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| David Bernier... |
 Posted: Sat Oct 31, 2009 10:33 am |
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Guest
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I'm interested in methods for computing the derivative of the
Riemann zeta function, zeta', in the critical strip
0<= Re(s) <= 1.
One of the equivalents of the Riemann Hypothesis is
that zeta' has no zero s with 0 < Re(s) < 1/2 .
Speiser [1934]:
http://www.aimath.org/WWN/rh/articles/html/89a/
There's a paper by Garaev and Yildirim, available in
it's pre-print form here:
http://arxiv.org/abs/math/0610377v2
"On small distances between ordinates of zeros of zeta(s) and zeta'(s)"
which seems like it could be a good source for previous results
on the zeros of zeta' and how they relate to the zeros of zeta,
in the critical strip.
Or else, maybe someone has already computed some of the zeros
of zeta' in the critical strip, but I didn't find anything like
that.
David Bernier |
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| Raymond Manzoni... |
| Posted: Sun Nov 01, 2009 4:28 am |
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Guest
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David Bernier a écrit :
[quote]I'm interested in methods for computing the derivative of the
Riemann zeta function, zeta', in the critical strip
0<= Re(s) <= 1.
One of the equivalents of the Riemann Hypothesis is
that zeta' has no zero s with 0 < Re(s) < 1/2 .
Speiser [1934]:
http://www.aimath.org/WWN/rh/articles/html/89a/
There's a paper by Garaev and Yildirim, available in
it's pre-print form here:
http://arxiv.org/abs/math/0610377v2
"On small distances between ordinates of zeros of zeta(s) and zeta'(s)"
which seems like it could be a good source for previous results
on the zeros of zeta' and how they relate to the zeros of zeta,
in the critical strip.
Or else, maybe someone has already computed some of the zeros
of zeta' in the critical strip, but I didn't find anything like
that.
David Bernier
[/quote]
Two recent papers :
Haseo Ki(2008) "The zeros of the derivative of the Riemann zeta
function near the critical line" <http://arxiv.org/pdf/math/0701726>
Yitang Zhang(2001) "On the zeros of zeta'(s) near the critical line"
<http://eduunix.ccut.edu.cn/index2/math/202.38.126.65/mathdoc/Duke.Mathematical.Journal/DMJ11003_4.pdf>
Some references provided in these papers could be interesting too.
Hoping it helped a little,
Raymond |
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| Marko Amnell... |
| Posted: Sun Nov 01, 2009 6:00 am |
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"Raymond Manzoni" <raymman at (no spam) free.fr> wrote:
[quote]David Bernier a écrit :
I'm interested in methods for computing the derivative of the
Riemann zeta function, zeta', in the critical strip
0<= Re(s) <= 1.
One of the equivalents of the Riemann Hypothesis is
that zeta' has no zero s with 0 < Re(s) < 1/2 .
Speiser [1934]:
http://www.aimath.org/WWN/rh/articles/html/89a/
There's a paper by Garaev and Yildirim, available in
it's pre-print form here:
http://arxiv.org/abs/math/0610377v2
"On small distances between ordinates of zeros of zeta(s) and zeta'(s)"
which seems like it could be a good source for previous results
on the zeros of zeta' and how they relate to the zeros of zeta,
in the critical strip.
Or else, maybe someone has already computed some of the zeros
of zeta' in the critical strip, but I didn't find anything like
that.
David Bernier
Two recent papers :
Haseo Ki(2008) "The zeros of the derivative of the Riemann zeta function
near the critical line" <http://arxiv.org/pdf/math/0701726
Yitang Zhang(2001) "On the zeros of zeta'(s) near the critical line"
http://eduunix.ccut.edu.cn/index2/math/202.38.126.65/mathdoc/Duke.Mathematical.Journal/DMJ11003_4.pdf
Some references provided in these papers could be interesting too.
Hoping it helped a little,
Raymond
[/quote]
Maybe Michael Rubinstein's L-function software will work.
The default L-function is the Riemann zeta function.
http://pmmac03.math.uwaterloo.ca/~mrubinst/L_function_public/L.html
There's an option for computing derivatives about which the Readme file
says:
"Presently the derivative option uses numeric differentiation, and one
loses about half the working precision for each successive derivative.
Multiprecision is still being implemented, so, for now, the derivative
option
only gives moderately reasonable output for the first derivative (about 6-7
digits),
and less for the second derivative (about 3 digits). Beyond this, one
needs to use the USE_LONG_DOUBLE compile option in the MAkefile or higher
precision." |
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| Raymond Manzoni... |
| Posted: Sun Nov 01, 2009 9:03 am |
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Guest
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Raymond Manzoni a écrit :
[quote]David Bernier a écrit :
I'm interested in methods for computing the derivative of the
Riemann zeta function, zeta', in the critical strip
0<= Re(s) <= 1.
(snip)[/quote]
[quote]
Some references provided in these papers could be interesting too.
[/quote]
For example :
Levinson+Montgomery(1974) "Zeros of the derivatives of the Riemann
zeta-function" :
<http://www.kryakin.com/files/Acta_Mat_(2_55)/acta150_107/133/133_03.pdf>
Conrey+Ghosh(1990?) "Zeros of derivatives of the Riemann
zeta-function near the critical line"
<http://books.google.fr/books?hl=fr&lr=&id=G02moOmuOX4C&oi=fnd&pg=PA95>
Mezzadri(2002) "Random matrix theory and the zeros of zeta'(s)" (with
numerical investigations) : <http://arxiv.org/pdf/math-ph/0207044>
Saidak(2004) "On the logarithmic derivative of the Euler product"
<http://tatra.mat.savba.sk/Full/29/14SAIDAK.ps>
[quote]
Or else, maybe someone has already computed some of the zeros
of zeta' in the critical strip, but I didn't find anything like
that.
[/quote]
I tried a numerical search (using pari/gp) and found two zeros of
zeta' around these values (may be...) :
0.96468562270568565 + 48.847159905068479085*I
0.864623222098647 + 76.362807896467*I
Hoping it helped,
Raymond |
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| Axel Vogt... |
| Posted: Sun Nov 01, 2009 10:07 am |
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Guest
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Raymond Manzoni wrote:
[quote]Raymond Manzoni a écrit :
David Bernier a écrit :
I'm interested in methods for computing the derivative of the
Riemann zeta function, zeta', in the critical strip
0<= Re(s) <= 1.
(snip)
Some references provided in these papers could be interesting too.
For example :
Levinson+Montgomery(1974) "Zeros of the derivatives of the Riemann
zeta-function" :
http://www.kryakin.com/files/Acta_Mat_(2_55)/acta150_107/133/133_03.pdf
Conrey+Ghosh(1990?) "Zeros of derivatives of the Riemann zeta-function
near the critical line"
http://books.google.fr/books?hl=fr&lr=&id=G02moOmuOX4C&oi=fnd&pg=PA95
Mezzadri(2002) "Random matrix theory and the zeros of zeta'(s)" (with
numerical investigations) : <http://arxiv.org/pdf/math-ph/0207044
Saidak(2004) "On the logarithmic derivative of the Euler product"
http://tatra.mat.savba.sk/Full/29/14SAIDAK.ps
Or else, maybe someone has already computed some of the zeros
of zeta' in the critical strip, but I didn't find anything like
that.
I tried a numerical search (using pari/gp) and found two zeros of
zeta' around these values (may be...) :
0.96468562270568565 + 48.847159905068479085*I
0.864623222098647 + 76.362807896467*I
Hoping it helped,
Raymond
[/quote]
Here 'are' some using Maple's command RootFinding[Analytic]:
.964685622705685650525780+48.8471599050684790854189*I
.848735328105403472052794+60.1408457820384239102073*I
.864622864426113300262053+76.3628078964670422358770*I
.780628004724644645328179+95.2929682713522169397106*I
.864103640598939499604406+88.1775174098810128722274*I
.856309339180055369726490+134.193836602386409228846*I
.943828539659771481951279+140.469959838197100688840*I
.662929906884329935358395+150.485953620246866996388*I
.966951342073371224843721+156.632667913413661808966*I
.863404697829980428874710+158.282522106715305651649*I
.635638410195870078269381+111.431017613746736506374*I
.847766864212029839013393+123.715269749934938660387*I |
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| Raymond Manzoni... |
| Posted: Sun Nov 01, 2009 11:12 am |
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Axel Vogt a écrit :
[quote]Raymond Manzoni wrote:
Raymond Manzoni a écrit :
David Bernier a écrit :
I'm interested in methods for computing the derivative of the
Riemann zeta function, zeta', in the critical strip
0<= Re(s) <= 1.
(snip)
Some references provided in these papers could be interesting too.
For example :
Levinson+Montgomery(1974) "Zeros of the derivatives of the Riemann
zeta-function" :
http://www.kryakin.com/files/Acta_Mat_(2_55)/acta150_107/133/133_03.pdf
Conrey+Ghosh(1990?) "Zeros of derivatives of the Riemann
zeta-function near the critical line"
http://books.google.fr/books?hl=fr&lr=&id=G02moOmuOX4C&oi=fnd&pg=PA95
Mezzadri(2002) "Random matrix theory and the zeros of zeta'(s)"
(with numerical investigations) : <http://arxiv.org/pdf/math-ph/0207044
Saidak(2004) "On the logarithmic derivative of the Euler product"
http://tatra.mat.savba.sk/Full/29/14SAIDAK.ps
Or else, maybe someone has already computed some of the zeros
of zeta' in the critical strip, but I didn't find anything like
that.
I tried a numerical search (using pari/gp) and found two zeros of
zeta' around these values (may be...) :
0.96468562270568565 + 48.847159905068479085*I
0.864623222098647 + 76.362807896467*I
Hoping it helped,
Raymond
Here 'are' some using Maple's command RootFinding[Analytic]:
.964685622705685650525780+48.8471599050684790854189*I
.848735328105403472052794+60.1408457820384239102073*I
.864622864426113300262053+76.3628078964670422358770*I
.780628004724644645328179+95.2929682713522169397106*I
.864103640598939499604406+88.1775174098810128722274*I
.856309339180055369726490+134.193836602386409228846*I
.943828539659771481951279+140.469959838197100688840*I
.662929906884329935358395+150.485953620246866996388*I
.966951342073371224843721+156.632667913413661808966*I
.863404697829980428874710+158.282522106715305651649*I
.635638410195870078269381+111.431017613746736506374*I
.847766864212029839013393+123.715269749934938660387*I
[/quote]
Nice! (my method was rather primitive...)
To the OP note that the zeta function has more zeros in the same
imaginary range so that I'll add some zeros with real part larger than 1 :
2.46316186945432128587439505331+23.2983204927628579020109616266i
1.28649682226904769704411427839+31.7082500831159086049543521423i
1.38276360571167457578453372043+42.2909645545967298190807460934i
(using this time the secant method on pari/gp)
Fine continuation!
Raymond |
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| David Bernier... |
| Posted: Mon Nov 02, 2009 12:01 am |
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Marko Amnell wrote:
[quote]"Raymond Manzoni" <raymman at (no spam) free.fr> wrote:
David Bernier a écrit :
I'm interested in methods for computing the derivative of the
Riemann zeta function, zeta', in the critical strip
0<= Re(s) <= 1.
One of the equivalents of the Riemann Hypothesis is
that zeta' has no zero s with 0 < Re(s) < 1/2 .
Speiser [1934]:
[...][/quote]
[quote]Two recent papers :
Haseo Ki(2008) "The zeros of the derivative of the Riemann zeta function
near the critical line" <http://arxiv.org/pdf/math/0701726
Yitang Zhang(2001) "On the zeros of zeta'(s) near the critical line"
http://eduunix.ccut.edu.cn/index2/math/202.38.126.65/mathdoc/Duke.Mathematical.Journal/DMJ11003_4.pdf
Some references provided in these papers could be interesting too.
Hoping it helped a little,
Raymond
Maybe Michael Rubinstein's L-function software will work.
The default L-function is the Riemann zeta function.
http://pmmac03.math.uwaterloo.ca/~mrubinst/L_function_public/L.html
There's an option for computing derivatives about which the Readme file
says:
"Presently the derivative option uses numeric differentiation, and one
loses about half the working precision for each successive derivative.
Multiprecision is still being implemented, so, for now, the derivative
option
only gives moderately reasonable output for the first derivative (about 6-7
digits),
and less for the second derivative (about 3 digits). Beyond this, one
needs to use the USE_LONG_DOUBLE compile option in the MAkefile or higher
precision."
[/quote]
Thanks to Raymond Manzoni and Marko Amnell for the useful information.
I've been experimenting with Michael Rubinstein's
lcalc command-line program.
For locating probable zeros of zeta', I've found
that knowing zeta'' is useful, once one
is near a probable zero of zeta'.
For the point s = 0.848735 + 60.140846*I,
I get:
bash$ ./lcalc -d 2 -v -x 0.848735 -y 60.140846
1.082 0.1029
which means zeta''(s) ~= 1.082 + 0.1029*I .
By using the approximation zeta''(s) ~~= 1
and Newton's method, it's quite easy to get closer and
closer to a probable zero of zeta' near s.
Below I give the smallest value of | zeta'(.)| I found
near s = 0.848735 + 60.140846*I:
bash$ ./lcalc -d 1 -v -x .84873530 -y 60.140845702
3.830269e-08 4.274359e-08 .
In other words,
zeta'(.8487353 + 60.140845702*I) ~= 0.00000004 + 0.00000004*I .
I haven't done so, but I think one could use the
Argument principle and numerically integrate zeta''/zeta'
in a small (but not too small) square or rectangle
around the probable zero to show that the square or rectangle
really does contain a zero of zeta' . I might try that with
a square of side about 0.001.
A link to a web page about the Argument Principle:
http://mathworld.wolfram.com/ArgumentPrinciple.html
David Bernier |
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| David Bernier... |
| Posted: Mon Nov 02, 2009 1:29 am |
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Guest
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David Bernier wrote:
[...]
[quote]Below I give the smallest value of | zeta'(.)| I found
near s = 0.848735 + 60.140846*I:
bash$ ./lcalc -d 1 -v -x .84873530 -y 60.140845702
3.830269e-08 4.274359e-08 .
In other words,
zeta'(.8487353 + 60.140845702*I) ~= 0.00000004 + 0.00000004*I .
[...][/quote]
Using PARI-gp I get further:
? (zeta(%62) - zeta(%62+delta))/delta
%65 = 1.2715731464878770130 E-11 + 4.321841984664728360 E-11*I
[derivative is very close to zero.]
? delta
%66 = 1.0000000000000000000000000000000000000 E-20
? %62
%67 = 0.84873532809000000000000000000000000000 +
60.140845782000000000000000000000000000*I
"%62" is the value on output-line number 62.
So zeta'(0.84873532809 + 60.140845782) ~= 0.
David Bernier
P.S. 60.140845782 is close to a local maximum of the
Riemann-Siegel Z(.) function, or
RiemannSiegelZ in Mathematica.
Cf.:
http://reference.wolfram.com/legacy/v5/Built-inFunctions/MathematicalFunctions/ZetaRelated/RiemannSiegelZ.html |
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| David Bernier... |
| Posted: Mon Nov 02, 2009 8:46 am |
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Guest
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David Bernier wrote:
[quote]David Bernier wrote:
[...]
Below I give the smallest value of | zeta'(.)| I found
near s = 0.848735 + 60.140846*I:
bash$ ./lcalc -d 1 -v -x .84873530 -y 60.140845702
3.830269e-08 4.274359e-08 .
In other words,
zeta'(.8487353 + 60.140845702*I) ~= 0.00000004 + 0.00000004*I .
[...]
Using PARI-gp I get further:
? (zeta(%62) - zeta(%62+delta))/delta
%65 = 1.2715731464878770130 E-11 + 4.321841984664728360 E-11*I
[derivative is very close to zero.]
? delta
%66 = 1.0000000000000000000000000000000000000 E-20
? %62
%67 = 0.84873532809000000000000000000000000000 +
60.140845782000000000000000000000000000*I
"%62" is the value on output-line number 62.
So zeta'(0.84873532809 + 60.140845782) ~= 0.
David Bernier
P.S. 60.140845782 is close to a local maximum of the
Riemann-Siegel Z(.) function, or
RiemannSiegelZ in Mathematica.
Cf.:
http://reference.wolfram.com/legacy/v5/Built-inFunctions/MathematicalFunctions/ZetaRelated/RiemannSiegelZ.html
[/quote]
Thanks additionally to Axel Vogt for the Maple computations.
The Riemann-Siegel Z function has a local minimum of -0.37 near t = 357.58.
I used Glen Pugh's Z-plotter to get this:
http://web.viu.ca/pughg/RiemannZeta/RiemannZetaLong.html
A local minimum of Z(t) where the absolute value of the minimum attained
is about 0.37 is quite small, for t ~ 357 .
Using lcalc and PARI-gp, I searched for a zero of zeta'
near s = 0.5 + 357.58*I .
After some time, I got:
? (zeta(%158 + 0.5 E-100) - zeta(%158 - 0.5 E-100))/(1.0 E-100)
%160 = -2.0529131546 E-34 -1.2558044177 E-34*I
? %158
%161 = 0.67492445948651172431513544516762825 +
357.57576692022870053439669037496992465*I
So for s = 0.6749244594865117 + 357.5757669202287005*I,
zeta'(s) ~= 0 .
PARI-gp has the advantage of being able to do
complex arithmetic, and also stores the output
of each command-line computation as %n, where
n --> the numeral for line number n, e.g.
%2 for output from line 2.
I searched for more points where Z(t) has a local
extremum whose value (in absolute value) is
quite small. If there are zeros of zeta' close by,
I wasn't able to find them with the limited methods and tools
I have, i.e. using lcalc at one point and PARI-gp.
A local minimum of Z(t) near
t = 376.079 attains about Z(t) = -0.19 .
Another local minimum of Z(t) is
for Z(946.222) ~= -0.081 .
From reading the hypothesis in Theorem 3 of Yitang Zhang's
2001 Duke Math. J. paper, with a link given by
Raymond Manzoni being:
<http://eduunix.ccut.edu.cn/index2/math/202.38.126.65/mathdoc/Duke.Mathematical.Journal/DMJ11003_4.pdf>
it could be that zeros of zeta' with real part close to 0.5
tend to occur near a pair of very close zeros of zeta;
this would correspond to very close zeros of the Z(t)
function. Also, in order to find zeros of zeta'
with with real part as close as possible to 0.5,
it seems reasonable to search for very close zeros
of Z(t), as I believe I read that
Z(t_0) = 0 and Z'(t_0) = 0 implies that zeta has
a double (or higher...) zero at 1/2 + i*t_0 .
David Bernier |
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| David Bernier... |
| Posted: Mon Nov 02, 2009 10:23 am |
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Guest
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David Bernier wrote:
[quote]David Bernier wrote:
David Bernier wrote:
[...]
Below I give the smallest value of | zeta'(.)| I found
near s = 0.848735 + 60.140846*I:
bash$ ./lcalc -d 1 -v -x .84873530 -y 60.140845702
3.830269e-08 4.274359e-08 .
In other words,
zeta'(.8487353 + 60.140845702*I) ~= 0.00000004 + 0.00000004*I .
[...]
Using PARI-gp I get further:
? (zeta(%62) - zeta(%62+delta))/delta
%65 = 1.2715731464878770130 E-11 + 4.321841984664728360 E-11*I
[derivative is very close to zero.]
? delta
%66 = 1.0000000000000000000000000000000000000 E-20
? %62
%67 = 0.84873532809000000000000000000000000000 +
60.140845782000000000000000000000000000*I
"%62" is the value on output-line number 62.
So zeta'(0.84873532809 + 60.140845782) ~= 0.
David Bernier
P.S. 60.140845782 is close to a local maximum of the
Riemann-Siegel Z(.) function, or
RiemannSiegelZ in Mathematica.
Cf.:
http://reference.wolfram.com/legacy/v5/Built-inFunctions/MathematicalFunctions/ZetaRelated/RiemannSiegelZ.html
Thanks additionally to Axel Vogt for the Maple computations.
The Riemann-Siegel Z function has a local minimum of -0.37 near t = 357.58.
I used Glen Pugh's Z-plotter to get this:
http://web.viu.ca/pughg/RiemannZeta/RiemannZetaLong.html
A local minimum of Z(t) where the absolute value of the minimum attained
is about 0.37 is quite small, for t ~ 357 .
Using lcalc and PARI-gp, I searched for a zero of zeta'
near s = 0.5 + 357.58*I .
After some time, I got:
? (zeta(%158 + 0.5 E-100) - zeta(%158 - 0.5 E-100))/(1.0 E-100)
%160 = -2.0529131546 E-34 -1.2558044177 E-34*I
? %158
%161 = 0.67492445948651172431513544516762825 +
357.57576692022870053439669037496992465*I
So for s = 0.6749244594865117 + 357.5757669202287005*I,
zeta'(s) ~= 0 .
PARI-gp has the advantage of being able to do
complex arithmetic, and also stores the output
of each command-line computation as %n, where
n --> the numeral for line number n, e.g.
%2 for output from line 2.
I searched for more points where Z(t) has a local
extremum whose value (in absolute value) is
quite small. If there are zeros of zeta' close by,
I wasn't able to find them with the limited methods and tools
I have, i.e. using lcalc at one point and PARI-gp.
A local minimum of Z(t) near
t = 376.079 attains about Z(t) = -0.19 .
Another local minimum of Z(t) is
for Z(946.222) ~= -0.081 .
From reading the hypothesis in Theorem 3 of Yitang Zhang's
2001 Duke Math. J. paper, with a link given by
Raymond Manzoni being:
http://eduunix.ccut.edu.cn/index2/math/202.38.126.65/mathdoc/Duke.Mathematical.Journal/DMJ11003_4.pdf
it could be that zeros of zeta' with real part close to 0.5
tend to occur near a pair of very close zeros of zeta;
this would correspond to very close zeros of the Z(t)
function. Also, in order to find zeros of zeta'
with with real part as close as possible to 0.5,
it seems reasonable to search for very close zeros
of Z(t), as I believe I read that
Z(t_0) = 0 and Z'(t_0) = 0 implies that zeta has
a double (or higher...) zero at 1/2 + i*t_0 .
[/quote]
This shows how one can accomplish the Newton iterative
method in PARI-gp: (for zeros of zeta')
%260 =
0.6159808652502243493441636295353630657935035232448022488850729948269200791600256469641388865822549070137942004625719591011444230932177007267705602939894378662566389476923435368427843967727976391216951517185305197348740676202437164879272912758855253667222428085147212409839813104516038844796851234321862426867417817534892046203133140476723510770506360409986523342535643727111436606614765236499423448487471861937818613282790249
+
185.2148123380504146026425868511807419367212456480013948970148740216439300819187581741693854389755434384956306427043055798877540530353880256171226710432704095174963732037243371458921372890739297231783479305479533665778338291236880768737749098457172586794949781316306785440490560424572325866953672145777350570622113387706746854837345900402663385674113926542553588966611225125836137545595347755103003717274167383786845386256705*I
? %260 -((zeta(%260+delta/2) - zeta(%260 - delta/2))/delta)/(((zeta(%260
+delta)-zeta(%260 ))/delta - (zeta(%260)-zeta(%260 - delta))/delta)/delta)
%261 =
0.61598086525022434934416362953536306579350352324480224888507299482692007916002564696413888658225490701379420046257195910114442309321767826762906274075509372456689106277299810
+
185.2148123380504146026425868511807419367212456480013948970148740216439300819187581741693854389755434384956306427043055798877540530353880210435451071431276191243312226292493623396089602295715887*I
(zeta(%261+0.0000000000000000000001)-zeta(%261-0.0000000000000000000001))/0.0000000000000000000002
%263 =
-2.0409330487129462454206541839119844680858556152243312780461944519282797989677857007170548820578865077668292759767906
E-44 +
1.1397362696963610746730853441636642275298134490298296710634666816089367900687222909370492667688224267939387852905115
E-44*I
? delta
%264 =
1.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
E-200
The basic iterative step in Newton's method to
locate a zero of zeta' is:
? %260 -((zeta(%260+delta/2) - zeta(%260 - delta/2))/
delta)/(((zeta(%260 +delta)-zeta(%260 ))/delta -
(zeta(%260)-zeta(%260 - delta))/delta)/delta) [ENTER]
-->
%261 [a number]
Above, %260 should be close enough to a zero of zeta' so that
the iterations converge.
Then, replace %260 by %261 in the long line after the question mark
above.
I eventually get "division by zero". PARI-gp "plays" with the
significant figures or something. Appending many 0s
to the real part, then the Imaginary part and combining
as: a + b*I seems to restore more significant digits.
David Bernier |
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| Raymond Manzoni... |
| Posted: Mon Nov 02, 2009 7:37 pm |
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Guest
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David Bernier a écrit :
(snip)
[quote]
Thanks additionally to Axel Vogt for the Maple computations.
The Riemann-Siegel Z function has a local minimum of -0.37 near t =
357.58.
I used Glen Pugh's Z-plotter to get this:
http://web.viu.ca/pughg/RiemannZeta/RiemannZetaLong.html
[/quote]
Another (possibly more confusing) applet is available on Matthew
Watkins' site about zeta :
<http://www.secamlocal.ex.ac.uk/people/staff/mrwatkin//zeta/CSExplorer/CSExplorer.htm>
(select Riemann Siegel at the bottom left, choose the Y offset
and... try to avoid using the scrollbar at the bottom or you'll get out
of the critical line, lose the fast Riemann-Siegel evaluation and have
to be patient...  )
[quote]A local minimum of Z(t) where the absolute value of the minimum attained
is about 0.37 is quite small, for t ~ 357 .
Using lcalc and PARI-gp, I searched for a zero of zeta'
near s = 0.5 + 357.58*I .
After some time, I got:
? (zeta(%158 + 0.5 E-100) - zeta(%158 - 0.5 E-100))/(1.0 E-100)
%160 = -2.0529131546 E-34 -1.2558044177 E-34*I
? %158
%161 = 0.67492445948651172431513544516762825 +
357.57576692022870053439669037496992465*I
So for s = 0.6749244594865117 + 357.5757669202287005*I,
zeta'(s) ~= 0 .
PARI-gp has the advantage of being able to do
complex arithmetic, and also stores the output
of each command-line computation as %n, where
n --> the numeral for line number n, e.g.
%2 for output from line 2.
I searched for more points where Z(t) has a local
extremum whose value (in absolute value) is
quite small.
[/quote]
Some of these points (search 'Lehmer's Phenomenon') are provided in
Edwards' excellent book about zeta :
<http://books.google.com/books?id=5uLAoued_dIC&pg=PA179>
Example : for t ~= 17143.8039 the maximum is around 0.002153
using the secant method (*) I found that
zeta'(0.5006167337067389436048937 + 17143.804216272698515881722i) was
nearly 0
If there are zeros of zeta' close by,
[quote]I wasn't able to find them with the limited methods and tools
I have, i.e. using lcalc at one point and PARI-gp.
A local minimum of Z(t) near
t = 376.079 attains about Z(t) = -0.19 .
Another local minimum of Z(t) is
for Z(946.222) ~= -0.081 .
From reading the hypothesis in Theorem 3 of Yitang Zhang's
2001 Duke Math. J. paper, with a link given by
Raymond Manzoni being:
http://eduunix.ccut.edu.cn/index2/math/202.38.126.65/mathdoc/Duke.Mathematical.Journal/DMJ11003_4.pdf
it could be that zeros of zeta' with real part close to 0.5
tend to occur near a pair of very close zeros of zeta;
this would correspond to very close zeros of the Z(t)
function. Also, in order to find zeros of zeta'
with with real part as close as possible to 0.5,
it seems reasonable to search for very close zeros
of Z(t), as I believe I read that
Z(t_0) = 0 and Z'(t_0) = 0 implies that zeta has
a double (or higher...) zero at 1/2 + i*t_0 .
This shows how one can accomplish the Newton iterative
method in PARI-gp: (for zeros of zeta')
%260 =
0.6159808652502243493441636295353630657935035232448022488850729948269200791600256469641388865822549070137942004625719591011444230932177007267705602939894378662566389476923435368427843967727976391216951517185305197348740676202437164879272912758855253667222428085147212409839813104516038844796851234321862426867417817534892046203133140476723510770506360409986523342535643727111436606614765236499423448487471861937818613282790249
+
185.2148123380504146026425868511807419367212456480013948970148740216439300819187581741693854389755434384956306427043055798877540530353880256171226710432704095174963732037243371458921372890739297231783479305479533665778338291236880768737749098457172586794949781316306785440490560424572325866953672145777350570622113387706746854837345900402663385674113926542553588966611225125836137545595347755103003717274167383786845386256705*I
? %260 -((zeta(%260+delta/2) - zeta(%260 - delta/2))/delta)/(((zeta(%260
+delta)-zeta(%260 ))/delta - (zeta(%260)-zeta(%260 - delta))/delta)/delta)
%261 =
0.61598086525022434934416362953536306579350352324480224888507299482692007916002564696413888658225490701379420046257195910114442309321767826762906274075509372456689106277299810
+
185.2148123380504146026425868511807419367212456480013948970148740216439300819187581741693854389755434384956306427043055798877540530353880210435451071431276191243312226292493623396089602295715887*I
(zeta(%261+0.0000000000000000000001)-zeta(%261-0.0000000000000000000001))/0.0000000000000000000002
%263 =
-2.0409330487129462454206541839119844680858556152243312780461944519282797989677857007170548820578865077668292759767906
E-44 +
1.1397362696963610746730853441636642275298134490298296710634666816089367900687222909370492667688224267939387852905115
E-44*I
? delta
%264 =
1.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
E-200
The basic iterative step in Newton's method to
locate a zero of zeta' is:
? %260 -((zeta(%260+delta/2) - zeta(%260 - delta/2))/
delta)/(((zeta(%260 +delta)-zeta(%260 ))/delta -
(zeta(%260)-zeta(%260 - delta))/delta)/delta) [ENTER]
--
%261 [a number]
Above, %260 should be close enough to a zero of zeta' so that
the iterations converge.
Then, replace %260 by %261 in the long line after the question mark
above.
I eventually get "division by zero". PARI-gp "plays" with the
significant figures or something. Appending many 0s
to the real part, then the Imaginary part and combining
as: a + b*I seems to restore more significant digits.
David Bernier
[/quote]
Yes pari/gp will remove the non-significative digits so that, for
example, if you evaluate zetap(z)= (zeta(z+eps/2)-zeta(z-eps/2))/eps
at a point z such that zeta'(z)=0 you'll lose nearly -log_10(eps)
digits (every time!).
A useful trick is to 'force' the default precision by replacing
zetap(z) with zetap(precision(z,default(realprecision)))
(by the way in a script default(realprecision, n) allows too to
change the default precision to n)
You'll still get "division by zero" at the end but probably because
you were subtracting two equal values at the denominator!
Pleasant Explorations!
Raymond
(*) Script I used in pari/gp <http://pari.math.u-bordeaux.fr/download.html>
eps=1e-40;
\p 200
zetap(x)=(zeta(x+eps/2)-zeta(x-eps/2))/eps;
zetas(x)=(zeta(x+eps/2)-2*zeta(x)+zeta(x-eps/2))/eps^2;
fn(x)=x-zetap(x)/zetas(x)
fs(x)=r=x-zetap(x)*(x-xp)/(zetap(x)-zetap(xp));xp=x;r
xp=0.5+17143.8*I
fn(%)
fs(precision(%,200))
(iterating the last line until 'division by zero' )
(iterating fn alone (Newton-Raphson) didn't converge most of the time...) |
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| David Bernier... |
| Posted: Mon Nov 02, 2009 8:58 pm |
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Guest
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Raymond Manzoni wrote:
[quote]David Bernier a écrit :
(snip)
Thanks additionally to Axel Vogt for the Maple computations.
The Riemann-Siegel Z function has a local minimum of -0.37 near t =
357.58.
I used Glen Pugh's Z-plotter to get this:
http://web.viu.ca/pughg/RiemannZeta/RiemannZetaLong.html
Another (possibly more confusing) applet is available on Matthew
Watkins' site about zeta :
http://www.secamlocal.ex.ac.uk/people/staff/mrwatkin//zeta/CSExplorer/CSExplorer.htm
(select Riemann Siegel at the bottom left, choose the Y offset and...
try to avoid using the scrollbar at the bottom or you'll get out of the
critical line, lose the fast Riemann-Siegel evaluation and have to be
patient... )
A local minimum of Z(t) where the absolute value of the minimum attained
is about 0.37 is quite small, for t ~ 357 .
Using lcalc and PARI-gp, I searched for a zero of zeta'
near s = 0.5 + 357.58*I .
After some time, I got:
? (zeta(%158 + 0.5 E-100) - zeta(%158 - 0.5 E-100))/(1.0 E-100)
%160 = -2.0529131546 E-34 -1.2558044177 E-34*I
? %158
%161 = 0.67492445948651172431513544516762825 +
357.57576692022870053439669037496992465*I
So for s = 0.6749244594865117 + 357.5757669202287005*I,
zeta'(s) ~= 0 .
PARI-gp has the advantage of being able to do
complex arithmetic, and also stores the output
of each command-line computation as %n, where
n --> the numeral for line number n, e.g.
%2 for output from line 2.
I searched for more points where Z(t) has a local
extremum whose value (in absolute value) is
quite small.
Some of these points (search 'Lehmer's Phenomenon') are provided in
Edwards' excellent book about zeta :
http://books.google.com/books?id=5uLAoued_dIC&pg=PA179
Example : for t ~= 17143.8039 the maximum is around 0.002153
using the secant method (*) I found that
zeta'(0.5006167337067389436048937 + 17143.804216272698515881722i) was
nearly 0
[/quote]
For the probable zero of zeta' you found, I get the approximation:
s~= 0.500616733706738943604893700414+17143.8042162726985158817223566*I
Then, 1/(Re(s) - 1/2) ~= 1621.445 .
I wonder if there are conjectures or guesses as to the true
asymptotics of
(Re(s) - 1/2) in terms of Im(s), for zeros s = beta' + gamma'
of zeta', where gamma' > 0 and letting gamma' become
arbitrarily large ...
Thanks for the info. on the behaviour of PARI-gp with
respect to significant digits in computations , left
unsnipped below.
David Bernier
[...]
[quote]Yes pari/gp will remove the non-significative digits so that, for
example, if you evaluate zetap(z)= (zeta(z+eps/2)-zeta(z-eps/2))/eps
at a point z such that zeta'(z)=0 you'll lose nearly -log_10(eps)
digits (every time!).
A useful trick is to 'force' the default precision by replacing
zetap(z) with zetap(precision(z,default(realprecision)))
(by the way in a script default(realprecision, n) allows too to
change the default precision to n)
You'll still get "division by zero" at the end but probably because
you were subtracting two equal values at the denominator!
Pleasant Explorations!
Raymond
(*) Script I used in pari/gp <http://pari.math.u-bordeaux.fr/download.html
eps=1e-40;
\p 200
zetap(x)=(zeta(x+eps/2)-zeta(x-eps/2))/eps;
zetas(x)=(zeta(x+eps/2)-2*zeta(x)+zeta(x-eps/2))/eps^2;
fn(x)=x-zetap(x)/zetas(x)
fs(x)=r=x-zetap(x)*(x-xp)/(zetap(x)-zetap(xp));xp=x;r
xp=0.5+17143.8*I
fn(%)
fs(precision(%,200))
(iterating the last line until 'division by zero' )
(iterating fn alone (Newton-Raphson) didn't converge most of the time...)[/quote] |
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| David Bernier... |
| Posted: Tue Nov 03, 2009 3:45 am |
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Guest
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David Bernier wrote:
[quote]Raymond Manzoni wrote:
David Bernier a écrit :
(snip)
Thanks additionally to Axel Vogt for the Maple computations.
The Riemann-Siegel Z function has a local minimum of -0.37 near t =
357.58.
I used Glen Pugh's Z-plotter to get this:
http://web.viu.ca/pughg/RiemannZeta/RiemannZetaLong.html
Another (possibly more confusing) applet is available on Matthew
Watkins' site about zeta :
http://www.secamlocal.ex.ac.uk/people/staff/mrwatkin//zeta/CSExplorer/CSExplorer.htm
(select Riemann Siegel at the bottom left, choose the Y offset
and... try to avoid using the scrollbar at the bottom or you'll get
out of the critical line, lose the fast Riemann-Siegel evaluation and
have to be patient... )
A local minimum of Z(t) where the absolute value of the minimum
attained
is about 0.37 is quite small, for t ~ 357 .
Using lcalc and PARI-gp, I searched for a zero of zeta'
near s = 0.5 + 357.58*I .
After some time, I got:
? (zeta(%158 + 0.5 E-100) - zeta(%158 - 0.5 E-100))/(1.0 E-100)
%160 = -2.0529131546 E-34 -1.2558044177 E-34*I
? %158
%161 = 0.67492445948651172431513544516762825 +
357.57576692022870053439669037496992465*I
So for s = 0.6749244594865117 + 357.5757669202287005*I,
zeta'(s) ~= 0 .
PARI-gp has the advantage of being able to do
complex arithmetic, and also stores the output
of each command-line computation as %n, where
n --> the numeral for line number n, e.g.
%2 for output from line 2.
I searched for more points where Z(t) has a local
extremum whose value (in absolute value) is
quite small.
Some of these points (search 'Lehmer's Phenomenon') are provided in
Edwards' excellent book about zeta :
http://books.google.com/books?id=5uLAoued_dIC&pg=PA179
[/quote]
Richard Brent mentioned a "Lehmer pair" in his 1979 article about
verifying RH for the first 75,000,000 non-trivial zeros.
As I understand it, with n = 41,820,581 the pair of zeros is the n'th
and the (n+1)st, where he found that
max_{t from Im(rho_n) to Im(rho_{n+1})} |Z(t)| < 0.00000248 .
[ rho_n is the n'th non-trivial zero].
I believe this is for t ~= 18882503.9 ,
and using a C program with Euler-MacLaurin summation,
I find that Z attains between Im(rho_n) and Im(rho_{n+1})
about as follows:
Z(18882503.90157) ~= 0.000002476
which is in line with Brent's results.
[quote]Example : for t ~= 17143.8039 the maximum is around 0.002153
using the secant method (*) I found that
zeta'(0.5006167337067389436048937 + 17143.804216272698515881722i) was
nearly 0
For the probable zero of zeta' you found, I get the approximation:
s~= 0.500616733706738943604893700414+17143.8042162726985158817223566*I
Then, 1/(Re(s) - 1/2) ~= 1621.445 .
I wonder if there are conjectures or guesses as to the true
asymptotics of
(Re(s) - 1/2) in terms of Im(s), for zeros s = beta' + gamma'
of zeta', where gamma' > 0 and letting gamma' become
arbitrarily large ...
Thanks for the info. on the behaviour of PARI-gp with
respect to significant digits in computations , left
unsnipped below.
David Bernier
[...]
Yes pari/gp will remove the non-significative digits so that, for
example, if you evaluate zetap(z)= (zeta(z+eps/2)-zeta(z-eps/2))/eps
at a point z such that zeta'(z)=0 you'll lose nearly -log_10(eps)
digits (every time!).
A useful trick is to 'force' the default precision by replacing
zetap(z) with zetap(precision(z,default(realprecision)))
(by the way in a script default(realprecision, n) allows too to
change the default precision to n)
You'll still get "division by zero" at the end but probably because
you were subtracting two equal values at the denominator!
Pleasant Explorations!
Raymond
(*) Script I used in pari/gp
http://pari.math.u-bordeaux.fr/download.html
eps=1e-40;
\p 200
zetap(x)=(zeta(x+eps/2)-zeta(x-eps/2))/eps;
zetas(x)=(zeta(x+eps/2)-2*zeta(x)+zeta(x-eps/2))/eps^2;
fn(x)=x-zetap(x)/zetas(x)
fs(x)=r=x-zetap(x)*(x-xp)/(zetap(x)-zetap(xp));xp=x;r
xp=0.5+17143.8*I
fn(%)
fs(precision(%,200))
(iterating the last line until 'division by zero' )
(iterating fn alone (Newton-Raphson) didn't converge most of the time...)[/quote] |
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| David Bernier... |
| Posted: Wed Nov 04, 2009 2:26 am |
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Guest
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David Bernier wrote:
[...]
[quote]Richard Brent mentioned a "Lehmer pair" in his 1979 article about
verifying RH for the first 75,000,000 non-trivial zeros.
As I understand it, with n = 41,820,581 the pair of zeros is the n'th
and the (n+1)st, where he found that
max_{t from Im(rho_n) to Im(rho_{n+1})} |Z(t)| < 0.00000248 .
[ rho_n is the n'th non-trivial zero].
I believe this is for t ~= 18882503.9 ,
and using a C program with Euler-MacLaurin summation,
I find that Z attains between Im(rho_n) and Im(rho_{n+1})
about as follows:
Z(18882503.90157) ~= 0.000002476
which is in line with Brent's results.
[...][/quote]
I have 'lcalc' pre-compiled, as distributed by M. Rubinstein. This uses
64-bit doubles, I believe. As for PARI-gp, evaluating
zeta(sigma +i*t) for t ~= 18882503.9 takes a lot of time
(maybe an hour or so). One result returned by PARI-gp
is this:
? zeta(0.5000227046 + 18882503.90177114*I)
%1 = 0.00000102513240401800620447999232271568910311260563490669721490611
- 0.000000247134107511231289945163352238926148478404827421149565046384*I
Euler-Maclaurin summation for zeta is described in Section 6.4
of Edwards' book, and programming it for
evaluating zeta(sigma +i*t) is just a bit more work than
programming it for evaluating zeta(1/2 +i*t).
Some compiled languages have a 'long double' floating point
type, with more significant bits than a 'double' type.
Perhaps | zeta'| is small at the point 1/2 +i*t where t
satisfies Z'(t) = 0, and t is in between two zeros of Z(.)
corresponding to a Lehmer pair:
Cf.:
http://en.wikipedia.org/wiki/Z_function
zeta(1/2 + i*t) = exp(-i theta(t)) Z(t),
then zeta'(1/2 + i*t) = ....
[ the problem lies in trying to take or taking complex derivatives
of the continuations of exp(-i theta(t)) and Z(t),
if they exist in some neighborhood.]
David Bernier |
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| David Bernier... |
| Posted: Wed Nov 04, 2009 5:31 am |
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Guest
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David Bernier wrote:
[...]
[quote]I have 'lcalc' pre-compiled, as distributed by M. Rubinstein. This uses
[/quote]
I was wrong about compilation. After unzipping and re-creating the
archived files and directories, the source code is compiled after
one gives the 'make' command.
David Bernier |
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