A Centennial History of the Prime Number Theorem

Among the thousands of discoveries made by mathematicians over the centuries, some stand out as significant landmarks. One of these is the prime number theorem, which describes the asymptotic distribution of prime numbers. It can be stated in various equivalent forms, two of which are: (1)<math display='block'> <mrow> <mi>&#x03C0;</mi><mrow><mo>(</mo> <mi>x</mi> <mo>)</mo></mrow><mo>&#x223C;</mo><mfrac> <mi>x</mi> <mrow> <mi>log</mi><mi>x</mi> </mrow> </mfrac> <mi>a</mi><mi>s</mi><mtext>&#x2003;</mtext><mi>x</mi><mo>&#x2192;</mo><mi>&#x221E;</mi><mo>,</mo> </mrow> </math>$$\pi \left( x \right) \sim \frac{x} {{\log x}}as\quad x \to \infty ,$$ and (2)<math display='block'> <mrow> <msub> <mi>p</mi> <mi>n</mi> </msub> <mo>&#x223C;</mo><mtext>&#x2009;</mtext><mi>n</mi><mtext>&#x2009;</mtext><mi>log</mi><mtext>&#x2009;</mtext><mi>n</mi><mtext>&#x2009;</mtext><mi>a</mi><mi>s</mi><mtext>&#x2009;</mtext><mi>n</mi><mtext>&#x2009;</mtext><mo>&#x2192;</mo><mi>&#x221E;</mi><mo>.</mo> </mrow> </math> $${p_n} \sim \,n\,\log \,n\,as\,n\, \to \infty.$$