[BUG] weird Zeta function

01122014, 10:38 PM
(This post was last modified: 01132014 01:40 PM by Tugdual.)
Post: #1




[BUG] weird Zeta function
Zeta(3) returns... Zeta(3). I tried Eval, no luck.
So I decided to define F1(X)=Zeta(X) but it doesn't help since F1(3) also returns Zeta(3). I plotted F1 and then I see the curve on screen. Using the touch screen, menu, I tried "go to" and "3" "Enter". Now I got an error message: function is not defined for x=3. I tried a manual approximation and entered sum(n^ 3,n,1,1000) which returns 1.20 which is close enough and consistent with what I see on the curve. So why zeta(3) fails? Also the zeta help says that Re(x) shall be >1 while zeta returns values for negative x? 

01122014, 10:55 PM
Post: #2




RE: weird Zeta function
(01122014 10:38 PM)Tugdual Wrote: Zeta(3) returns... Zeta(3). In CAS mode, forcing the argument to be real does provide an approximation of Apéry's constant. Zeta(3.) = 1.20205690316 But this doesn't work in Home mode. Mark Hardman Ceci n'est pas une signature. 

01132014, 06:55 AM
(This post was last modified: 01132014 06:56 AM by Tugdual.)
Post: #3




RE: weird Zeta function
(01122014 10:55 PM)Mark Hardman Wrote: In CAS mode, forcing the argument to be real does provide an approximation of Apéry's constant. Hi Mark, thanks for the hint! I find it even more bizarre that the only way to get an approximation is in CAS which seems to be dedicated to symbolic calculations. Also in the meantime I tried Zeta(1+i) it says Eta(1+i)/(12^i) I think this is correct but suprisingly we see Dirichelt's Eta function popping up while it is neither documented nor listed in the catalog. Note: I have edited the first post to call it a BUG 

01132014, 04:58 PM
Post: #4




RE: [BUG] weird Zeta function
Yes,
This was already noted a while back (if you peruse the archives). Dirichlet's eta function also won't get evaluated numerically. 

01132014, 07:51 PM
Post: #5




RE: [BUG] weird Zeta function
(01132014 06:55 AM)Tugdual Wrote: I find it even more bizarre that the only way to get an approximation is in CAS which seems to be dedicated to symbolic calculations. Zeta(approx(3)) returns a real in Home. So does turning off "Change apparent integers into exact integers" in the CAS Settings (page 1, end of 3rd line), but since that's not programmable, I suggest using the Zeta(approx(3)) method. These two methods are related. If you type Zeta(3.) in Home, CAS turns the real 3 into an integer automatically (not a bug; it's due to that CAS setting), hence the symbolic result. But approx is a CAS function, so it turns approx(3) into a real 3 in the CAS, and feeds that to the Zeta function. OR you can turn off the "convert whole numbers into integers" mode, and avoid using approx. X<> c Joe 

01132014, 09:48 PM
Post: #6




RE: [BUG] weird Zeta function
(01132014 07:51 PM)Joe Horn Wrote:Hi Joe, that's interesting, I didn't even notice this option in CAS parameters (a simple check box with no further explanation...). I understand what you say but I'm happy I don't have to enter sin(approx(3)) everytime I need to do a sinus calculation ;)(01132014 06:55 AM)Tugdual Wrote: I find it even more bizarre that the only way to get an approximation is in CAS which seems to be dedicated to symbolic calculations. 

01162014, 07:23 PM
Post: #7




RE: [BUG] weird Zeta function
(01132014 09:48 PM)Tugdual Wrote:The explanation is at the bottom of the screen, immediately above the soft keys. Those prompts are often the only explanation offered by input forms.(01132014 07:51 PM)Joe Horn Wrote: ... "Change apparent integers into exact integers" in the CAS Settings (page 1, end of 3rd line) ...... I didn't even notice this option in CAS parameters (a simple check box with no further explanation...) X<> c Joe 

01172014, 09:33 PM
Post: #8




RE: [BUG] weird Zeta function
I had noticed this phenomenon with the Prime. My own opinion FWIW is that Zeta(r) is evaluated in terms of Zeta(r2), recursively until a value between 0 and 2 is reached. Faced with Zeta() of an odd integer, it will backtrack successfully until it reaches Zeta(1) and boggles.


01182014, 07:17 AM
(This post was last modified: 01182014 07:18 AM by parisse.)
Post: #9




RE: [BUG] weird Zeta function
Zeta(n) is left as is if n>0 is odd, and computed using bernoulli numbers if n is even. You can't do any better if you want an exact answer. Of course, you can enter Zeta(d) where d is a floating point number to get an approximation.
Code:


01182014, 11:01 AM
Post: #10




RE: [BUG] weird Zeta function
Yet if I ask for Zeta(3) on my WP34S, I get the perfectly sensible answer of 1.20205690316, the same answer that Zeta(approx(3)) gives on the Prime.


01192014, 01:16 PM
Post: #11




RE: [BUG] weird Zeta function
So on top of hidden Eta (see my previous posts) I see that we also have Benouilli numbers. Why don't we have access to those functions?
Also I agree with Curlytop, why don't you simply cast to real and return an estimate instead of "symbolic(at_Zeta,x);"? 

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