The Riemann zeta function, denoted by ζ(s), is equal to zero for certain values of 's'. Specifically, for a particular set of inputs, the function's output is exactly 0.
Understanding the Trivial Zeros
Based on the provided reference, the Riemann zeta function ζ(s) is zero when the input 's' is a negative even integer. These specific zero points are known as the "trivial zeros" of the zeta function.
Why ζ(s) is Zero for Negative Even Integers
The reason ζ(s) is zero when 's' is a negative even integer is directly related to a factor in one of the formulations of the function. According to the reference:
- If s is a negative even integer then ζ(s) = 0 because the factor sin(πs/2) vanishes.
Let's break this down:
- Negative Even Integers: These are numbers like -2, -4, -6, -8, and so on.
- sin(πs/2): This is a mathematical expression involving the sine function.
- Vanishes: This means the value of the expression becomes zero.
When you substitute a negative even integer for 's' in sin(πs/2), the term πs/2 becomes an integer multiple of π. For example:
- If s = -2, then πs/2 = π(-2)/2 = -π. The sine of -π is 0.
- If s = -4, then πs/2 = π(-4)/2 = -2π. The sine of -2π is 0.
- If s = -6, then πs/2 = π(-6)/2 = -3π. The sine of -3π is 0.
The sine function is zero for any integer multiple of π (..., -2π, -π, 0, π, 2π, ...). Since s is a negative even integer, s/2 is always a negative integer. Therefore, πs/2 is always a negative integer multiple of π, causing sin(πs/2) to be zero.
This vanishing sin(πs/2) factor, appearing in certain formulas for the zeta function (like the functional equation relating ζ(s) to ζ(1-s)), effectively forces ζ(s) to be zero at these points.
Examples of Trivial Zeros
The trivial zeros occur at all negative even integers:
- s = -2
- s = -4
- s = -6
- s = -8
- s = -10
- ...and so on indefinitely.
These are considered "trivial" because their location is relatively easy to determine and explain compared to the function's "non-trivial" zeros, which lie on a critical line in the complex plane and are central to the famous Riemann Hypothesis.
Table of Trivial Zeros
| Value of s | s is a Negative Even Integer | Value of πs/2 | sin(πs/2) | Value of ζ(s) (Trivial Zero) |
|---|---|---|---|---|
| -2 | Yes | -π | 0 | 0 |
| -4 | Yes | -2π | 0 | 0 |
| -6 | Yes | -3π | 0 | 0 |
| -8 | Yes | -4π | 0 | 0 |
In summary, one specific reason the Riemann zeta function is zero is when its input is a negative even integer. This happens because a crucial factor, sin(πs/2), which is part of the function's definition or functional equation, becomes zero for these specific inputs.
