Timeline for Integrals involving $1/|\zeta(1+i t)|^2$: closed expressions?
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Aug 28, 2021 at 11:03 | answer | added | juan | timeline score: 1 | |
| Aug 27, 2021 at 22:15 | comment | added | Terry Tao | Sure, but they are pretty disgusting: you've already basically expressed your integral in terms of the (logarithmically weighted) Mertens function, and that function can be written (assuming simple zeroes) as a sum over zeroes (but with coefficients that are reciprocals of products of differences of zeroes, so not super pleasant to deal with). After interchanging some sums you should be able to write your expression as a sum over pairs of zeroes of some nasty product over zeroes. Can't say that this will be too enlightening, though. | |
| Aug 27, 2021 at 22:13 | comment | added | H A Helfgott | Well, do you have some such expressions? | |
| Aug 27, 2021 at 22:11 | comment | added | Terry Tao | Given that the finiteness of this integral implies the prime number theorem, it is unlikely that there is going to be a closed form for this expression that is manifestly finite without the assistance of this theorem, unless you are willing to allow expressions that depend on the location of the zeroes of the zeta function as a "closed form". | |
| Aug 27, 2021 at 21:55 | comment | added | juan | Making some transformations from the integral, transformations that I will be ashamed to confess, I get to this expression $$-\frac12\sum_{n=1}^\infty \frac1n\Bigl(\sum_{ab=n}\mu(a)\mu(b)|\log(a/b)|\Bigr).$$ Numerically it appear to have some sense. | |
| Aug 27, 2021 at 17:30 | comment | added | H A Helfgott | @davidlowryduda Interesting - what do you have in mind? A single sum would probably count as "closed enough". | |
| Aug 27, 2021 at 17:29 | comment | added | H A Helfgott | Well, this one seems like a toss-up between the two sites, so I let the universe decide. | |
| Aug 27, 2021 at 16:39 | comment | added | davidlowryduda | About the question itself: it might be possible to adapt some of the work of KST (or earlier related work by Perelli) to get a related expression with $\sum \mu^2(n)/n$ appearing, but maybe this is also not very closed. It seems hard for me to imagine an expression without some sum over $\mu(n)$ appearing. | |
| Aug 27, 2021 at 16:32 | comment | added | davidlowryduda | Generally I'd recommend choosing the site that seems like the best fit and just posting it there. (In this case, that seems like MO). If that doesn't yield answers after a bit, then perhaps ask somewhere else and link between them. This is basically what's recommended here. Some people are really sensitive to people posting the same question on different forums, thinking that they're quickly trying to get an answer and not worrying that this might encourage duplication of effort. | |
| Aug 27, 2021 at 16:21 | comment | added | H A Helfgott | (Should I have stated that, and, if so, how?) | |
| Aug 27, 2021 at 16:11 | comment | added | davidlowryduda | Note that this has been crossposted to MSE: math.stackexchange.com/q/4234446/9754 | |
| Aug 27, 2021 at 16:00 | history | asked | H A Helfgott | CC BY-SA 4.0 |