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| David Bernier... |
 Posted: Thu Nov 05, 2009 5:22 am |
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David Bernier wrote:
[quote]David Bernier wrote:
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
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]
In the Newton-Raphson method, the value (A) (or %2) below
was my starting point as an approximation to
a zero of zeta' . Using finite difference quotients,
I got the approximation to the second derivative of
zeta, at A, which appears in (C) below.
I also had an approximation to zeta' at A [not shown].
So applying finite differences Newton's method to
the approximate probable zero in (A) [ for the
function zeta' ] , one step led to (B) below,
which differs from (A) in absolute value by
about 6 E-12 . The approximation (B) to
a probable zero of zeta' should be better
than approximation (A), if all went well.
I used PARI-gp, and each zeta evaluation seemed to
take an hour or two, if not more.
? %2
0.500000252347038803 + 18882503.901568975845293608*I (A)
? %20
0.50000025235197704296 + 18882503.90156897584828762548*I (B)
zeta''(%2) ~= 62.035774 - 38.815900*I (C)
David Bernier |
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| Raymond Manzoni... |
| Posted: Sun Nov 08, 2009 4:15 pm |
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David Bernier a écrit :
(snip)
[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]
Perhaps that some recent papers of Nathan Ng could interest you :
<http://arxiv.org/find/math/1/au:+Ng_N/0/1/0/all/0/1>
especially his discussion in "Extreme values of zeta prime rho"
See too Garaev's "On small distances between ordinates of zeros of
zeta(s) and zeta'(s)"
<http://www.math.boun.edu.tr/instructors/yildirim/paper/OnSmallDistancesBtwOrdinates.pdf>
Concerning fast evaluation of Riemann zeta you may look at Hiary's
"Fast methods to compute the Riemann zeta function" and references
provided there : <http://fr.arxiv.org/abs/0711.5005v1>
Euler MacLaurin is easy to implement but requires evaluation of about
t terms (t= Im(s)) of the partial zeta sum. I think it is often used to
find the small zeros with high precision (and probably in pari/gp...).
Riemann-Siegel is not very accurate for small values of t but pretty
good for large values since needing only around sqrt(t/(2 pi)) terms of
the zeta sum.
Both methods are described in the Edwards book even if Riemann-Siegel
is harder to implement (the error term is the hard part!) and restricted
there to the case Re(s)= 1/2. The two applets referenced use R-S on the
critical line (I think it could be implemented for Re(s)<>1).
For an implementation see Ken Takusagawa's "Tabulating values of the
Riemann-Siegel Z function along the critical line" :
<http://web.mit.edu/kenta/www/six/parallel/2-Final-Report.html>
see too Tuck's "Riemann-Siegel sums via stationary phase"
<https://www.austms.org.au/Publ/Bulletin/V72P2/pdf/722-5212-Tuck-v.pdf>
Regards,
Raymond |
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| David Bernier... |
| Posted: Mon Nov 09, 2009 5:47 am |
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Guest
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Raymond Manzoni wrote:
[quote]David Bernier a écrit :
(snip)
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
Perhaps that some recent papers of Nathan Ng could interest you :
http://arxiv.org/find/math/1/au:+Ng_N/0/1/0/all/0/1
especially his discussion in "Extreme values of zeta prime rho"
See too Garaev's "On small distances between ordinates of zeros of
zeta(s) and zeta'(s)"
http://www.math.boun.edu.tr/instructors/yildirim/paper/OnSmallDistancesBtwOrdinates.pdf
Concerning fast evaluation of Riemann zeta you may look at Hiary's
"Fast methods to compute the Riemann zeta function" and references
provided there : <http://fr.arxiv.org/abs/0711.5005v1
Euler MacLaurin is easy to implement but requires evaluation of about
t terms (t= Im(s)) of the partial zeta sum. I think it is often used to
find the small zeros with high precision (and probably in pari/gp...).
Riemann-Siegel is not very accurate for small values of t but pretty
good for large values since needing only around sqrt(t/(2 pi)) terms of
the zeta sum.
[/quote]
For now, I'm interested mostly in numerical approximation of zeros
beta' + i*gamma' of the derivative of Riemann zeta, with
beta' very close to 1/2. Empirically, good places to start
seem to include the vicinity of Lehmer pairs.
For the Lehmer pair with imaginary part ~= 17143.8, PARI-gp
took perhaps 2 or 3 hours per 38-digit zeta computation at
a height t ~= 17143.8.
Moving on to the Lehmer pair with imaginary parts
about 388,588,886 mentioned in Odlyzko et al, the
zeta computations become more time-consuming if
one wants 12+ digits accuracy.
"A New Lehmer pair of zeros and a new lower bound for the de
Bruijn-Newman constant LAMBDA" [1993]
authors: G. Csordas, A. M. Odlyzko, W. Smith, and R. S. Varga.
That was based on "by-products" of RH verification for hundreds
of millions of non-trivial zeros in the 1980's by the Dutch,
e.g.
LRW86:
http://en.wikipedia.org/wiki/Riemann_hypothesis#Numerical_calculations
[ Australian R. Brent is linked to J. van de Lune, H. J. J. te Riele, D.
T. Winter via RH verification before 1986].
Cf.:
< http://www.dtc.umn.edu/~odlyzko/doc/cnt.html > , 10th paper or so ...
The Lehmer pair appears there as
t = 3.888 588 860 022 851 203 e+08,
t = 3.888 588 860 023 936 899 e+08
equivalent to
t = 388,858,886.0022851203 and
t = 388,858,886.0023936899 .
PARI-gp's built-in zeta(.) can probably do the zeta evaluations, but
the time it took for a Lehmer pair with t ~= 18,000,000 (about 3 hours)
doesn't bode well for t ~= 388,858,886.002.
It seems to me that for numerical computation of zeros of zeta',
a lot of accuracy in the zeta function values is desirable, since
zeta varies slowly near a zero of zeta' such as the one
with imaginary part about 18,000,000 . I think this argues for
Euler-Maclaurin summation. Even "long doubles" seem to give
only about 11 or 12 decimal digits (after the decimal point)
for the two zeta zeros at height ~= 388,858,886.02 .
One workable option is to sum 1 billion or so terms in
PARI-gp of both cos(t log(n))/sqrt(n) [n = 1 ... 10^9]
and sin(t log(n))/sqrt(n) [n = 1 ... 10^9]
and add a few terms in the Euler-Maclaurin expansion,
in the PARI-gp environment.
I've done the 1 billion cosine partial sum, and it took a few hours.
Another possibility for 20 decimal+ zeta evaluations is through
the use of Victor Shoup's NTL library:
Cf.:
< http://www.shoup.net/ntl/ >
So far, I've been able to gunzip the *.gz file, extract the *.tar
archive, run ./configure [ default] , 'make', and 'make check'
[ Tests Ok ]. Then, 'make install' as root: # make install .
Next, I'd want to write a program that uses NTL to do simple
transcendental function computations using "quad_floats",
which offer 106-bit precision.
[quote]Both methods are described in the Edwards book even if Riemann-Siegel
is harder to implement (the error term is the hard part!) and restricted
there to the case Re(s)= 1/2. The two applets referenced use R-S on the
critical line (I think it could be implemented for Re(s)<>1).
[/quote]
Since for the time being I just want to compute zeta' zeros
which are or could be near a few selected Lehmer pairs of zeros,
I find Euler-Maclaurin summation more appealing than the
Riemann-Siegel formula.
----
In the article
http://sci.tech-archive.net/Archive/sci.math/2009-10/msg02124.html ,
I was wondering if Z(t) having no negative local maximum when t>100
& Z(t) having no positive local minimum when t>100
was enough to imply the Riemann Hypothesis.
In the problem statement of RH at ClayMath, Bombieri mentions
an RH equivalent:
"The Riemann hypothesis is equivalent to the statement that
all local maxima of xi(t) are positive and
all local minima are negative, [...] "
Cf.:
< http://www.claymath.org/millennium/Riemann_Hypothesis/ >
---> Official Problem Description, page 6 of 11.
Starting from:
"all local maxima of xi(t) are positive and
all local minima are negative"
I've been thinking about relations with:
"Z(t) has no negative local maximum when t>100
& Z(t) has no positive local minimum when t>100"
I know well that the average of |Z(t)| grows slowly
as t>0 increases. I don't know xi(t) so well,
however if the average value of |xi(t)| near t
changes fast as the point on the critical line
corresponding to t moves up the line Im(s) = 1/2,
perhaps one can't rule out
"all local maxima of xi(t) are positive and
all local minima are negative" Failing,
while "Z(t) has no negative local maximum when t>100
& Z(t) has no positive local minimum when t>100" might still
Hold ...
I'd be rather interested in knowing if:
Z(t) has no negative local maximum when t>100
& Z(t) has no positive local minimum when t>100" implies RH ...
Regards,
David Bernier
[quote]For an implementation see Ken Takusagawa's "Tabulating values of the
Riemann-Siegel Z function along the critical line" :
http://web.mit.edu/kenta/www/six/parallel/2-Final-Report.html
see too Tuck's "Riemann-Siegel sums via stationary phase"
https://www.austms.org.au/Publ/Bulletin/V72P2/pdf/722-5212-Tuck-v.pdf
Regards,
Raymond
[/quote] |
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| David Bernier... |
| Posted: Fri Nov 13, 2009 9:17 pm |
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Guest
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David Bernier wrote:
[quote]Raymond Manzoni wrote:
David Bernier a écrit :
(snip)
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
Perhaps that some recent papers of Nathan Ng could interest you :
http://arxiv.org/find/math/1/au:+Ng_N/0/1/0/all/0/1
especially his discussion in "Extreme values of zeta prime rho"
See too Garaev's "On small distances between ordinates of zeros of
zeta(s) and zeta'(s)"
http://www.math.boun.edu.tr/instructors/yildirim/paper/OnSmallDistancesBtwOrdinates.pdf
Concerning fast evaluation of Riemann zeta you may look at Hiary's
"Fast methods to compute the Riemann zeta function" and references
provided there : <http://fr.arxiv.org/abs/0711.5005v1
Euler MacLaurin is easy to implement but requires evaluation of
about t terms (t= Im(s)) of the partial zeta sum. I think it is often
used to find the small zeros with high precision (and probably in
pari/gp...).
Riemann-Siegel is not very accurate for small values of t but pretty
good for large values since needing only around sqrt(t/(2 pi)) terms
of the zeta sum.
For now, I'm interested mostly in numerical approximation of zeros
beta' + i*gamma' of the derivative of Riemann zeta, with
beta' very close to 1/2. Empirically, good places to start
seem to include the vicinity of Lehmer pairs.
For the Lehmer pair with imaginary part ~= 17143.8, PARI-gp
took perhaps 2 or 3 hours per 38-digit zeta computation at
a height t ~= 17143.8.
Moving on to the Lehmer pair with imaginary parts
about 388,588,886 mentioned in Odlyzko et al, the
zeta computations become more time-consuming if
one wants 12+ digits accuracy.
"A New Lehmer pair of zeros and a new lower bound for the de
Bruijn-Newman constant LAMBDA" [1993]
authors: G. Csordas, A. M. Odlyzko, W. Smith, and R. S. Varga.
That was based on "by-products" of RH verification for hundreds
of millions of non-trivial zeros in the 1980's by the Dutch,
e.g.
LRW86:
http://en.wikipedia.org/wiki/Riemann_hypothesis#Numerical_calculations
[ Australian R. Brent is linked to J. van de Lune, H. J. J. te Riele, D.
T. Winter via RH verification before 1986].
Cf.:
http://www.dtc.umn.edu/~odlyzko/doc/cnt.html > , 10th paper or so ...
The Lehmer pair appears there as
t = 3.888 588 860 022 851 203 e+08,
t = 3.888 588 860 023 936 899 e+08
equivalent to
t = 388,858,886.0022851203 and
t = 388,858,886.0023936899 .
PARI-gp's built-in zeta(.) can probably do the zeta evaluations, but
the time it took for a Lehmer pair with t ~= 18,000,000 (about 3 hours)
doesn't bode well for t ~= 388,858,886.002.
It seems to me that for numerical computation of zeros of zeta',
a lot of accuracy in the zeta function values is desirable, since
zeta varies slowly near a zero of zeta' such as the one
with imaginary part about 18,000,000 . I think this argues for
Euler-Maclaurin summation. Even "long doubles" seem to give
only about 11 or 12 decimal digits (after the decimal point)
for the two zeta zeros at height ~= 388,858,886.02 .
One workable option is to sum 1 billion or so terms in
PARI-gp of both cos(t log(n))/sqrt(n) [n = 1 ... 10^9]
and sin(t log(n))/sqrt(n) [n = 1 ... 10^9]
and add a few terms in the Euler-Maclaurin expansion,
in the PARI-gp environment.
I've done the 1 billion cosine partial sum, and it took a few hours.
Another possibility for 20 decimal+ zeta evaluations is through
the use of Victor Shoup's NTL library:
Cf.:
http://www.shoup.net/ntl/
So far, I've been able to gunzip the *.gz file, extract the *.tar
archive, run ./configure [ default] , 'make', and 'make check'
[ Tests Ok ]. Then, 'make install' as root: # make install .
Next, I'd want to write a program that uses NTL to do simple
transcendental function computations using "quad_floats",
which offer 106-bit precision.
[...][/quote]
I don't know much about C++, and I didn't find sample code
with commented examples that really helped me.
David Bailey and others have developed MPFUN90 and other
Fortran or C++ libraries for high-precision arithmetic,
and I'll probably experiment with one of the libraries
some time.
Link to David Bailey web page:
< http://crd.lbl.gov/~dhbailey/mpdist/ >
David Bernier |
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| Dave Seaman... |
| Posted: Sat Nov 14, 2009 4:19 am |
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Guest
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On Fri, 13 Nov 2009 21:17:34 -0500, David Bernier wrote:
[quote]I don't know much about C++, and I didn't find sample code
with commented examples that really helped me.
David Bailey and others have developed MPFUN90 and other
Fortran or C++ libraries for high-precision arithmetic,
and I'll probably experiment with one of the libraries
some time.
Link to David Bailey web page:
http://crd.lbl.gov/~dhbailey/mpdist/
David Bernier
[/quote]
GMP is an extended-precision library that can be used with C or C++.
It claims to be faster than any other bignum library.
<http://gmplib.org/>
--
Dave Seaman
Third Circuit ignores precedent in Mumia Abu-Jamal ruling.
<http://www.indybay.org/newsitems/2008/03/29/18489281.php> |
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| David Bernier... |
| Posted: Sun Nov 15, 2009 12:45 am |
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Guest
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Dave Seaman wrote:
[quote]On Fri, 13 Nov 2009 21:17:34 -0500, David Bernier wrote:
I don't know much about C++, and I didn't find sample code
with commented examples that really helped me.
David Bailey and others have developed MPFUN90 and other
Fortran or C++ libraries for high-precision arithmetic,
and I'll probably experiment with one of the libraries
some time.
Link to David Bailey web page:
http://crd.lbl.gov/~dhbailey/mpdist/
David Bernier
GMP is an extended-precision library that can be used with C or C++.
It claims to be faster than any other bignum library.
http://gmplib.org/
[/quote]
Thanks. I downloaded mpfun90.tar.gz from David Bailey's web site.
Unzipping and extracting the archive went without a problem.
As building, I changed the name of the Fortran compiler to
"gfortran", I think, and removed any calls to timing functions.
Since I also tried the ARPREC package, it could be something
like removing calls to "etime" for mpfun90, and calls to
"second()" for the ARPREC package.
The problem I had with ARPREC was getting the "includes" right for my
own programs.
For mpfun90, I was able to run test programs that get built when
one does "make" [ this uses what's in the furnished Makefile ].
One of these is the executable quadts, which does
15 numerical integration problems. I'm quite impressed:
about 400D precision except Problem 15 (something not
quite right there) in a few minutes.
I uploaded the output here:
< http://berniermath.net/mp90quadrature.txt >
The CPU times mean nothing, as I disabled timing.
So I'm thinking about getting the right compiler options
for mpfun90, for my own source code. AFAIK, the executable
_quadts_ was built using directives, such as those in the
Makefile.
Perhaps there's a "verbose" option with GNU make, so that
I could see what the compiler options were when building
_quadts_ or other included test programs ...
David Bernier |
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| David Bernier... |
| Posted: Mon Nov 23, 2009 8:47 am |
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Guest
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Dave Seaman wrote:
[quote]On Fri, 13 Nov 2009 21:17:34 -0500, David Bernier wrote:
I don't know much about C++, and I didn't find sample code
with commented examples that really helped me.
David Bailey and others have developed MPFUN90 and other
Fortran or C++ libraries for high-precision arithmetic,
and I'll probably experiment with one of the libraries
some time.
Link to David Bailey web page:
http://crd.lbl.gov/~dhbailey/mpdist/
David Bernier
GMP is an extended-precision library that can be used with C or C++.
It claims to be faster than any other bignum library.
http://gmplib.org/
[/quote]
I installed GMP and it was quite easy. It is easy to do
multi-precision basic arithmetic; certainly +, - , * and '/' .
Maybe also sqrt...
But I didn't find a multi-precision log function, sine or cosine
functions.
There are C++ libraries that can work in standalone or in
coordination with GMP and have transcendental multi-precision
functions.
One I tried is CLN (Class Library for Numbers):
< http://www.ginac.de/CLN/cln.html#SEC_Top >
One problem for me is my unfamiliarity with C++ : classes,
input/output , etc. Also, the "make and Makefile" do
their best to build everything right and put lib...so.6 in
the right place. But my g++ compiler only found
lib...so.6 after I put it in /usr/lib64 .
[my computer has an x86_64 architecture ].
For now, I'm going back to PARI-gp and the Euler-Maclaurin
formula.
To check on accuracy of C arithmetic with "long floats",
I did the sum (for t = 388858886.0023394051)
sum_{n = 1 ... 10^9} cos(t*log(n))/sqrt(n) (***)
both in C with long floats and with pari-gp:
bash$ ./a.out
Main term:
t= 388858886.002339405094971880 sigma = 0.5000 :
real_part = 0.000081980021093364, ( C result for (***))
term1_real = 0.000080353148421926 // other term in E.M.
term2_real = -0.000002433374635325 // other term in E.M.
term3_real = 0.000001012524886827 // other term in E.M.
term4_real = 0.000000002551752271 // other term in E.M.
PARI-gp gives, for the main term, (***)
0.0000819799059123200468 [ result A ]
So the C program is off by 1.15 E-10 .
That's too large to locate zeros of zeta' within
about 1.0 E-8 (or better), I think.
As for timing, PARI-gp returned "result A" in
one or two hours, which seems Ok if I just want
to locate the probable zero of zeta' near
0.5 + i*388858886.0023394051 .
David Bernier |
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| David Bernier... |
| Posted: Mon Nov 23, 2009 7:52 pm |
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Guest
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David Bernier wrote:
[...]
[quote]For now, I'm going back to PARI-gp and the Euler-Maclaurin
formula.
To check on accuracy of C arithmetic with "long floats",
I did the sum (for t = 388858886.0023394051)
sum_{n = 1 ... 10^9} cos(t*log(n))/sqrt(n) (***)
both in C with long floats and with pari-gp:
bash$ ./a.out
Main term:
t= 388858886.002339405094971880 sigma = 0.5000 :
real_part = 0.000081980021093364, ( C result for (***))
term1_real = 0.000080353148421926 // other term in E.M.
term2_real = -0.000002433374635325 // other term in E.M.
term3_real = 0.000001012524886827 // other term in E.M.
term4_real = 0.000000002551752271 // other term in E.M.
PARI-gp gives, for the main term, (***)
0.0000819799059123200468 [ result A ]
So the C program is off by 1.15 E-10 .
That's too large to locate zeros of zeta' within
about 1.0 E-8 (or better), I think.
As for timing, PARI-gp returned "result A" in
one or two hours, which seems Ok if I just want
to locate the probable zero of zeta' near
0.5 + i*388858886.0023394051 .
[/quote]
Using formula (1) in section 6.4 of Edwards
(2nd printing, 1974; unabridged republication by Dover, 2001),
with N = 10^9 + 1, and up to the B_14 term where
B_14 = 7/6 (Bernoulli numbers), I get
zeta(0.5 + 388858886.0023394051*i) ~=
0.00000507521799042249 + 0.00002463889615635353*i.
I got the "Lehmer pair" of zeros from:
Csordas, Odlyzko, Smith and Varga;
"A new Lehmer pair of zeros and a new lower
bound for the de Bruijn-Newman constant Lambda",
Electronic Transactions on Numerical Analysis, vol. 1,
pp. 104-111, Dec. 1993.
By averaging the two zeros in the Lehmer pair, we
get s = 0.5 + 388858886.0023394051 as above.
David Bernier |
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| David Bernier... |
| Posted: Tue Nov 24, 2009 11:16 am |
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Guest
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David Bernier wrote:
[quote]David Bernier wrote:
[...]
For now, I'm going back to PARI-gp and the Euler-Maclaurin
formula.
To check on accuracy of C arithmetic with "long floats",
I did the sum (for t = 388858886.0023394051)
sum_{n = 1 ... 10^9} cos(t*log(n))/sqrt(n) (***)
both in C with long floats and with pari-gp:
bash$ ./a.out
Main term:
t= 388858886.002339405094971880 sigma = 0.5000 :
real_part = 0.000081980021093364, ( C result for (***))
term1_real = 0.000080353148421926 // other term in E.M.
term2_real = -0.000002433374635325 // other term in E.M.
term3_real = 0.000001012524886827 // other term in E.M.
term4_real = 0.000000002551752271 // other term in E.M.
PARI-gp gives, for the main term, (***)
0.0000819799059123200468 [ result A ]
So the C program is off by 1.15 E-10 .
That's too large to locate zeros of zeta' within
about 1.0 E-8 (or better), I think.
As for timing, PARI-gp returned "result A" in
one or two hours, which seems Ok if I just want
to locate the probable zero of zeta' near
0.5 + i*388858886.0023394051 .
Using formula (1) in section 6.4 of Edwards
(2nd printing, 1974; unabridged republication by Dover, 2001),
with N = 10^9 + 1, and up to the B_14 term where
B_14 = 7/6 (Bernoulli numbers), I get
zeta(0.5 + 388858886.0023394051*i) ~=
0.00000507521799042249 + 0.00002463889615635353*i.
[/quote]
There were some mistakes ...
I've tried to reconcile Euler-Mac. for N = 10^9 + 1 with the value for
N = 2*10^9 + 1.
Now I get zeta(0.5 + 388858886.0023394051*i) ~=
//for N = 10^9 + 1
0.00000020846871558245436915586964904969378368 +
0.000000075820596896391683884239455999894392459*I
// for N = 2*10^9 + 1
0.00000020846871558245437662886707468701281518 +
0.000000075820596896391682720123370911320226157*I
[quote]I got the "Lehmer pair" of zeros from:
Csordas, Odlyzko, Smith and Varga;
"A new Lehmer pair of zeros and a new lower
bound for the de Bruijn-Newman constant Lambda",
Electronic Transactions on Numerical Analysis, vol. 1,
pp. 104-111, Dec. 1993.
By averaging the two zeros in the Lehmer pair, we
get s = 0.5 + 388858886.0023394051 as above.
David Bernier
[/quote] |
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