The Sum of Natural Numbers and the Role of Zero
The Short Answer
No, it does not matter whether zero is included in the natural numbers. The -1/12 result is independent of whether you define the natural numbers starting at 0 or at 1.
This is an excellent question that gets to the heart of what the "-1/12" result actually means and how it relates to the definition of natural numbers.
The Core of the "-1/12" Result
First, it's crucial to understand that the equation:
is not true in the sense of standard, classical summation where you add numbers step-by-step. The series of natural numbers is divergent—its partial sums go to infinity.
This result comes from a more advanced mathematical concept known as analytic continuation, specifically of the Riemann zeta function.
The Role of the Riemann Zeta Function
The Riemann zeta function is initially defined for complex numbers with a real part greater than 1 by the following series:
For example, ζ(2) = 1⁻² + 2⁻² + 3⁻² + ... = 1 + 1/4 + 1/9 + ... = π²/6, which is a famous result. This series only converges when the real part of s is greater than 1.
However, mathematicians have found a way to define a unique function that agrees with this series where it converges but is also defined for all other complex numbers (except s=1). This process is called analytic continuation.
When we plug s = -1 into this analytically continued function, we get:
Now, if we were to formally substitute s = -1 into the original series definition, we would get:
This is the origin of the provocative "equality." It's a value assigned by the powerful and rigorous process of analytic continuation, not by conventional addition.
Where Does Zero Fit In?
Now, let's address your specific question about zero.
If your natural numbers start at 1, the series is 1 + 2 + 3 + 4 + ... and its associated zeta function is ζ(s) = 1⁻ˢ + 2⁻ˢ + 3⁻ˢ + ... As above, ζ(-1) = -1/12.
If your natural numbers start at 0, the series would be 0 + 1 + 2 + 3 + 4 + ... Let's see what happens if we try to connect this to the zeta function:
The zeta function is defined as starting from 1, so 0 + 1 + 2 + 3 + ... = 0 + ζ(s). But we have to be careful. For the analytic continuation to work at s=-1, we need to assign a value to 0⁻ˢ as well. For s = -1, this is 0¹, which is 0.
Therefore, the assigned value would be: 0 + ζ(-1) = 0 + (-1/12) = -1/12.
Summary
| Definition of Naturals | Series | Value from Analytic Continuation |
|---|---|---|
| Starting at 1 | 1 + 2 + 3 + 4 + ... | -1/12 |
| Starting at 0 | 0 + 1 + 2 + 3 + 4 + ... | 0 + (-1/12) = -1/12 |
Conclusion
The famous (and often misunderstood) result that the "sum of all natural numbers is -1/12" is a statement about the analytic continuation of the Riemann zeta function evaluated at s=-1. Since adding zero does not change the value, the debate over whether zero is a natural number is irrelevant in this specific context.
The result remains -1/12 regardless of whether you include zero in the natural numbers or not.
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