# Definition of heavy-tailed distribution:

The distribution of a random variable $$X$$ with distribution function $$F$$ is said to have a heavy right tail if the moment generating function of $$X$$, $$M_X(t)$$ is infinite for all $$t > 0$$.

This implies that:

$$\forall t > 0, \mathbb{E}[e^{tX}] = \int_{-\infty}^{\infty} e^{tx} dF(x) = \infty$$

and given that $$dF(x) = f(x)dx$$ this is reducible to:

$$\int_{-\infty}^{\infty} e^{tx} f(x)dx = \infty$$

# The distribution of prime numbers is heavy-tailed:

Due to the Prime Number Theorem, given the set of primes $$\mathbb{P} \subset \mathbb{N}$$ for large $$N$$ we have:

$$\frac{\pi(N)}{N} \sim \frac{1}{\ln(N)}$$

$$\forall x \in [1,N], P(N \in \mathbb{P}) \leq P(x \in \mathbb{P})$$

and since there is at most one prime number in each interval $$(n,n+1]$$ we have:

$$\frac{1}{N \cdot \ln(N)} \leq P(N \in \mathbb{P}) \leq \frac{1}{\ln(N)}$$

so if we define:

$$f(x) := P(x \in \mathbb{P})$$

we find:

$$\forall t > 0, \int_{2}^{\infty} e^{tx} f(x)dx \geq \lim_{N \to \infty} \int_{2}^{N} e^{tx} \cdot \frac{1}{x \cdot \ln(N)} dx = \infty$$

so the prime numbers have a heavy-tailed distribution.

# References:

1. Doron Zagier. Newman’s short proof of the Prime Number Theorem. The American Mathematical Monthly, Vol. 104, No. 8 (Oct., 1997), pp. 705-708

2. Embrechts P.; Klueppelberg C.; Mikosch T. (1997). Modelling extremal events for insurance and finance. Stochastic Modelling and Applied Probability. 33. Berlin: Springer