[math-fun] please let this *not* be pi
what is the limit of the sum of derivatives of the zeta function taken at all its zero’s? Shouldn’t it be a real number because of symmetry? for the first 1000 zero’s I get 2.86778 -0.00254555 i for the next 1001-2000 it’s 3.43164 +0.00472815 i for 2001-3000 3.64089 -0.00544207 i for 3001-4000 3.80755 -0.00334517 i and so the running average up to n goes like n=1000 : 2.86778 -0.00254555 i 2000 : 3.14971 +0.0010913 i 3000 : 3.31343 -0.00108649 i 4000 : 3.43696 -0.00165116 i no sweet convergence sofar just curious, Wouter
I don't see why it has to converge. The zeta function slowly gets "wrinklier" as you go up the critical line. and (as you have observed) the derivatives at the zeroes tend to cluster around the real line, so I am guessing that the derivatives gradually crawl rightward. If you did a million zeroes, you might see values around 4. (I think the tendency is logarithmic, that is, you have to double the distance out the critical line to see an increment in average derivative.) On Fri, Sep 14, 2018 at 11:26 AM Wouter Meeussen <wouter.meeussen@telenet.be> wrote:
what is the limit of the sum of derivatives of the zeta function taken at all its zero’s? Shouldn’t it be a real number because of symmetry?
for the first 1000 zero’s I get 2.86778 -0.00254555 i for the next 1001-2000 it’s 3.43164 +0.00472815 i for 2001-3000 3.64089 -0.00544207 i for 3001-4000 3.80755 -0.00334517 i
and so the running average up to n goes like
n=1000 : 2.86778 -0.00254555 i 2000 : 3.14971 +0.0010913 i 3000 : 3.31343 -0.00108649 i 4000 : 3.43696 -0.00165116 i
no sweet convergence sofar
just curious, Wouter _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
This reminds me: I would like to see a plot of the complex plane, with the locus of Re(Zeta(z)) = 0 in blue and the Im(Zeta(z)) = 0 in red. The red and blue lines would cross many times along the critical line, but would avoid each other elsewhere. On Fri, Sep 14, 2018 at 11:45 AM Allan Wechsler <acwacw@gmail.com> wrote:
I don't see why it has to converge. The zeta function slowly gets "wrinklier" as you go up the critical line. and (as you have observed) the derivatives at the zeroes tend to cluster around the real line, so I am guessing that the derivatives gradually crawl rightward. If you did a million zeroes, you might see values around 4. (I think the tendency is logarithmic, that is, you have to double the distance out the critical line to see an increment in average derivative.)
On Fri, Sep 14, 2018 at 11:26 AM Wouter Meeussen < wouter.meeussen@telenet.be> wrote:
what is the limit of the sum of derivatives of the zeta function taken at all its zero’s? Shouldn’t it be a real number because of symmetry?
for the first 1000 zero’s I get 2.86778 -0.00254555 i for the next 1001-2000 it’s 3.43164 +0.00472815 i for 2001-3000 3.64089 -0.00544207 i for 3001-4000 3.80755 -0.00334517 i
and so the running average up to n goes like
n=1000 : 2.86778 -0.00254555 i 2000 : 3.14971 +0.0010913 i 3000 : 3.31343 -0.00108649 i 4000 : 3.43696 -0.00165116 i
no sweet convergence sofar
just curious, Wouter _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
See https://arxiv.org/abs/math/0309433 and for a much nicer set of curves, http://www.numbertheory.org/pdfs/xrays.pdf Steve
On Sep 14, 2018, at 3:23 PM, Allan Wechsler <acwacw@gmail.com> wrote:
This reminds me: I would like to see a plot of the complex plane, with the locus of Re(Zeta(z)) = 0 in blue and the Im(Zeta(z)) = 0 in red. The red and blue lines would cross many times along the critical line, but would avoid each other elsewhere.
On Fri, Sep 14, 2018 at 11:45 AM Allan Wechsler <acwacw@gmail.com> wrote:
I don't see why it has to converge. The zeta function slowly gets "wrinklier" as you go up the critical line. and (as you have observed) the derivatives at the zeroes tend to cluster around the real line, so I am guessing that the derivatives gradually crawl rightward. If you did a million zeroes, you might see values around 4. (I think the tendency is logarithmic, that is, you have to double the distance out the critical line to see an increment in average derivative.)
On Fri, Sep 14, 2018 at 11:26 AM Wouter Meeussen < wouter.meeussen@telenet.be> wrote:
what is the limit of the sum of derivatives of the zeta function taken at all its zero’s? Shouldn’t it be a real number because of symmetry?
for the first 1000 zero’s I get 2.86778 -0.00254555 i for the next 1001-2000 it’s 3.43164 +0.00472815 i for 2001-3000 3.64089 -0.00544207 i for 3001-4000 3.80755 -0.00334517 i
and so the running average up to n goes like
n=1000 : 2.86778 -0.00254555 i 2000 : 3.14971 +0.0010913 i 3000 : 3.31343 -0.00108649 i 4000 : 3.43696 -0.00165116 i
no sweet convergence sofar
just curious, Wouter _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://urldefense.proofpoint.com/v2/url?u=https-3A__mailman.xmission.com_cg...
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Thank you, those are exactly what I was looking for. On Fri, Sep 14, 2018 at 3:41 PM Lucas, Stephen K - lucassk <lucassk@jmu.edu> wrote:
See https://arxiv.org/abs/math/0309433 and for a much nicer set of curves, http://www.numbertheory.org/pdfs/xrays.pdf
Steve
On Sep 14, 2018, at 3:23 PM, Allan Wechsler <acwacw@gmail.com> wrote:
This reminds me: I would like to see a plot of the complex plane, with the locus of Re(Zeta(z)) = 0 in blue and the Im(Zeta(z)) = 0 in red. The red and blue lines would cross many times along the critical line, but would avoid each other elsewhere.
On Fri, Sep 14, 2018 at 11:45 AM Allan Wechsler <acwacw@gmail.com> wrote:
I don't see why it has to converge. The zeta function slowly gets "wrinklier" as you go up the critical line. and (as you have observed) the derivatives at the zeroes tend to cluster around the real line, so I am guessing that the derivatives gradually crawl rightward. If you did a million zeroes, you might see values around 4. (I think the tendency is logarithmic, that is, you have to double the distance out the critical line to see an increment in average derivative.)
On Fri, Sep 14, 2018 at 11:26 AM Wouter Meeussen < wouter.meeussen@telenet.be> wrote:
what is the limit of the sum of derivatives of the zeta function taken at all its zero’s? Shouldn’t it be a real number because of symmetry?
for the first 1000 zero’s I get 2.86778 -0.00254555 i for the next 1001-2000 it’s 3.43164 +0.00472815 i for 2001-3000 3.64089 -0.00544207 i for 3001-4000 3.80755 -0.00334517 i
and so the running average up to n goes like
n=1000 : 2.86778 -0.00254555 i 2000 : 3.14971 +0.0010913 i 3000 : 3.31343 -0.00108649 i 4000 : 3.43696 -0.00165116 i
no sweet convergence sofar
just curious, Wouter _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com
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At end of sect. 8 on p. 27 of https://arxiv.org/abs/math/0309433 is a permutation of positive integers (provided RH true), not in OEIS: 1,2,3,4,5,7,6,8,10,9,11,13,12,14,16,15,17,18,20,19,21,23,24,22,26,25,27,28,30,31,29,32,34,33,35,36,... WFL On 9/14/18, Allan Wechsler <acwacw@gmail.com> wrote:
Thank you, those are exactly what I was looking for.
On Fri, Sep 14, 2018 at 3:41 PM Lucas, Stephen K - lucassk <lucassk@jmu.edu> wrote:
See https://arxiv.org/abs/math/0309433 and for a much nicer set of curves, http://www.numbertheory.org/pdfs/xrays.pdf
Steve
On Sep 14, 2018, at 3:23 PM, Allan Wechsler <acwacw@gmail.com> wrote:
This reminds me: I would like to see a plot of the complex plane, with the locus of Re(Zeta(z)) = 0 in blue and the Im(Zeta(z)) = 0 in red. The red and blue lines would cross many times along the critical line, but would avoid each other elsewhere.
On Fri, Sep 14, 2018 at 11:45 AM Allan Wechsler <acwacw@gmail.com> wrote:
I don't see why it has to converge. The zeta function slowly gets "wrinklier" as you go up the critical line. and (as you have observed) the derivatives at the zeroes tend to cluster around the real line, so I am guessing that the derivatives gradually crawl rightward. If you did a million zeroes, you might see values around 4. (I think the tendency is logarithmic, that is, you have to double the distance out the critical line to see an increment in average derivative.)
On Fri, Sep 14, 2018 at 11:26 AM Wouter Meeussen < wouter.meeussen@telenet.be> wrote:
what is the limit of the sum of derivatives of the zeta function taken at all its zero’s? Shouldn’t it be a real number because of symmetry?
for the first 1000 zero’s I get 2.86778 -0.00254555 i for the next 1001-2000 it’s 3.43164 +0.00472815 i for 2001-3000 3.64089 -0.00544207 i for 3001-4000 3.80755 -0.00334517 i
and so the running average up to n goes like
n=1000 : 2.86778 -0.00254555 i 2000 : 3.14971 +0.0010913 i 3000 : 3.31343 -0.00108649 i 4000 : 3.43696 -0.00165116 i
no sweet convergence sofar
just curious, Wouter _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com
https://urldefense.proofpoint.com/v2/url?u=https-3A__mailman.xmission.com_cg...
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Fred said: "At end of sect. 8 on p. 27 of https://arxiv.org/abs/math/0309433 is a permutation of positive integers (provided RH true), not in OEIS: 1,2,3,4,5,7,6,8,10,9,11,13,12,14,16,15,17,18,20,19,21,23,24, 22,26,25,27,28,30,31,29,32,34,33,35,36,..." Me: I have added it to the OEIS as https://oeis.org/A318936. However, here is a 2003 sequence, A088750, from the same author with the same date: %I A088750 %S A088750 1,2,3,4,5,7,6,8,10,9,11,13,12,14,16,15,17,18,20,19,21,24,22,23,25,27, %T A088750 26,28,29,32,30,31,33,35,34,36,37,40,38,39,41,44,42,43,45,46,48,47,49, %U A088750 50,53,51,52,54,55,57,56,58 %N A088750 a(n) = number of the zero of the Riemann zeta-function on the same line as the Gram point g(n-2). It is only well-defined if the Riemann hypothesis is true. %C A088750 To make the relation between zeros and Gram points bijective we must associate the Gram points on a parallel line with the zero on the next parallel line above it. n->a(n) is a bijection of the natural numbers. For some absolute constant C and every n we have |n-a(n)|<C log n. By a theorem of Speiser the sequence is well-defined if and only if the hypothesis of Riemann is true. Some relations with the sequence A088749 that appear to be true for the first terms are not true in general. The sequence is given with some mistakes in the reference arXiv:math.NT/0309433. %C A088750 The only way I know to obtain the sequence is to draw the curves Re zeta(s)=0 and Im zeta(s)=0. %H A088750 J. Arias-de-Reyna, <a href="https://arxiv.org/abs/math/0309433">X-Ray of Riemann zeta-function</a>, arXiv:math/0309433 [math.NT], 2003. %H A088750 A. Speiser, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002277352">Geometrisches zur Riemannschen Zetafunktion</a>, Math. Ann., Vol. 110 (1934), pp. 514-521. %e A088750 a(9)=10 because the Gram point g(7)=g(9-2) is on the same sheet Im zeta(s)=0 that the tenth nontrivial zero of Riemann zeta function. %Y A088750 Cf. A088749. %K A088750 hard,nonn %O A088750 1,2 %A A088750 _Juan Arias-de-Reyna_, Oct 15 2003 Are they really different, or is one an incorrect version of the other? Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Fri, Sep 14, 2018 at 7:03 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
At end of sect. 8 on p. 27 of https://arxiv.org/abs/math/0309433 is a permutation of positive integers (provided RH true), not in OEIS: 1,2,3,4,5,7,6,8,10,9,11,13,12,14,16,15,17,18,20,19,21,23,24, 22,26,25,27,28,30,31,29,32,34,33,35,36,...
WFL
On 9/14/18, Allan Wechsler <acwacw@gmail.com> wrote:
Thank you, those are exactly what I was looking for.
On Fri, Sep 14, 2018 at 3:41 PM Lucas, Stephen K - lucassk < lucassk@jmu.edu> wrote:
See https://arxiv.org/abs/math/0309433 and for a much nicer set of curves, http://www.numbertheory.org/pdfs/xrays.pdf
Steve
On Sep 14, 2018, at 3:23 PM, Allan Wechsler <acwacw@gmail.com> wrote:
This reminds me: I would like to see a plot of the complex plane, with the locus of Re(Zeta(z)) = 0 in blue and the Im(Zeta(z)) = 0 in red. The red and blue lines would cross many times along the critical line, but would avoid each other elsewhere.
On Fri, Sep 14, 2018 at 11:45 AM Allan Wechsler <acwacw@gmail.com> wrote:
I don't see why it has to converge. The zeta function slowly gets "wrinklier" as you go up the critical line. and (as you have observed) the derivatives at the zeroes tend to cluster around the real line, so I am guessing that the derivatives gradually crawl rightward. If you did a million zeroes, you might see values around 4. (I think the tendency is logarithmic, that is, you have to double the distance out the critical line to see an increment in average derivative.)
On Fri, Sep 14, 2018 at 11:26 AM Wouter Meeussen < wouter.meeussen@telenet.be> wrote:
what is the limit of the sum of derivatives of the zeta function taken at all its zero’s? Shouldn’t it be a real number because of symmetry?
for the first 1000 zero’s I get 2.86778 -0.00254555 i for the next 1001-2000 it’s 3.43164 +0.00472815 i for 2001-3000 3.64089 -0.00544207 i for 3001-4000 3.80755 -0.00334517 i
and so the running average up to n goes like
n=1000 : 2.86778 -0.00254555 i 2000 : 3.14971 +0.0010913 i 3000 : 3.31343 -0.00108649 i 4000 : 3.43696 -0.00165116 i
no sweet convergence sofar
just curious, Wouter _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com
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NS: "Are they really different, or is one an incorrect version of the other?" The 2003 A088750 *mentions* the article on which your just-added A318936 is based. Specifically: "The sequence is given with some mistakes in the reference arXiv:math.NT/0309433".
Hans, Thanks! "The sequence is given with some mistakes in the reference arXiv:math.NT/0309433". - I missed that. Yes, that would explain the discrepancies. I will mark the new version as "erroneous". Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Sat, Sep 15, 2018 at 2:25 PM, Hans Havermann <gladhobo@bell.net> wrote:
NS: "Are they really different, or is one an incorrect version of the other?"
The 2003 A088750 *mentions* the article on which your just-added A318936 is based. Specifically: "The sequence is given with some mistakes in the reference arXiv:math.NT/0309433".
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When I posted that remark about the sequence being absent from OEIS, a little voice kept nudging me to try searching also for a shorter segment; though I was busy elsewhere, and ignored it. Hans evidently followed the same path rather more assiduously. Coincidence? If not, what might have triggered our suspicions? WFL On 9/15/18, Neil Sloane <njasloane@gmail.com> wrote:
Hans, Thanks! "The sequence is given with some mistakes in the reference arXiv:math.NT/0309433". - I missed that. Yes, that would explain the discrepancies. I will mark the new version as "erroneous".
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Sat, Sep 15, 2018 at 2:25 PM, Hans Havermann <gladhobo@bell.net> wrote:
NS: "Are they really different, or is one an incorrect version of the other?"
The 2003 A088750 *mentions* the article on which your just-added A318936 is based. Specifically: "The sequence is given with some mistakes in the reference arXiv:math.NT/0309433".
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participants (6)
-
Allan Wechsler -
Fred Lunnon -
Hans Havermann -
Lucas, Stephen K - lucassk -
Neil Sloane -
Wouter Meeussen