The Mathematical Intelligencer

, Volume 28, Issue 3, pp 6–9

# A Heuristic for the Prime Number Theorem

Article

## Conclusion

Might there be a chance of proving in a simple way thatx/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. LetL(x) be the upper convex hull of the full graph ofxπ(x) (precise definition to follow). The piecewise linear functionL(x) is increasing becausex/π(x) → ∞ asx → ∞. Moreover, using PNT, we can give a proof thatL(x) is indeed asymptotic tox/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

## Refferences

1. 1.
D. Bressoud and S. Wagon,A Course in Computational Number Theory, Key College, San Francisco, 2000.
2. 2.
R. Courant and H. Robbins,What is Mathematics?, Oxford University Press, London, 1941.Google Scholar
3. 3.
J. Friedlander, A. Granville, A. Hildebrand, and H. Maier, Oscillation theorems for primes in arithmetic progressions and for sifting functions,J. Amer. Math. Soc. 4 (1991) 25–86.
4. 4.
X. Gourdon and P. Sebah. The π(x) project, http://numbers. computation.free.fr/Constants/constants.html.Google Scholar
5. 5.
G. H. Hardy and E. M. Wright,An Introduction to the Theory of Numbers, 4th ed., Oxford University Press, London, 1965.Google Scholar
6. 6.
I. Niven, H. S. Zuckerman, and H. L. Montgomery,An Introduction to the Theory of Numbers, 2nd ed., Wiley, New York, 1991.Google Scholar
7. 7.
G. Tenenbaum and M. Mendès France,The Prime Numbers and their Distribution, Amer. Math. Soc., Providence, R.I., 2000.Google Scholar
8. 8.
S. Wagon, It’s only natural,Math Horizons 13:1 (2005) 26–28.Google Scholar
9. 9.
D. Zagier, The first 50,000,000 prime numbers,The Mathematical Intelligencer 0 (1977) 7–19.