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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)[reply]

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)[reply]

Resolved

Thanks Bubba73 ^{You talkin' to me?} 06:32, 22 January 2022 (UTC)[reply]

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)[reply]
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,^[1] see the full text of Rigorous high speed separation of zeros of Riemann's zeta function, 2. --Lambiam 11:56, 23 January 2022 (UTC)[reply]

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