# 113: Riemann-Zeta

Riemann-Zeta |

Title text: The graph is of the magnitude of the function with the real value between 0 and 2 and the imaginary between about 35 and 40. I've misplaced the exact parameters I used. |

## Explanation[edit]

A prime number is any positive whole number greater than 1 that cannot be formed by multiplying two smaller whole numbers. There are an infinite number of prime numbers, but they are somewhat elusive, since there is no known function that yields all, and only, prime numbers. The Riemann zeta function begins with the infinite series of 1/n^s, where s is a complex number (i.e., s = x + iy where x > 1, y is real, and i is the imaginary number), and is summed from n = 1 to infinity. It is then extended, by some advanced techniques, to be defined for all complex numbers, except for s = 1. As the image text indicates, the graph in the picture is of the Riemann-Zeta function in the complex plane. Leonhard Euler proved that there is a relationship between the Riemann-Zeta function and prime numbers, which explains the reference to the deep ties between the two.

Here, Randall appears to be talking to his significant other, comparing her to prime numbers and himself to the Riemann-Zeta function. It is mathematically correct and quite poetic, until he mentions that his relationship differs from the comparison because "The Riemann-Zeta function couldn't have given you herpes." This implies that he has infected his lover with an incurable venereal disease. The comic effect of an abrupt change in tone like this is known as bathos.

The title text explains that Randall has forgotten the value of the complex number "s" for which he plotted the Riemann zeta function, but the "real" part (the "x" when s is written x + iy) is between 0 (the square root of 0) and +2 (the positive square root of +4), and the "imaginary" part (the "iy" when s is written x + iy) is between about 35i (the square root of -1225) and 40i (the square root of -1600).

## Transcript[edit]

- [A z = fn(x, y) plot, with pointy spikes on the back sloping to a relatively flat front.]

- You are like the prime numbers. Unpredictable turns, unconstrainable. Tantalizingly regular but never quite the same.
- I am like the Riemann-zeta function. A rippled curtain of the imagined and real. Deeply tied with you in ways incomprehensible.
- Although, strictly speaking, The Riemann-zeta function couldn't have given you herpes.

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