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#1 2016-11-18 22:23:10

zetafunc
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Registered: 2014-05-21
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A zeta-like sum

I would like to consider the following sum:

where each
is a non-zero vector in
and
denotes the usual Euclidean norm on
I would like to know under which conditions this sum converges (dependent on
and the dimension
).

If
then this sum converges only if
(You can see this just by replacing
with
). But what about for other values of
? For instance, does the series

converge in any dimension
?

In the case where
the condition
lets us replace at most one of
in the sum, but from that point, it doesn't seem like there is anything obvious you can do (mainly due to the
term appearing in the denominator, if one chooses to replace
). You can obtain other expressions involving terms like
by using the triangle inequality, but I don't know what to do with the resulting sums.

One idea I've had is to break up each
and consider regions where
for some
and see if this yields anything in terms of zeta functions (for normed arguments).

A remark: for anyone interested in lattice point problems, I have managed to show that

where
is the
-norm of the remainder in the Gauss circle problem (and its generalisations).

Last edited by zetafunc (2016-11-18 22:51:56)

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#2 2016-11-19 00:01:46

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,407

Re: A zeta-like sum

Hi:

Do you wish this to be done by computer, if not, it belongs in the highly acclaimed Euler Avenue.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#3 2016-11-19 00:08:17

zetafunc
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Registered: 2014-05-21
Posts: 1,849
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Re: A zeta-like sum

I would like to see if a computer can do anything with small values of r. I'd like to try Mathematica, but unfortunately my licence expired a few days ago, so I'm currently trying to get a new one.

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#4 2016-11-19 00:09:32

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,407

Re: A zeta-like sum

Hi;

Could you put it into the form of a command?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#5 2016-11-19 00:21:45

zetafunc
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Registered: 2014-05-21
Posts: 1,849
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Re: A zeta-like sum

Can Boole[] be used with multiple conditions, e.g. Boole[a+b=0,c+d=0,e+f=0]? Or is it better to use If[] statements?

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#6 2016-11-19 00:23:50

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,407

Re: A zeta-like sum

Hi;

I never did get the hang of Boole, George or the M command.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#7 2016-11-19 20:58:47

zetafunc
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Registered: 2014-05-21
Posts: 1,849
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Re: A zeta-like sum

Why does this yield "Indeterminate"?

Sum[1/Abs[a] Boole[a != 0], {a, -10, 10}]

Similarly, why does the same thing happen here?

f[a_] := (Abs[a])^(-1)
Sum[f[a]  If[a == 0, 0, 1] , {a, -10, 10}]

Last edited by zetafunc (2016-11-19 21:09:16)

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#8 2016-11-19 21:19:00

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,407

Re: A zeta-like sum

Hi;

For the first one:

1/Abs[a] Boole[a != 0]

When a = 0 you are multiplying (1/0) by 0, which is indeterminate.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#9 2016-11-19 21:23:45

zetafunc
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Registered: 2014-05-21
Posts: 1,849
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Re: A zeta-like sum

Oh, I see. How do I impose conditions on the sum?

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#10 2016-11-19 21:32:21

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,407

Re: A zeta-like sum

The first way that comes to mind would be:

Sum[1/Abs[a], {a, Range[-10, -1]}] + Sum[1/Abs[a], {a, Range[1, 10]}]

Or even:

2 Sum[1/a, {a, 1,10}]

Or you could define a piecewise function which are like the one we use in math.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#11 2016-11-19 22:02:14

zetafunc
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Registered: 2014-05-21
Posts: 1,849
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Re: A zeta-like sum

Then, the sum I would like to investigate for now is:

f[a_] := Piecewise[{{(Abs[a])^(-1), a != 0}, {0, a = 0}}]
g[b_] := Piecewise[{{(Abs[b])^(-1), b != 0}, {0, b = 0}}]
h[c_] := Piecewise[{{(Abs[c])^(-1), c != 0}, {0, c = 0}}]
Sum[f[a]*g[b]*h[c] Boole[a + b + c == 0], {a, -10000, 10000}, {b, -10000, 
  10000}, {c, -10000, 10000}]

where the limits can be changed.

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#12 2016-11-19 22:12:32

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,407

Re: A zeta-like sum

Yes, I would change the limits because 20000^3 = 8 000 000 000 000 which is going to take a long time.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#13 2016-11-19 22:17:29

zetafunc
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Registered: 2014-05-21
Posts: 1,849
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Re: A zeta-like sum

I am currently running it with limits from {-100,100}. No results yet.

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#14 2016-11-19 22:21:08

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,407

Re: A zeta-like sum

Even that is around 8 million iterations, not a trivial amount.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#15 2016-11-19 22:24:27

zetafunc
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Registered: 2014-05-21
Posts: 1,849
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Re: A zeta-like sum

Is there a better way to do this? (I'm also trying NSum.)

Last edited by zetafunc (2016-11-19 22:25:49)

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#16 2016-11-19 22:29:57

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,407

Re: A zeta-like sum

I do not know yet. For one thing it is okay to define sin(x)/x at x = 0 as one in a piecewise function because sin(0)/0 is what they call a removable singularity. But is it okay to define 1 / 0 as 0 as you have done in your piecewise functions? See you later, I have to get dressed and head out.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#17 2016-11-19 22:36:56

zetafunc
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Registered: 2014-05-21
Posts: 1,849
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Re: A zeta-like sum

I want the sum to range over all non-zero integers, and with the extra Boole condition. This should be fine as it is a sum rather than an integral.

Last edited by zetafunc (2016-11-19 22:37:24)

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#18 2016-11-19 22:38:50

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,407

Re: A zeta-like sum

Hi;

Unfortunately I have to get dressed and head out, see you later.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#19 2016-11-19 22:40:35

zetafunc
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Registered: 2014-05-21
Posts: 1,849
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Re: A zeta-like sum

OK, see you later.

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#20 2016-11-19 22:56:04

zetafunc
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Registered: 2014-05-21
Posts: 1,849
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Re: A zeta-like sum

Something strange is happening here. The sum

f[a_] := Piecewise[{{(Abs[a])^(-1), a != 0}, {0, a = 0}}]
g[b_] := Piecewise[{{(Abs[b])^(-1), b != 0}, {0, b = 0}}]
h[c_] := Piecewise[{{(Abs[c])^(-1), c != 0}, {0, c = 0}}]
NSum[f[a]*g[b]*h[c] Boole[a + b + c == 0], {a, -10, 10}, {b, -10, 
  10}, {c, -10, 10}]

yields 9.96439. However, changing the limits to -15, 15 gives a bunch of error messages, followed by a ridiculously large number which cannot possibly be correct (of order 10^10). It works fine up to -14,14 but at that point it started spitting out nonsense (-16,16 gives a negative number, which is impossible because the summand is positive).

Last edited by zetafunc (2016-11-19 22:58:11)

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#21 2016-11-20 06:59:56

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,407

Re: A zeta-like sum

Hi;

Do you already know the exact answer?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#22 2016-11-20 07:16:25

zetafunc
Member
Registered: 2014-05-21
Posts: 1,849
Website

Re: A zeta-like sum

No, I have no idea if it even converges. My instinct says it will converge for d = 1 and diverge for d = 2.

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#23 2016-11-20 07:25:59

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,407

Re: A zeta-like sum

You have some syntax errors.

(a = 0 assigns 0 to a, a == 0 tests whether a is 0. You want the test.)

Start a new session and then try changing your a = 0 , b = 0, c = 0 to a == 0, b == 0, c == 0.

Then change NSum to Sum when using small values. But you will still get an answer close to 10.

Based on your code I would say getting an answer of around 2 as you suggest is not correct.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#24 2016-11-20 08:07:36

zetafunc
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Registered: 2014-05-21
Posts: 1,849
Website

Re: A zeta-like sum

I did not suggest the answer would be around 2, I said that it would diverge for d = 2. I corrected the syntax. I get an answer close to 10 for smaller values though.

EDIT: Changed NSum to Sum and it worked well, even with -50,50 for all the ranges. Trying -100,100 now.

Last edited by zetafunc (2016-11-20 08:11:16)

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#25 2016-11-20 08:11:33

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,407

Re: A zeta-like sum

Please let me see your code.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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