# A Heuristic for the Prime Number Theorem

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## Conclusion

Might there be a chance of proving in a simple way that*x*/π(*x*) is asymptotic to an increasing function, thus getting another proof of PNT? This is probably wishful thinking. However, there is a natural candidate for the increasing function. Let*L*(*x*) be the upper convex hull of the full graph of*x*π(*x*) (precise definition to follow). The piecewise linear function*L*(*x*) is increasing because*x*/π(*x*) → ∞ as*x* → ∞. Moreover, using PNT, we can give a proof that*L*(*x*) is indeed asymptotic to*x*/π(*x*). But the point of our work in this article is that for someone who wishes to understand why the growth of primes is governed by natural logarithms, a reasonable approach is to convince oneself via computation that the convex hull just mentioned satisfies the hypothesis of our theorem, and then use the relatively simple proof to show that this hypothesis rigorously implies the prime number theorem.

## Keywords

Convex Hull Mathematical Intelligencer Piecewise Linear Function Wishful Thinking Curve Part## Preview

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