Riemann-Siegel Z function and Lehmer's phenomenon
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From: David Bernier on 20 Oct 2009 03:25 Z(t) := exp( i theta(t) ) zeta(1/2 + it) for t>0. Cf.: http://web.viu.ca/pughg/RiemannZeta/RiemannZetaLong.html or http://en.wikipedia.org/wiki/Z_function Glen Pugh writes on the 1st web page above: "One interesting fact about the Z(t) curve is that the absence of a zero between consecutive local extrema would signal a counter-example to the Riemann Hypothesis (RH)." [say for t>15, due to an exceptional extremum for t~~= 10 ]. In the article: http://mathforum.org/kb/message.jspa?messageID=6079385 I wrote: "I found a reference to a precise statement and a proof assuming several advanced results on the zeta function in Chapter 11 by Ivic in: Yoichi Motohashi (Editor): ``Analytic Number Theory", London Math. Society Lecture Note Series 247, pp. 143-144 . Cf.: < google.com.au/books?id=OacqLxcMos8C" target="_blank">http://books.google.com.au/books?id=OacqLxcMos8C > " Two nearby zeros of the Z function where Z(t) "barely crosses the t-axis" is a case of Lehmer's phenomenon. If, for some t>20, we had consecutive local extrema at t_1 and t_2 of Z without a zero of Z in between, then RH would be false. What I'd like to know is if, in the scenario above, there must be non-trivial zeros of zeta within some known bound of 1/2 +i t_1 or 1/2 + i t_2 ( such as a distance of O(log(t)) ). David Bernier
From: David Bernier on 20 Oct 2009 10:04 David Bernier wrote: > Z(t) := exp( i theta(t) ) zeta(1/2 + it) for t>0. > > Cf.: > http://web.viu.ca/pughg/RiemannZeta/RiemannZetaLong.html > > or > http://en.wikipedia.org/wiki/Z_function > > Glen Pugh writes on the 1st web page above: > > "One interesting fact about the Z(t) curve is that the > absence of a zero between consecutive local extrema > would signal a counter-example to the Riemann Hypothesis (RH)." > > [say for t>15, due to an exceptional extremum for t~~= 10 ]. > > In the article: > > http://mathforum.org/kb/message.jspa?messageID=6079385 > > I wrote: > > "I found a reference to a precise statement and a > proof assuming several advanced results on the > zeta function in Chapter 11 by Ivic in: > Yoichi Motohashi (Editor): ``Analytic Number Theory", > London Math. Society Lecture Note Series 247, pp. 143-144 . > > Cf.: > < google.com.au/books?id=OacqLxcMos8C" > target="_blank">http://books.google.com.au/books?id=OacqLxcMos8C > " > > Two nearby zeros of the Z function where Z(t) "barely crosses the t-axis" > is a case of Lehmer's phenomenon. > > If, for some t>20, we had consecutive local extrema at t_1 and t_2 of Z > without a zero of Z in between, then RH would be false. > > What I'd like to know is if, in the scenario above, there must be > non-trivial zeros of > zeta within some known bound of 1/2 +i t_1 or 1/2 + i t_2 > ( such as a distance of O(log(t)) ). H. Edwards has a section on Lehmer's phenomenon in his well-known book: it's section 8.3, pp. 175-179. He shows that assuming RH, for large enough t and between two consecutive zeros of Z(t), -Z'/Z is a strictly increasing function of t. The proof appears on pages 176 and 177. David Bernier
From: David Bernier on 24 Oct 2009 22:24 David Bernier wrote: > David Bernier wrote: >> Z(t) := exp( i theta(t) ) zeta(1/2 + it) for t>0. >> >> Cf.: >> http://web.viu.ca/pughg/RiemannZeta/RiemannZetaLong.html >> >> or >> http://en.wikipedia.org/wiki/Z_function >> >> Glen Pugh writes on the 1st web page above: >> >> "One interesting fact about the Z(t) curve is that the >> absence of a zero between consecutive local extrema >> would signal a counter-example to the Riemann Hypothesis (RH)." >> >> [say for t>15, due to an exceptional extremum for t~~= 10 ]. More precisely, Aleksandar Ivic writes: "The function Z(t), defined by (1.10), has a negative local maximum −0.52625 . . . at t = 2.47575" Cf.: < http://arxiv.org/abs/math/0311162v1 > "On some reasons for doubting the Riemann hypothesis", page 4. There is also a negative local minimum near t = 10, so it's the negative local maximum near t = 2.47575 which is an "exceptional" extremum. >> In the article: >> >> http://mathforum.org/kb/message.jspa?messageID=6079385 >> >> I wrote: >> >> "I found a reference to a precise statement and a >> proof assuming several advanced results on the >> zeta function in Chapter 11 by Ivic in: >> Yoichi Motohashi (Editor): ``Analytic Number Theory", >> London Math. Society Lecture Note Series 247, pp. 143-144 . >> >> Cf.: >> < google.com.au/books?id=OacqLxcMos8C" >> target="_blank">http://books.google.com.au/books?id=OacqLxcMos8C > " >> >> Two nearby zeros of the Z function where Z(t) "barely crosses the t-axis" >> is a case of Lehmer's phenomenon. >> >> If, for some t>20, we had consecutive local extrema at t_1 and t_2 of Z >> without a zero of Z in between, then RH would be false. >> >> What I'd like to know is if, in the scenario above, there must be >> non-trivial zeros of >> zeta within some known bound of 1/2 +i t_1 or 1/2 + i t_2 >> ( such as a distance of O(log(t)) ). I looked at what Ivic says in his paper "On some reasons for doubting the Riemann hypothesis" about Lehmer's phenomenon. Section 2 is about Lehmer's phenomenon. In Proposition 1, Ivic states that if RH is true, then for some t_0, t > t_0 implies that Z'(t)/Z(t) is strictly decreasing between consecutive zeros 1/2 + i*gamma_1 and 1/2 +i*gamma_2, t_0 < gamma_1 < gamma_2 . What is shown in fact is that (Z'(t)/Z(t))' < 0 if t >= t0 , and this is enough to show Proposition 1. Ivic shows, using known properties of zeta, that (Z'(t)/Z(t))' can be evaluated exactly as [-sum_{gamma in R s.t. 1/2 + i*gamma is a zero of zeta} (gamma - t)^(-2) ] - d/dt(f'(t)/f(t))_{evaluated at 't'}, where d/dt ( f'(t)/f(t) )_{at t} << 1/t, so I guess the second term is O(1/t) ... The expression for (Z'(t)/Z(t))' assumes RH and that gamma_1 < t < gamma_2 [i.e. 1/2 + it isn't a zero of zeta]. --- Suppose we have a non-trivial zero at alpha + i*gamma_0, where gamma_0 > 0 , alpha > 1/2 . Then there is also a non-trivial zero at (1- alpha ) + i*gamma_0 . We can use (2.1) and (2.2) to find (Z'(t)/Z(t))' evaluated at t = gamma_0 . What I get is (Z'(t)/Z(t))'| t = gamma_0 = P + Rem. , where P is a positive term involving the non-trivial zeros, and Rem. is Rem. = [-sum_{gamma in R s.t. 1/2 + i*gamma is a zero of zeta} (gamma - gamma_0)^(-2) ] - d/dt(f'(t)/f(t))_{evaluated at gamma_0}. From the pair of non-trivial zeros at alpha + i*gamma_0 and (1- alpha ) + i*gamma_0, I get a contribution to the positive term P of 2 (alpha - 1/2)^(-2) . If alpha is close enough to 1/2, 2 (alpha - 1/2)^(-2) could be large enough to outweigh Rem. above, in which case we would have: (Z'(t)/Z(t))'| (t = gamma_0) > 0 . On the other hand, if alpha is close to 1 (such as alpha = 0.75 ), then 2 (alpha - 1/2)^(-2) = 32. I suppose it could be interesting to get a few values of (Z'(t)/Z(t))' near (positive) local maxima and (negative) local minima of Z(t) to get an idea of statistics around some large height t ~= T, with T = 10^6 or maybe larger ... One thing I'm not sure about is what we can say if (Z'(t)/Z(t))'| (t = t_0) > 0 , where gamma_1 < t_0 < gamma_2, 1/2 + i*gamma_1 and 1/2 +i*gamma_2 being consecutive zeros of zeta on the critical line. If gamma_1 > 10, then I think we have RH false. But if (Z'(t)/Z(t))'| (t = t_0) > 0, would we see a negative local maximum or a positive local minimum of Z(t) near t = t_0 ? David Bernier > H. Edwards has a section on Lehmer's phenomenon in his > well-known book: it's section 8.3, pp. 175-179. > > He shows that assuming RH, for large enough t and between > two consecutive zeros of Z(t), -Z'/Z is a strictly > increasing function of t. > > The proof appears on pages 176 and 177.
From: David Bernier on 25 Oct 2009 05:32 David Bernier wrote: > David Bernier wrote: >> David Bernier wrote: >>> Z(t) := exp( i theta(t) ) zeta(1/2 + it) for t>0. >>> >>> Cf.: >>> http://web.viu.ca/pughg/RiemannZeta/RiemannZetaLong.html >>> >>> or >>> http://en.wikipedia.org/wiki/Z_function >>> >>> Glen Pugh writes on the 1st web page above: >>> >>> "One interesting fact about the Z(t) curve is that the >>> absence of a zero between consecutive local extrema >>> would signal a counter-example to the Riemann Hypothesis (RH)." >>> >>> [say for t>15, due to an exceptional extremum for t~~= 10 ]. > > More precisely, Aleksandar Ivic writes: > "The function Z(t), defined by (1.10), has a negative local maximum > −0.52625 . . . at t = 2.47575" > > Cf.: > < http://arxiv.org/abs/math/0311162v1 > > "On some reasons for doubting the Riemann hypothesis", page 4. > > There is also a negative local minimum near t = 10, so > it's the negative local maximum near t = 2.47575 which is > an "exceptional" extremum. > > > >>> In the article: >>> >>> http://mathforum.org/kb/message.jspa?messageID=6079385 >>> >>> I wrote: >>> >>> "I found a reference to a precise statement and a >>> proof assuming several advanced results on the >>> zeta function in Chapter 11 by Ivic in: >>> Yoichi Motohashi (Editor): ``Analytic Number Theory", >>> London Math. Society Lecture Note Series 247, pp. 143-144 . >>> >>> Cf.: >>> < google.com.au/books?id=OacqLxcMos8C" >>> target="_blank">http://books.google.com.au/books?id=OacqLxcMos8C > " >>> >>> Two nearby zeros of the Z function where Z(t) "barely crosses the >>> t-axis" >>> is a case of Lehmer's phenomenon. >>> >>> If, for some t>20, we had consecutive local extrema at t_1 and t_2 of Z >>> without a zero of Z in between, then RH would be false. >>> >>> What I'd like to know is if, in the scenario above, there must be >>> non-trivial zeros of >>> zeta within some known bound of 1/2 +i t_1 or 1/2 + i t_2 >>> ( such as a distance of O(log(t)) ). > > > I looked at what Ivic says in his paper > "On some reasons for doubting the Riemann hypothesis" about > Lehmer's phenomenon. > > Section 2 is about Lehmer's phenomenon. > > In Proposition 1, Ivic states that if RH is true, then for some t_0, > t > t_0 implies that > > Z'(t)/Z(t) is strictly decreasing between consecutive zeros > 1/2 + i*gamma_1 and 1/2 +i*gamma_2, t_0 < gamma_1 < gamma_2 . > > What is shown in fact is that > (Z'(t)/Z(t))' < 0 if t >= t0 , and this is enough to show Proposition 1. > > Ivic shows, using known properties of zeta, that > > (Z'(t)/Z(t))' can be evaluated exactly > as > [-sum_{gamma in R s.t. 1/2 + i*gamma is a zero of zeta} > (gamma - t)^(-2) ] - d/dt(f'(t)/f(t))_{evaluated at 't'}, > > where d/dt ( f'(t)/f(t) )_{at t} << 1/t, so I guess the second > term is O(1/t) ... > > The expression for (Z'(t)/Z(t))' assumes RH and that > gamma_1 < t < gamma_2 [i.e. 1/2 + it isn't a zero of zeta]. > > --- > > Suppose we have a non-trivial zero at alpha + i*gamma_0, > where gamma_0 > 0 , alpha > 1/2 . Then there is also a non-trivial > zero at (1- alpha ) + i*gamma_0 . > > We can use (2.1) and (2.2) to find (Z'(t)/Z(t))' evaluated at > t = gamma_0 . > > What I get is (Z'(t)/Z(t))'| t = gamma_0 = P + Rem. , > > where P is a positive term involving the non-trivial zeros, > and Rem. is > Rem. = [-sum_{gamma in R s.t. 1/2 + i*gamma is a zero of zeta} > (gamma - gamma_0)^(-2) ] - d/dt(f'(t)/f(t))_{evaluated at gamma_0}. > > From the pair of non-trivial zeros at alpha + i*gamma_0 and > (1- alpha ) + i*gamma_0, I get a contribution to the positive term P > of 2 (alpha - 1/2)^(-2) . > > If alpha is close enough to 1/2, > 2 (alpha - 1/2)^(-2) could be large enough to outweigh Rem. above, > in which case we would have: > (Z'(t)/Z(t))'| (t = gamma_0) > 0 . > > On the other hand, if alpha is close to 1 (such as alpha = 0.75 ), > then 2 (alpha - 1/2)^(-2) = 32. I suppose it could be interesting > to get a few values of (Z'(t)/Z(t))' near (positive) local maxima > and (negative) local minima of Z(t) to get an idea of statistics > around some large height t ~= T, with T = 10^6 or maybe larger ... > > > One thing I'm not sure about is what we can say if > (Z'(t)/Z(t))'| (t = t_0) > 0 , where > gamma_1 < t_0 < gamma_2, > 1/2 + i*gamma_1 and 1/2 +i*gamma_2 being consecutive > zeros of zeta on the critical line. > > If gamma_1 > 10, then I think we have RH false. > > But if (Z'(t)/Z(t))'| (t = t_0) > 0, would we see > a negative local maximum or a positive > local minimum of Z(t) near t = t_0 ? At least if gamma_1 and gamma_2 correspond to simple zeros, then I think we must have at least two points where Z' is zero between gamma_1 and gamma_2, because Z'/Z (t) > M for some t> gamma_1, t< gamma_2 (t near gamma_1) for any real M, and similarly Z'/Z (t) < M for t< gamma_2, t ~= gamma_2, for any real M. The known zero-free regions for non-trivial zeros off the critical line get thinner and thinner as t>0 increases. Cf.: < http://en.wikipedia.org/wiki/Riemann_hypothesis#Zero-free_regions > ---- If the conclusion of Proposition 1 in Ivic's "On some reasons for doubting the Riemann hypothesis" is true , (cf. page 5) namely that Z'(t)/Z(t) is strictly decreasing between consecutive zeros of Z(t), as long as t > some t_0, I'm wondering if it's still possible that RH is false ... For example, suppose Z(t) has no negative-valued local maxima or positive-valued local minima when t>10; then I don't see that this would imply RH. David Bernier
From: David Bernier on 25 Oct 2009 12:23
David Bernier wrote: [...] >> --- >> >> Suppose we have a non-trivial zero at alpha + i*gamma_0, >> where gamma_0 > 0 , alpha > 1/2 . Then there is also a non-trivial >> zero at (1- alpha ) + i*gamma_0 . >> >> We can use (2.1) and (2.2) to find (Z'(t)/Z(t))' evaluated at >> t = gamma_0 . >> >> What I get is (Z'(t)/Z(t))'| t = gamma_0 = P + Rem. , >> >> where P is a positive term involving the non-trivial zeros, [ Rather, instead of "P is a positive term", "the pair of non-trivial zeros at alpha + i*gamma_0 and (1- alpha ) + i*gamma_0 make a positive contribution to P" .] >> and Rem. is >> Rem. = [-sum_{gamma in R s.t. 1/2 + i*gamma is a zero of zeta} >> (gamma - gamma_0)^(-2) ] - d/dt(f'(t)/f(t))_{evaluated at gamma_0}. >> >> From the pair of non-trivial zeros at alpha + i*gamma_0 and >> (1- alpha ) + i*gamma_0, I get a contribution to the positive term P >> of 2 (alpha - 1/2)^(-2) . >> >> If alpha is close enough to 1/2, >> 2 (alpha - 1/2)^(-2) could be large enough to outweigh Rem. above, >> in which case we would have: >> (Z'(t)/Z(t))'| (t = gamma_0) > 0 . >> >> On the other hand, if alpha is close to 1 (such as alpha = 0.75 ), >> then 2 (alpha - 1/2)^(-2) = 32. I suppose it could be interesting >> to get a few values of (Z'(t)/Z(t))' near (positive) local maxima >> and (negative) local minima of Z(t) to get an idea of statistics >> around some large height t ~= T, with T = 10^6 or maybe larger ... >> >> >> One thing I'm not sure about is what we can say if >> (Z'(t)/Z(t))'| (t = t_0) > 0 , where >> gamma_1 < t_0 < gamma_2, >> 1/2 + i*gamma_1 and 1/2 +i*gamma_2 being consecutive >> zeros of zeta on the critical line. >> >> If gamma_1 > 10, then I think we have RH false. >> >> But if (Z'(t)/Z(t))'| (t = t_0) > 0, would we see >> a negative local maximum or a positive >> local minimum of Z(t) near t = t_0 ? > > At least if gamma_1 and gamma_2 correspond to > simple zeros, then I think we must have at least > two points where Z' is zero between gamma_1 and gamma_2, > because Z'/Z (t) > M for some t> gamma_1, t< gamma_2 > (t near gamma_1) for any real M, and > similarly Z'/Z (t) < M for t< gamma_2, t ~= gamma_2, > for any real M. > > The known zero-free regions for non-trivial zeros > off the critical line get thinner and thinner as t>0 > increases. > > Cf.: > < http://en.wikipedia.org/wiki/Riemann_hypothesis#Zero-free_regions > > > ---- > > If the conclusion of Proposition 1 in Ivic's > "On some reasons for doubting the Riemann hypothesis" is true , > (cf. page 5) namely that > Z'(t)/Z(t) is strictly decreasing between consecutive zeros > of Z(t), as long as t > some t_0, > > I'm wondering if it's still possible that RH is false ... > > For example, suppose Z(t) has no negative-valued local maxima > or positive-valued local minima when t>10; then I don't see that > this would imply RH. I still don't know. I looked at "Solved and Unsolved Problems in Number Theory", the edition by Daniel Shanks around 1983 or 1984, < http://books.google.com/books?id=KjhM9pZEGCkC&pg=PA264#v=onepage&q=&f=false > On page 285, he writes: "Exercise 186. Define A(t) = (Z'(t))^2 - Z(t) Z''(t). and show that it is positive for all t ". [ Say with t>10 ]. We have d/dt (Z'/Z) = (Z Z'' - (Z')^2)/ Z^2 when Z is non-zero. When Z(t) is non-zero, d/dt (Z'/Z) < 0 <==> (Z')^2 - Z Z'' > 0. This is consistent with the conclusion of Proposition 1 from Ivic, < http://arxiv.org/abs/math/0311162v1 > "On some reasons for doubting the Riemann hypothesis" Section 2 on Lehmer's phenomenon. The hypothesis for Proposition 1 is RH. So suppose we assume what has to be proved in Exercise 186, that A(t) = (Z'(t))^2 - Z(t) Z''(t) is positive for all t>10. Does that imply RH ? If not, how much would it count as (additional) evidence for RH ? David Bernier |