# The Philosophy of Square-Root Cancellation

In this note, I’ll discuss why square-root cancellation is so typical in problems in number theory and give a quick survey of important sums known or widely conjectured to satisfy bounds of this form.

The prime number theorem (PNT) was not proven until 1896, but a weaker form (up to constants) was established decades earlier. The earliest proof was due to Chebyshev in 1852, and his work inspired others to take up the mantle and inch towards what they thought would be a proof of the PNT. Here, we show the strength of the Chebyshev method and ask whether it had the power to prove the PNT after all.