From: 3blue1brown
Complex functions, which take complex numbers as inputs and produce complex numbers as outputs, can be effectively understood through visualization as transformations [07:54:00]. This approach involves imagining every possible input moving to its corresponding output [07:54:00].
Visualizing Transformations
To illustrate, consider the function f(s) = s² [08:06:00].
- An input of s = 2 yields 4, so the point at 2 moves to 4 [08:09:00].
- An input of s = -1 yields 1, moving the point at -1 to 1 [08:16:00].
- An input of s = i yields -1, moving the point at i to -1 [08:24:00].
When visualized, applying f(s) = s² to an entire grid of complex numbers shows a dynamic transformation [08:40:00]. This provides a rich picture of the function’s behavior in two dimensions [09:10:00].
The Riemann Zeta Function
The Riemann zeta function, often denoted ζ(s), is a significant object in modern mathematics [00:00:04]. It is primarily known for the Riemann hypothesis, an unsolved problem regarding when the function equals zero, carrying a one-million-dollar prize [00:00:15].
Initial Definition and Convergence
The zeta function is initially defined as an infinite sum for a given input s: ζ(s) = 1/1ˢ + 1/2ˢ + 1/3ˢ + 1/4ˢ + … [01:32:00]. This sum includes all natural numbers [01:46:00].
For example, when s = 2, the sum is 1 + 1/4 + 1/9 + 1/16 + …, which converges to π²/6 (approximately 1.645) [01:50:00]. This sum converges to a specific number as long as the real part of s is greater than 1 [03:31:00]. In this domain, the function is well-defined [03:36:00].
Complex Inputs
Mathematicians, notably Bernhard Riemann, focused on understanding the zeta function when s is a complex number [03:50:00]. Raising a number to a complex power involves both scaling and rotation [04:41:00]. Specifically, raising a base to a pure imaginary number results in a complex number on the unit circle [05:02:00]. The further the base is from 1, the more quickly the output walks around the unit circle as the imaginary input changes [05:26:00].
When a complex input like 2 + i is plugged into the zeta function, each term in the sum (e.g., 1/nˢ) gets rotated by some amount, but its length does not change [06:34:00]. This means the sum still converges, typically in a spiral, to a specific point on the complex plane [06:40:00]. This demonstrates that the zeta function is a reasonable complex function as long as the real part of the input is greater than 1 [07:12:00].
The visualization of the zeta function shows how points on the right half of the complex plane (where the real part of numbers is greater than 1) are transformed into corresponding outputs [09:39:00].
Analytic Continuation
For inputs where the real part of s is less than or equal to 1, the original infinite sum definition of the zeta function breaks down, producing nonsensical results like 1 + 2 + 3 + … = -1/12 [02:34:00], [03:41:00], [11:02:00]. However, the transformed grid from the domain where the sum does make sense “begs” to be extended [10:10:00], [11:16:00].
The Property of “Analytic” Functions
This extension is achieved through a concept called analytic continuation [00:50:00], which relies on the property of a function being “analytic” [14:02:00]. A complex function is considered analytic if it has a derivative everywhere [12:19:00]. Geometrically, this means that the function preserves angles [13:37:00]. If any two lines in the input space intersect at a certain angle, they will still intersect at the same angle after the transformation, even if the lines themselves become curved [13:08:00]. A simple way to observe this is that grid lines, which initially intersect at right angles, continue to intersect at right angles after the transformation [13:47:00].
This angle-preserving property is highly restrictive [14:42:00]. The surprising fact is that if an analytic function is extended beyond its original domain, and the new extended function is also required to be analytic, there is only one possible extension, if one exists at all [15:28:00]. This is akin to an infinite, continuous jigsaw puzzle where the angle-preserving requirement locks the extension into a unique solution [15:53:00].
Defining the Full Riemann Zeta Function
The full Riemann zeta function is defined as follows [16:14:00]:
- For values of s where the real part is greater than 1, it’s defined by the converging infinite sum [16:18:00].
- For the rest of the complex plane, it is defined as the unique analytic continuation of that sum [16:33:00]. This is a valid definition because there is only one possible analytic continuation [16:52:00].
Zeros and the Riemann Hypothesis
The points where the Riemann zeta function equals zero are particularly important [17:24:00].
- Trivial Zeros: The negative even numbers (e.g., -2, -4, -6…) are mapped to zero [17:36:00]. These are called “trivial” because mathematicians understand them well [17:44:00].
- Non-trivial Zeros: The remaining zeros lie within a region called the critical strip [17:54:00]. Their specific placement contains surprising information about prime numbers [18:02:00].
The Riemann Hypothesis states that all non-trivial zeros lie on the critical line, which is the line of numbers s whose real part is exactly one half [18:19:00]. Proving this hypothesis would imply a remarkably tight grasp on the pattern of prime numbers and many other mathematical patterns [18:31:00]. Visualizing the critical line under the zeta function’s transformation shows it passing through zero many times [18:50:00].
Another known property of this extended function is that it maps the point negative one (-1) to negative one twelfth (-1/12) [19:26:00]. While this seems to imply that 1 + 2 + 3 + 4 + … equals -1/12, it’s crucial to remember that this value comes from the analytic continuation, not the original diverging sum [19:42:00]. However, the uniqueness of this analytic continuation suggests a deep connection between these extended values and the original sum [20:01:00].