## Longest World's Complex Math's Formula - Riemann Hypothesis

World's longest mathematics formula. Mathematics formulas can range in length from just a few symbols to several pages long, depending on the complexity of the problem they are trying to solve.

However, one example of a particularly long mathematical formula is the Riemann Hypothesis. The Riemann Hypothesis is a conjecture about the distribution of prime numbers among the positive integers, and it has been one of the most important open problems in mathematics for over 150 years.

The Riemann Hypothesis is typically stated in terms of the Riemann zeta function, which is defined as:

**ζ(s) = 1 + 1/2^s + 1/3^s + 1/4^s + ...**

where s is a complex number with real part greater than 1. The Riemann Hypothesis states that all non-trivial zeros of the Riemann zeta function lie on the critical line Re(s) = 1/2, where "non-trivial" means that the zeros are not located at the negative even integers -2, -4, -6, ... (known as the "trivial zeros").

The Riemann Hypothesis has many important consequences for number theory and has been the subject of intense study and research by mathematicians around the world. While the formula itself is not particularly long, the problem it represents is considered one of the most challenging in mathematics.

he Riemann zeta function is a mathematical function defined as:

ζ(s) = 1^(-s) + 2^(-s) + 3^(-s) + 4^(-s) + ...

where s is a complex number with real part greater than 1. This series converges for values of s with real part greater than 1, and the Riemann zeta function can be analytically continued to the whole complex plane, except for a simple pole at s = 1.

The Riemann zeta function has many important properties and connections to other areas of mathematics, such as number theory, complex analysis, and algebraic geometry. In particular, the Riemann Hypothesis is a conjecture about the distribution of the non-trivial zeros of the Riemann zeta function, and has been one of the most important open problems in mathematics for over 150 years.

### The full formula for the Riemann zeta function is:

**ζ(s) = ∑ (n=1 to ∞) n^(-s)**

where the sum is taken over all positive integers n.

The Riemann zeta function can be used to solve various problems in number theory. However, the Riemann Hypothesis itself has not been proven, so any use of the formula would be based on assumptions and conjectures.

That being said, here is an example of a problem that can be solved using the Riemann zeta function:

Problem: Find the sum of the reciprocals of the first 10 positive integers.

Solution: We can use the Riemann zeta function to find this sum. Specifically, we can use the fact that:

ζ(s) = 1 + 1/2^s + 1/3^s + 1/4^s + ...

to find the sum of the reciprocals of the first n positive integers. To do this, we simply plug in s = 1 into the formula, since we want to find the sum of the reciprocals:

ζ(1) = 1 + 1/2 + 1/3 + 1/4 + ...

= ∞

Unfortunately, we see that the sum diverges to infinity, which means that there is no finite sum of the reciprocals of the positive integers. However, this is still an interesting result, as it shows that the harmonic series diverges, despite the fact that each term gets smaller and smaller.

So while we can't use the Riemann zeta function to find the sum of the first 10 positive integers, we can use it to prove that there is no finite sum.

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