I was recently reading Conrey and Young's 2011 paper "More than 41% of the zeros of the zeta function are on the critical line", in which they improve the lower bound on the proportion of zeros on the critical line of the Riemann $\zeta$-function. In order to achieve this, they define the (monstrous) mollifier:

$$ \psi=\sum_{n\leq y_1}\frac{\mu(n)P_1[n]n^{\sigma_0-\frac{1}{2}}}{n^s}+\chi(s+\frac{1}{2}-\sigma_0)\sum_{hk\leq y_2}\frac{\mu_2(h)h^{\sigma_0-\frac{1}{2}}k^{\frac{1}{2}-\sigma_0}}{h^sk^{1-s}}P_2(hk). $$

This is followed by a nearly 20-page tour-de-force of grueling analytic number theory, coming to a grinding halt by improving the already known bounds from $40.88\%$ to... $41.05\%$.

Of course any improvement of this lower bound is a highly important achievement, and the methods and techniques used in the paper are deeply inspirational. However, given the highly complex nature of the proof and the small margin of improvement obtained over already existing results, I was inspired to ask the following question:

What are other examples of notably long or difficult proofs that only improve upon existing results by a small amount?

I was recently reading Bui, Conrey and Young's 2011 paper "More than 41% of the zeros of the zeta function are on the critical line", in which they improve the lower bound on the proportion of zeros on the critical line of the Riemann $\zeta$-function. In order to achieve this, they define the (monstrous) mollifier:

$$ \psi=\sum_{n\leq y_1}\frac{\mu(n)P_1[n]n^{\sigma_0-\frac{1}{2}}}{n^s}+\chi(s+\frac{1}{2}-\sigma_0)\sum_{hk\leq y_2}\frac{\mu_2(h)h^{\sigma_0-\frac{1}{2}}k^{\frac{1}{2}-\sigma_0}}{h^sk^{1-s}}P_2(hk). $$

This is followed by a nearly 20-page tour-de-force of grueling analytic number theory, coming to a grinding halt by improving the already known bounds from $40.88\%$ to... $41.05\%$.

Of course any improvement of this lower bound is a highly important achievement, and the methods and techniques used in the paper are deeply inspirational. However, given the highly complex nature of the proof and the small margin of improvement obtained over already existing results, I was inspired to ask the following question:

What are other examples of notably long or difficult proofs that only improve upon existing results by a small amount?

I was recently reading Conrey and Young's 2011 paper "More than 41% of the zeros of the zeta function are on the critical line", in which they improve the lower bound on the proportion of zeros on the critical line of the Riemann $\zeta$-function. In order to achieve this, they define the (monstrous) mollifier:

$$ \psi=\sum_{n\leq y_1}\frac{\mu(n)P_1[n]n^{\sigma_0-\frac{1}{2}}}{n^s}+\chi(s+\frac{1}{2}-\sigma_0)\sum_{hk\leq y_2}\frac{\mu_2(h)h^{\sigma_0-\frac{1}{2}}k^{\frac{1}{2}-\sigma_0}}{h^sk^{1-s}}P_2(hk). $$

This is followed by a nearly 20-page tour-de-force of grueling analytic number theory, coming to a grinding halt by improving the already known bounds from $40.88\%$ to... $41.05\%$.

Of course any improvement of this lower bound is a highly important achievement, and the methods and techniques used in the paper are deeply inspirational. However, given the highly complex nature of the proof and the small margin of improvement obtained over already existing results, I was inspired to ask the following question:

What are other examples of notably long or difficult proofs that only improve upon existing results by a small amount?

I was recently reading Conrey and Young's 2011 paper "More than 41% of the zeros of the zeta function are on the critical line", in which they improve the lower bound on the proportion of zeros on the critical line of the Riemann $\zeta$-function. In order to achieve this, they define the (monstrous) mollifier:

$$ \psi=\sum_{n\leq y_1}\frac{\mu(n)P_1[n]n^{\sigma_0-\frac{1}{2}}}{n^s}+\chi(s+\frac{1}{2}-\sigma_0)\sum_{hk\leq y_2}\frac{\mu_2(h)h^{\sigma_0-\frac{1}{2}}k^{\frac{1}{2}-\sigma_0}}{h^sk^{1-s}}P_2(hk). $$

This is followed by a nearly 20-page tour-de-force of grueling analytic number theory, coming to a grinding halt by improving the already known bounds from $40.88\%$ to... $41.05\%$.

Of course any improvement of this lower bound is a highly important achievement, and the methods and techniques used in the paper are deeply inspirational. However, given the highly complex nature of the proof and the small margin of improvement obtained over already existing results, I was inspired to ask the following question:

What are other examples of notably long or difficult proofs that only improve upon existing results by a small amount?