Wikipedia:Reference desk/Archives/Mathematics/2022 January 22

= January 22 =

Zeros of the Riemann zeta function
I've read that computers have shown that trillions of zeros of the Riemann zeta function have real part 0.5. Since a computer has a finite precision, how can it verify that the real part is exactly 0.5, as opposed to $$0.5+\epsilon$$, for some tiny, but non-negative $$\epsilon$$? Bubba73 You talkin' to me? 05:18, 22 January 2022 (UTC)


 * Here you can find an exposition of a remarkable method to verify this computationally in a rigorous manner. --Lambiam 06:20, 22 January 2022 (UTC)

Thanks Bubba73 You talkin' to me? 06:32, 22 January 2022 (UTC)


 * I looked at how it actually works, and the key is when you divide a value by 2$$\pi$$i and take the nearest integer. Since there is only a finite precision approximation to $$\pi$$, that could possibly be 1 off from the true value, but in that unlikely case, it would indicate a zero way off the critical line, so you would know to look more closely at that one.  Bubba73 You talkin' to me? 23:07, 22 January 2022 (UTC)
 * The value to be divided by $$2\pi i$$ is the result of a contour integral and much more difficult to compute to a high precision than $$\pi.$$ If you are interested in the gory details of the numerical methods used to show that the first 1,500,000,001 zeros in the critical strip are simple and have real part $1/2$, see the full text of Rigorous high speed separation of zeros of Riemann's zeta function, 2. --Lambiam 11:56, 23 January 2022 (UTC)


 * Thanks, I downloaded it to maybe look at the gory details. Bubba73 You talkin' to me? 16:21, 24 January 2022 (UTC)