The Mathematical Intelligencer

, Volume 28, Issue 3, pp 6–9 | Cite as

A Heuristic for the Prime Number Theorem

  • Hugh L. Montgomery
  • Stan Wagon


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.


Convex Hull Mathematical Intelligencer Piecewise Linear Function Wishful Thinking Curve Part 
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Copyright information

© Springer Science + Business Media Inc. 2006

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA
  2. 2.Department of MathematicsMacalester CollegeSt. PaulUSA

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