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May 11

zeros of the Riemann zeta function

Computers have determined that trillions of zeros of Riemann zeta function are on the critical line. How do they determine that it is exactly on the critical line? Bubba73 ^{You talkin' to me?} 00:34, 11 May 2016 (UTC)[reply]

There seem to be some good answers here. -- BenRG (talk) 05:46, 11 May 2016 (UTC)[reply]

To summarize the above, since apparently no one read it: you can count zeroes on the critical strip (where all nontrivial zeroes provably reside) with a contour integral, and there is a real-valued continuous function Z(t) which is zero iff ζ(1/2+it) is zero. So to prove that there are no zeroes off the critical line in a portion of the critical strip, it's sufficient to bound the value of the contour integral to a region containing no integers larger than n, and find n+1 points where Z provably alternates between positive and negative. This only requires proving rough bounds, not computing exact values. It gets more complicated if Z has any multiple zeroes, but that's a safe-side failure, it hasn't happened yet, and it's conjectured that it won't happen. Also, there's a link to Andrew Odlyzko's papers. -- BenRG (talk) 21:45, 12 May 2016 (UTC)[reply]

For what it's worth, I did read through the page you linked ... thank you for providing it! Nimur (talk) 18:10, 13 May 2016 (UTC)[reply]

Contour integrals! You can integrate around a closed curve that encloses a point, and using the residue theorem you can analytically relate the value of the integral to the number of zeros enclosed.

In the general case, this doesn't make your job any easier: an arbitrary contour around an arbitrary point might not have an easy or analytical solution; but some (many!) of them do; and contour integration is a great mathematical tool to have in your toolbox. Before digital computers made numerical solutions cheap and easy, contour integration was a primary method used for stability and control analysis; everybody learned how to do it. (Well, almost everybody). It's sort of become an esoteric art in recent years, along with the circular slide rule ... for analytical math, they keep you honest and generally promote a style of thinking that can help deflate the false precision that creeps into computational solutions. There are a zillion techniques borrowed from control theory that will help you count the number of poles and zeros inside a closed contour. (...Even if it encloses infinity).

Here's some more reading from MathWorld: on the Hankel contour as it applies to evaluating properties of the Riemann Zeta function. That's a great and easy contour that you can trivially construct to encircle an arbitrary point - (and every other point to the right of it) - provided that you're comfortable traveling to infinity and back. This is a more restrictive contour than the Nyquist contour, but you can use the exact same techniques as controls folks use for stability analysis - and you can enforce the limit as the imaginary part goes to zero (which is what you want in your special case)!

So, that's how I would do it - by combining mathematical analyis tools and then doing some gorey and difficult integration. It would be hard and would probably take hours to set up and do it properly. I also have the luxury of not really "needing" the answer.

But you asked about how a computer would do it! I'm not sure that a computer can think so creatively! I'm also not sure you could do this kind of work numerically - not even with arbitrary precision arithmetic.

Nimur (talk) 15:24, 11 May 2016 (UTC)[reply]

Aren't you kind of burying the lede? Am I misunderstanding something, or is your answer basically "I can conceive of analytic methods that might in principle suffice but don't know how this would be done via numeric/computational methods? Because I think OP's implicit assumption is correct - that this huge but finite number of checks has been accomplished with some level of rather uncharacteristically rigorous numeric work, that is, computational techniques, that, when applied carefully and cleverly and with some human-powered analytic work in between, can indeed confer rigorous results that the imaginary part is exactly zero. But I don't know for sure, that's just my impression. Maybe they just get it down to 10e-100 and call it a day. But if I had to wager a beer or a coffee, I'd wager against that, at least for work in done in the last few decades. For example, this [1] paper contains a nice history of computational approaches to this issue. It does explicitly say the first computational work (in 1903) only had the values of the first 10 zeros correct to the first 6 decimal places, but I'm not sure if that applies to total error or error on the real part. Anyway, that ref has some very good further refs for OP to look into, and that's the main reason I posted :) SemanticMantis (talk) 20:39, 11 May 2016 (UTC)[reply]

Ah, yes. I was thinking that the general root-finding algorithms would not be sufficient to prove that a zero was exactly on the critical line, due to the imprecision of floating-point numbers. Bubba73 ^{You talkin' to me?} 20:34, 11 May 2016 (UTC)[reply]

In addition to the Odlyzko paper I linked above, here [2] is a nice list of some of the most important papers, curated and authored by the same guy. SemanticMantis (talk) 20:43, 11 May 2016 (UTC)[reply]

Thanks for tracking down Odlyzko, his work made for some entertaining reading material! That guy is surely the world's expert in solving the exact problem that Bubba73 is looking for. He also seems to be a pretty all-around awesome guy who I could probably get along with.

I'll grant SemanticMantis credit for calling me out on "burying the lede." I don't write with the intent to be obfuscatory, but I do think one of our great advantages at Wikipedia, and at the Reference Desk in particular, is that we can take readers in directions that a search-engine will never show them: we can cross boundaries and demonstrate subtle-but-still-relevant connections to topics that a "search query" will not turn up. We can provide a very broad contextual base, crossing boundaries and using human intelligence to link to items that would not match any keywords... in that sense, we're entirely better than a search-engine, which will send you straight to a completely-context-free search-result.

In this specific case, I think my obfuscatory post has a little merit: why in the world would anybody study this crazy and bizarre branch of mathematics? Well, it's true that some mathematicians simply love the pure and unadulterated theory. But in this specific case, the specific problem - which superficially appears to be entirely the domain of mathematical ascetics - is also studied by people who use applied mathematical theory to design and analyze engineered systems, like radio circuits and bridge pillars. Even the most detached mathematical researcher can benefit from the knowledge that other people in different fields are solving similar problems. You wouldn't get that insight if your search-query sent you straight to the Methods section of some 1988 numerical algorithm paper, even if it exactly and accurately answered the query. Here on the reference desk, we can give you both: context and depth, with aspirations to the style of James Burke.

Nimur (talk) 14:43, 12 May 2016 (UTC)[reply]

From what I can understand from these articles, they aren't sure that they are exactly on the critical line. They talk about "medium accuracy", "high accuracy", etc. Bubba73 ^{You talkin' to me?} 03:55, 16 May 2016 (UTC)[reply]

Flash player PARTIAL failure (Game menus not loading)

Hello,

I have noticed on certain flash games on this computer, said flash game does not work properly. I know it is not the game itself, because it works fine on another computer. I am using Windows 8 with Firefox 64 bit and adobe flash.

The game "Loads", and proceeds to show introductory logos and the like, but when it gets to where the main game menu ought to be, there is no menu... but rather JUST the background color of the menu can be seen, with no objects.

Examples:

1 - Shows a creamy off white background only, after "CrescentYR" logo.

2 - Shows pure black background after a quote and an "Idle Adventures" logo.

My browser and flash version are both completely up to date. What could possibly cause this strange situation where the flash content PARTLY works?! I was thinking something goes wrong when it tries to make user interactive parts, but this doesn't seem true since in both games you have to click "Play" after the loading screen, which seems to work fine!

I can't find anyone with this issue via Google searching. Any thoughts?

216.173.144.188 (talk) 17:09, 11 May 2016 (UTC)[reply]

Possibly a corrupted version. Try an uninstall and reinstall. StuRat (talk) 18:54, 11 May 2016 (UTC)[reply]

First, download:
http://fpdownload.macromedia.com/get/flashplayer/pdc/21.0.0.242/install_flash_player.exe
http://download-installer.cdn.mozilla.net/pub/firefox/releases/46.0.1/win32/de/Firefox%20Setup%2046.0.1.exe 32 Bit English (only when using a 32 Bit Version of Windows)
http://download-installer.cdn.mozilla.net/pub/firefox/releases/46.0.1/win64/de/Firefox%20Setup%2046.0.1.exe 64 Bit English (only when using a 64 Bit Version of Windows)

Start %APPDATA% and Backup the folder Mozilla and delete the folder Macromedia.
Change to .. to leave the folder Roaming and enter the folder Local.
Inside the folder Local, delete the folder Mozilla.
Start APPWIZ.CPL and there, uninstall all You see from Flash Player and Firefox.
In the explorer, rightclick on MyComputer to see You have 32 oder 64 Bit Windows, else start CMD and enter set p if ther is a ProgramFiles(x86)= in the list, You have 64 Bit Windows.
From Links above, install only the one downloaded Firefox Versions, that fit to Your Windows. Never install both versions same time. A 32 Bit version will run on a 64 Bit Windows, but not the other way.
Wait until the install is completed, then install the downloaded flash player.
The links point to the recent version when this answer was given. Note for future use to get the newest version only which includes all security updates. --Hans Haase (有问题吗) 20:34, 11 May 2016 (UTC)[reply]

The OP already said they were running Firefox 64 bit. As you said, Firefox x86-64 won't work at all on Windows x86-32 so we can assume the OP is running Windows x86-64 unless they're mistaken about what version of Firefox they were running. Nil Einne (talk) 07:07, 12 May 2016 (UTC)[reply]

Changed from 21.0.0.213 to 21.0.0.242 wich was released recently. Btw. maybe two parallel installations can cause similar problems. --Hans Haase (有问题吗) 11:45, 13 May 2016 (UTC)[reply]