div>
33. Let's now revisit the case where $s=0$. It was previously undefined using the original formulation of the zeta function (see [[Riemann hypothesis#^6fc7cc|§21]]), but we now have Riemann's analytical continuation tool at our disposal that extends the domain of the function. Using the functional equation from [[Riemann hypothesis#^8b2a36|§24]], we replace $s$ with $0$ and we get: $\zeta(0) = \sum_{n=1}^{\infty} \frac{1}{n^0} = 2^0 \pi^{-1} \sin(\frac{\pi \cdot 0}{2}) \Gamma(1) \zeta(1)$^2b81de
34. Let's consider how each term evaluates individually:
$2^0=1$
$\pi^{-1}=1/\pi$
$\sin(\frac{\pi \cdot 0}{2}) = 0$
$\Gamma(1)=1$
$\zeta(1) \rightarrow \infty$
^7a0de5
35. Note that, when $s=0$, the third term $\sin(\pi s/2)$ evaluates to $0$, and the fifth term $\zeta(1-s)$ diverges to infinity (as we saw in [[Riemann hypothesis#^29620c|§20]]). So, it looks like we end up multiplying $0$ and $\infty$, which is typically an undefined operation. The subtle trick Riemann used to avoid this singularity at $s=0$ was to treat this operation not as a direct multiplication, but rather as a *limiting process* by looking at how each of those two terms behave as they *approach* their respective limits. Let's do just that using the arrow notation $s \rightarrow 0$ to mean "as $s$ approaches $0
quot; instead of $s=0$.
<div style="font-size: x-large; text-align:center;">∴</div>
36. Let's start with the third term $\sin(\pi s/2)$. As $s \rightarrow 0$, $\pi s/2 \rightarrow 0$, and thus the $\sin$ becomes increasingly irrelevant, such that $\sin(\pi s/2) \approx \pi s/2$. We will reuse that simple form $\pi s/2$ later when we rewrite the equation for $\zeta(0)$ from [[Riemann hypothesis#^2b81de|§33]]. Note here that the term converges to $0$ at a *linear* rate (there are no exponents involved). ^0e64fe
<div style="font-size: x-large; text-align:center;">∴</div>
37. As for the fifth term $\zeta(1-s)$, Riemann used the *Laurent series expansion* to show that as $s \rightarrow 1$: $\zeta(s) \approx \frac{1}{s-1}$ ^de4cc0
38. I won't explain the Laurent series here, but the key insight here is that both $\zeta(s)$ and $1/(s-1)$ behave *similarly* as $s$ approaches $1$, in the sense that they both diverge to infinity at the same linear rate.
<div style="font-size: x-large; text-align:center;">∴</div>
39. So, we now have this new function $1/(s-1)$ that behaves just like $\zeta(s)$ in the vicinity of $s=1$. However, we are interested not in the behavior of $\zeta(s)$, but in that of $\zeta(1-s)$, since it is the term that appears in the functional equation from [[Riemann hypothesis#^8b2a36|§24]]. To solve this, all we need to do is replace $s$ with $1-s$ in our new function from [[Riemann hypothesis#^de4cc0|§37]], which gives us:$\zeta(1-s) \approx \frac{1}{(1-s)-1}=\frac{1}{-s}=-\frac{1}{s}$ ^c1aa19
40. We can now rewrite the functional equation in [[Riemann hypothesis#^2b81de|§33]] using the simplified third term ($\pi s/2$) from [[Riemann hypothesis#^0e64fe|§36]] and fifth term ($-1/s$) from [[Riemann hypothesis#^c1aa19|§39]], and plug in the remaining terms from [[Riemann hypothesis#^7a0de5|§34]]:$\zeta(0) = \sum_{n=1}^{\infty} \frac{1}{n^0} = 2^0 \cdot \pi^{-1} \cdot \sin(\frac{\pi \cdot 0}{2}) \cdot \Gamma(1-0) \cdot \zeta(1-s)$
$= 1 \cdot \frac{1}{\pi} \cdot \frac{\pi \cdot s}{2} \cdot 1 \cdot (-\frac{1}{s})$
^bb4463
41. Now that's a more palatable equation. Notice how we have deliberately *not yet* replaced $s$ by $0$ in the third and fifth terms. This is because they cancel out nicely (as numerator and denominator, respectively) *before* the fifth term gets a chance to cause an undefined result (a division by zero). Likewise, the two $\pi$ also cancel out, which leaves us with:$\zeta(0) = -\frac{1}{2}$ ^c07bb4
42. This is a *highly unintuitive* result, and you may want to pause here for a moment to consider its extraordinary nature. Remember the earlier result for $\zeta(0)$ from [[Riemann hypothesis#^6fc7cc|§21]]:
$\zeta(0) = \sum_{n=1}^{\infty} \frac{1}{n^0} = \frac{1}{1^0} + \frac{1}{2^0} + \frac{1}{3^0} + \frac{1}{4^0} + \cdots = 1 + 1 + 1 + 1 + \cdots \rightarrow \infty$
43. How can $\zeta(0)$ be *both* $\infty$ and $-1/2$? Does it mean that $1+1+1+1+\cdots=-1/2$? Obviously not in a direct sense. The infinity arises from trying to apply the original formulation of the zeta function (from [[Riemann hypothesis#^ea46f6|§4]]) to $s=0$, *which is outside of its domain of definition* ($s \gt 1$). Riemann had to find and use *another* equation (the functional equation in [[Riemann hypothesis#^8b2a36|§24]]) that both *preserves* the properties of the original for $s \gt 1$ and *extends* its domain of definition to provide well-defined, converging results $s \le 1$.
<div style="font-size: x-large; text-align:center;">∴</div>
44. We are nearly done with our visit of the newly extended domain of $\zeta(s)$ for $s \le 1$. We already know from [[Riemann hypothesis#^c07bb4|§41]] that $\zeta(0)=-1/2$, and from [[Riemann hypothesis#^d3c5c7|§31]] that $\zeta(-2k)=0$ (every negative even integer is a trivial zero of the zeta function). That leaves only the negative odd integers to explore.
<div style="font-size: x-large; text-align:center;">∴</div>
45. Let's explore $\zeta(-1)$ and plug $-1$ into Riemann's functional equation from [[Riemann hypothesis#^8b2a36|§24]]:$\zeta(-1) = \sum_{n=1}^{\infty} \frac{1}{n^{-1}} = 2^{-1} \cdot \pi^{-2} \cdot \sin(\frac{-\pi}{2}) \cdot \Gamma(2) \cdot \zeta(2)$
46. Let's consider how each term evaluates individually (the last one being Euler's result from [[Riemann hypothesis#^ab9768|§9]]):
$2^{-1}=1/2$
$\pi^{-2}=1/\pi^2$
$\sin(-\pi/2) = -1$
$\Gamma(2)=1$
$\zeta(2)=\pi^2/6$
47. We can now evaluate $\zeta(-1)$:$\zeta(-1)=\frac{1}{2} \cdot \frac{1}{\pi^2} \cdot (-1) \cdot \frac{\pi^2}{6}=-\frac{\pi^2}{12\pi^2}=-\frac{1}{12}$
48. Another stunning result, reminiscent of $\zeta(0)=-1/2$ just earlier. Remember from [[Riemann hypothesis#^6fc7cc|§21]] that $\zeta(-1)$ can be expressed as an infinite series of $1+2+3+4+\cdots \rightarrow \infty$. Thanks to Riemann's analytical continuation, we are now able to ascribe the finite value $-1/12$ to that previously undefined result. I am working on a separate note on this particular case of divergent series summation.
<div style="font-size: x-large; text-align:center;">∴</div>
49. Let's do one last zeta evaluation for completeness:$\zeta(-3) = \sum_{n=1}^{\infty} \frac{1}{n^{-1}} = 2^{-3} \cdot \pi^{-4} \cdot \sin(\frac{-3\pi}{2}) \cdot \Gamma(4) \cdot \zeta(4)$
50. Let's consider how each term evaluates individually (the last one being Euler's result from [[Riemann hypothesis#^ab9768|§9]]):
$2^{-3}=1/8$
$\pi^{-4}=1/\pi^4$
$\sin(-3\pi/2) = 1$
$\Gamma(4)=3!=6$
$\zeta(4)=\pi^4/90$
51. We can now evaluate $\zeta(-3)$:$\zeta(-3)=\frac{1}{8} \cdot \frac{1}{\pi^4} \cdot 1 \cdot 6 \cdot \frac{\pi^4}{90}=\frac{6\pi^4}{720\pi^4}=\frac{1}{120}$
52. Here again, the counter-intuitive implication is that the finite value of $1/120$ maps onto the original series of $1^1+1^2+1^3+1^4+\cdots \rightarrow \infty$ shown in [[Riemann hypothesis#^6fc7cc|§21]].
53. These last few examples of "visiting the domain" of the zeta function that was previously inaccessible to us (at least not without invoking singularities) should illustrate well enough the power of Riemann's analytical continuation. What Riemann did was unearth a deeper and more general structure that was implicitly buried within the zeta function, one that applied not just to the real number set $\mathbb{R}$ but the whole complex number set $\mathbb{C}$ (which is a [[Nine Zulu Queens Ruled China|superset of the former]]).
<div style="font-size: x-large; text-align:center;">∴</div>
54. As a final recapitulation of this section on analytical continuation of the zeta function's domain, here is a table relating zeta values under both forms of the function:
| | $\zeta(-4)$ | $\zeta(-3)$ | $\zeta(-2)$ | $\zeta(-1)$ | $\zeta(0)$ | $\zeta(1)$ | $\zeta(2)$ | $\zeta(3)$ | $\zeta(4)$ |
| --------------------------------------------------------- | ----------------------- | ----------------------- | ----------------------- | ----------------------- | ----------------------- | ----------------------- | ---------- | ---------------------------- | ---------- |
| Dirichlet series ([[Riemann hypothesis#^ea46f6\|§4]]) | Undefined<br>($\infty$) | Undefined<br>($\infty$) | Undefined<br>($\infty$) | Undefined<br>($\infty$) | Undefined<br>($\infty$) | Undefined<br>($\infty$) | $\pi^2/6$ | $1.20205\cdotslt;br>(Apéry's) | $\pi^4/90$ |
| Functional equation ([[Riemann hypothesis#^8b2a36\|§24]]) | 0<br>(trivial) | $1/120$ | 0<br>(trivial) | $-1/12$ | $-1/2$ | Undefined<br>($\infty$) | $\pi^2/6$ | $1.20205\cdotslt;br>(Apéry's) | $\pi^4/90$ |
## Non-trivial zeros
55. We have now seen how the Riemann zeta function has evolved from its original Dirichlet series form ([[Riemann hypothesis#^ea46f6|§4]]), to Euler's generalization for positive even integer values of $s$ ([[Riemann hypothesis#^26dab0|§14]]), to Riemann's functional equation in [[Riemann hypothesis#^8b2a36|§24]] for all complex values of $s$. We have also seen how Riemann's analytic continuation of the function expanded the domain of definition of the zeta function, and gave new finite values to summations that were otherwise diverging to infinity. Among those values are trivial zeros at every negative even integer value of $s$. Are there other zeros of the zeta function anywhere else?
<div style="font-size: x-large; text-align:center;">∴</div>
56. The functional equation in [[Riemann hypothesis#^8b2a36|§24]] gives us a clue to the affirmative. Notice how the equation (restated below for clarity) links $\zeta(s)$ to the left, with $\zeta(1-s)$ to the right.$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} = 2^s \cdot \pi^{s-1} \cdot \sin({\frac{\pi s}{2}}) \cdot \Gamma(1-s) \cdot \zeta(1-s)$
57. For $\zeta(s)$ on the left to be equal to $0$, at least one of the five product terms to the right must also be equal to $0$. It clearly cannot be the first term, $2^s$; nor the second term, $\pi^{s-1}$, neither of which can be zero regardless of $s$. The third term, $\sin(\pi s/2)$, can only return (trivial) zeros at negative even integer values of $s$, as we have seen in [[Riemann hypothesis#^d3c5c7|§31]]. The fourth term is the gamma function, which has infinitely many poles along the real number line, but does not have any zero. This leaves only the fifth term $\zeta(1-s)$; if the Riemann zeta function is to have any non-trivial zeros, it must be when $\zeta(s)$ and $\zeta(1-s)$ *both* equal zero.
<div style="font-size: x-large; text-align:center;">∴</div>
58. In other words, Riemann's functional equation for the zeta function introduces a *symmetry* between $s$ and $1-s$. That symmetry is already observed between the non-trivial zeros, which come in pairs: $\zeta(-2)$ and $\zeta(-4)$ are symmetrical zeros about $s=-3$, $\zeta(-4)$ and $\zeta(-6)$ about $s=-5$, etc. But what about non-trivial zeros elsewhere? ^2735ee
<div style="font-size: x-large; text-align:center;">∴</div>
59. Consider the transformation $s \rightarrow 1-s$. At what point along the number line is that transformation *invariant*? The table below shows a few example values for $s$ centered around $0$, and shows their transformation to $1-s$:
| $s$ | $\cdots$ | -3 | -2.5 | -2 | -1.5 | -1 | -0.5 | 0 | **0.5** | 1 | 1.5 | 2 | 2.5 | 3 | $\cdots$ |
| ----- | -------- | --- | ---- | --- | ---- | --- | ---- | --- | ------- | --- | ---- | --- | ---- | --- | -------- |
| $1-s$ | $\cdots$ | 4 | 3.5 | 3 | 2.5 | 2 | 1.5 | 1 | **0.5** | 0 | -0.5 | -1 | -1.5 | -2 | $\cdots$ |
60. Notice from this table that there is only one value on the number line where the equality $s = 1-s$ holds true, and that is when $s=1/2$. This implies that any non-trivial zeros of the Riemann zeta function *must* be distributed symmetrically about a vertical line that intersects the real number line at $1/2$. That line has a name: it is called the *critical line*.
<div style="font-size: x-large; text-align:center;">∴</div>
61. How far to the left and right of the critical line must those non-trivial zeros be? We already know from [[Riemann hypothesis#^d3c5c7|§31]] that the only zeros in the domain $s \lt 0$ are the trivial zeros, which can be found at negative even integer values of zeta due to the sine function. So, the furthest to the left of the critical line that any *non-trivial* zeros can reside is within the $0 \lt s \le 1/2$ domain.
<div style="font-size: x-large; text-align:center;">∴</div>
62. By virtue of symmetry about the critical line, this implies that their counterparts must reside in the $1/2 \le s \lt 1$ domain. The combined domain of $0 \lt s \lt 1$ has a name: the *critical strip*.
<div style="font-size: x-large; text-align:center;">∴</div>
63. It is believed from his notes (not his 1859 paper) that Riemann himself searched for the first few non-trivial zeros of the zeta function within the critical strip using numerical integration, and found the first one located on the critical line. That first zero is:$s=\frac{1}{2}+14.134725i$ ^4b1592
64. I have mentioned before that the functional equation in [[Riemann hypothesis#^8b2a36|§24]] extended the domain of definition of the zeta function to the complex plane. Here we are starting to see those complex numbers of the form $s = \sigma+ti$ where $\sigma$ and $t$ are real numbers and $i$ is an imaginary number. Until now, I had only referred to the real part of $s$, such as $Re(s)=-2$ being a trivial zero. The imaginary part was implied to be null, as in $s = \sigma + 0 \cdot i$. This simplification worked well until now because the trivial zeros of the zeta function were all located on the horizontal real number axis.
<div style="font-size: x-large; text-align:center;">∴</div>
65. However, now that we are looking at non-trivial zeros located up or down the vertical critical strip in the domain $0 \lt Re(s) \lt 1$, the imaginary part of $s$ becomes necessary to surface. The first non-trivial zero shown in [[Riemann hypothesis#^4b1592|§63]] has real part $Re(s)=1/2$ (i.e., it is located on the critical line intersecting the horizontal axis of the real number line at value $1/2$) *and* an imaginary part $14.134725i$, meaning that it is roughly $14$ "units" above the horizontal axis.
<div style="font-size: x-large; text-align:center;">∴</div>
66. We saw in [[Riemann hypothesis#^2735ee|§58]] how Riemann's functional equation introduced a *horizontal* symmetry between the non-trivial zeros of the zeta function about the critical line. There is *another* symmetry which we have to consider. This one is a *vertical* symmetry between the non-trivial zeros, this one stemming from something called the Schwarz reflection principle (coincidentally, I find that [Hermann Schwarz](https://upload.wikimedia.org/wikipedia/commons/c/c7/Hermann_Amand_Schwarz_%281843-1921%29_by_Louis_Zipfel.jpg) looks remarkably like [Bernard Riemann](https://upload.wikimedia.org/wikipedia/commons/8/82/Georg_Friedrich_Bernhard_Riemann.jpeg) — yet another symmetry!). Without going into too many details, it states that the zeros that exist in the upper half-plane will have their conjugate in the lower-half plane. More rigorously, if $s = \sigma + t \cdot i$ is a zero of the zeta function, then so is $s = \sigma - t \cdot i$. This means that, based on the non-trivial zero from [[Riemann hypothesis#^4b1592|§63]], this one is *also* a non-trivial zero:$s=\frac{1}{2}-14.134725i$
## The conjecture
67. We now have all the tools that we need to understand the Riemann hypothesis stated in the abstract: **that all non-trivial zeros of the Riemann zeta function have real part $1/2$** (sometimes written as: all non-trivial zeros of the Riemann zeta function are on the critical line).
<div style="font-size: x-large; text-align:center;">∴</div>
68. So far, computations have found $1,236,300,000,000$ (1.2 trillion) such non-trivial zeros on the critical line, and *none* outside the critical line. We can continue finding additional zeros on the critical line forever, but that is of course not enough to prove the hypothesis. Conversely, it would take finding only *one* non-trivial zero outside the critical line to invalidate the hypothesis.
<div style="font-size: x-large; text-align:center;">∴</div>
69. Recent theoretical advances (see [[New large value estimates for Dirichlet polynomials#^cbb9d3|Guth & Maynard, 2024]]) have lowered the bound for how many non-trivial zeros may exist outside the critical line, but we are still nowhere near having proven the conjecture.
<div style="font-size: x-large; text-align:center;">∴</div>
70. I realize I have barely mentioned primes and their distribution at all, and instead focused on "rebuilding" the Riemann hypothesis from first principles. I should write another note about the prime number theorem, the prime-counting function, Euler's product, etc. as well as implications of either proving or disproving the Riemann hypothesis.
## References
- Clay Mathematics Institute. (2025). *Riemann Hypothesis*. https://www.claymath.org/millennium/riemann-hypothesis/
- Derbyshire, J. (2003). *Prime obsession: Bernhard Riemann and the greatest unsolved problem in mathematics*. Joseph Henry Press.
- Glas, J., & Yeh, K. (2019). The classical theta function and the Riemann zeta function. https://metaphor.ethz.ch/x/2019/fs/401-4110-19L/sc/talk4.pdf
- Havil, J. (2003). *Gamma: Exploring Euler's Constant*. Princeton University Press. https://ia801407.us.archive.org/15/items/X8QX87XSQY7QX7XSQXS98XQX98/%5BJulian_Havil%5D_Gamma_Exploring_Euler%27s_Constant%28BookZZ.org%29.pdf
- Riemann, B. (1859). On the number of prime numbers less than a given quantity. https://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/EZeta.pdf