Difference between revisions of "De Bruijn-Newman constant"

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or
 
or
  
:<math>\displaystyle H_t(z) = \frac{1}{2} \int_{\bf R} e^{t\log^2 x} \Phi(\log x) e^{iz \log x}\ \frac{dx}{x}.</math>
+
:<math>\displaystyle H_t(z) = \frac{1}{2} \int_0^\infty e^{t\log^2 x} \Phi(\log x) e^{iz \log x}\ \frac{dx}{x}.</math>
  
 
In the notation of [KKL2009], one has
 
In the notation of [KKL2009], one has
  
:<math>\displaystyle H_t(z) = \Xi_{4t}(2z).</math>
+
:<math>\displaystyle H_t(z) = \frac{1}{8} \Xi_{t/4}(z/2).</math>
  
 
De Bruijn [B1950] and Newman [N1976] showed that there existed a constant, the ''de Bruijn-Newman constant'' <math>\Lambda</math>, such that <math>H_t</math> has all zeroes real precisely when <math>t \geq \Lambda</math>.  The Riemann hypothesis is equivalent to the claim that <math>\Lambda \leq 0</math>.  Currently it is known that <math>0 \leq \Lambda < 1/2</math> (lower bound in [RT2018], upper bound in [KKL2009]).
 
De Bruijn [B1950] and Newman [N1976] showed that there existed a constant, the ''de Bruijn-Newman constant'' <math>\Lambda</math>, such that <math>H_t</math> has all zeroes real precisely when <math>t \geq \Lambda</math>.  The Riemann hypothesis is equivalent to the claim that <math>\Lambda \leq 0</math>.  Currently it is known that <math>0 \leq \Lambda < 1/2</math> (lower bound in [RT2018], upper bound in [KKL2009]).
 +
 +
The '''Polymath15''' project seeks to improve the upper bound on <math>\Lambda</math>.  The current strategy is to combine the following three ingredients:
 +
 +
# Numerical zero-free regions for <math>H_t(x+iy)</math> of the form <math>\{ x+iy: 0 \leq x \leq T; y \geq \varepsilon \}</math> for explicit <math>T, \varepsilon, t > 0</math>.
 +
# Rigorous asymptotics that show that <math>H_t(x+iy)</math> whenever <math>y \geq \varepsilon</math> and <math>x \geq T</math> for a sufficiently large <math>T</math>.
 +
# Dynamics of zeroes results that control <math>\Lambda</math> in terms of the maximum imaginary part of a zero of <math>H_t</math>.
  
 
== <math>t=0</math> ==
 
== <math>t=0</math> ==
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where
 
where
  
:<math>\displaystyle \xi(s) := \frac{s(s-1)}{2} \pi^{s/2} \Gamma(s/2) \zeta(s)</math>
+
:<math>\displaystyle \xi(s) := \frac{s(s-1)}{2} \pi^{-s/2} \Gamma(s/2) \zeta(s)</math>
  
 
is the Riemann xi function.  In particular, <math>z</math> is a zero of <math>H_0</math> if and only if <math>\frac{1}{2} + \frac{iz}{2}</math> is a non-trivial zero of the Riemann zeta function.  Thus, for instance, the Riemann hypothesis is equivalent to all the zeroes of <math>H_0</math> being real, and [https://en.wikipedia.org/wiki/Riemann%E2%80%93von_Mangoldt_formula Riemann-von Mangoldt formula] (in the explicit form given by Backlund) gives
 
is the Riemann xi function.  In particular, <math>z</math> is a zero of <math>H_0</math> if and only if <math>\frac{1}{2} + \frac{iz}{2}</math> is a non-trivial zero of the Riemann zeta function.  Thus, for instance, the Riemann hypothesis is equivalent to all the zeroes of <math>H_0</math> being real, and [https://en.wikipedia.org/wiki/Riemann%E2%80%93von_Mangoldt_formula Riemann-von Mangoldt formula] (in the explicit form given by Backlund) gives
  
:<math>\displaystyle N_0(T) - (\frac{T}{4\pi} \log \frac{T}{4\pi} - \frac{T}{4\pi} - \frac{7}{8})| <  0.137 \log (T/2) + 0.443  \log\log(T/2) + 4.350 </math>
+
:<math>\displaystyle \left|N_0(T) - (\frac{T}{4\pi} \log \frac{T}{4\pi} - \frac{T}{4\pi} - \frac{7}{8})\right| <  0.137 \log (T/2) + 0.443  \log\log(T/2) + 4.350 </math>
  
 
for any <math>T > 4</math>, where <math>N_0(T)</math> denotes the number of zeroes of <math>H_0</math> with real part between 0 and T.
 
for any <math>T > 4</math>, where <math>N_0(T)</math> denotes the number of zeroes of <math>H_0</math> with real part between 0 and T.
  
The first <math>10^{13}</math> zeroes of <math>H_0</math> (to the right of the origin) are real [G2004].  This numerical computation uses the Odlyzko-Schonhage algorithm.
+
The first <math>10^{13}</math> zeroes of <math>H_0</math> (to the right of the origin) are real [G2004].  This numerical computation uses the Odlyzko-Schonhage algorithm.  In [P2017] it was independently verified that all zeroes of <math>H_0</math> between 0 and 61,220,092,000 were real.
  
 
== <math>t>0</math> ==
 
== <math>t>0</math> ==
  
For any <math>t>0</math>, it is known that all but finitely many of the zeroes of <math>H_t</math> are real and simple [KKL2009, Theorem 1.3]
+
For any <math>t>0</math>, it is known that all but finitely many of the zeroes of <math>H_t</math> are real and simple [KKL2009, Theorem 1.3].  In fact, assuming the Riemann hypothesis, ''all'' of the zeroes of <math>H_t</math> are real and simple [CSV1994, Corollary 2].
  
Let <math>\sigma_{max}(t)</math> denote the largest imaginary part of a zero of <math>H_t</math>, thus <math>\sigma_{max}(t)=0</math> if and only if <math>t \geq \Lambda</math>.  It is known that the quantity <math>\frac{1}{2} \sigma_{max}(t)^2 + t</math> is non-decreasing in time whenever <math>\sigma_{max}(t)>0</math> (see [KKL2009, Proposition A].  In particular we have
+
It is known that <math>\xi</math> is an entire function of order one ([T1986, Theorem 2.12]).  Hence by the fundamental solution for the heat equation, the <math>H_t</math> are also entire functions of order one for any <math>t</math>.
 +
 
 +
Because <math>\Phi</math> is positive, <math>H_t(iy)</math> is positive for any <math>y</math>, and hence there are no zeroes on the imaginary axis.
 +
 
 +
Let <math>\sigma_{max}(t)</math> denote the largest imaginary part of a zero of <math>H_t</math>, thus <math>\sigma_{max}(t)=0</math> if and only if <math>t \geq \Lambda</math>.  It is known that the quantity <math>\frac{1}{2} \sigma_{max}(t)^2 + t</math> is non-increasing in time whenever <math>\sigma_{max}(t)>0</math> (see [KKL2009, Proposition A].  In particular we have
  
 
:<math>\displaystyle \Lambda \leq t + \frac{1}{2} \sigma_{max}(t)^2</math>
 
:<math>\displaystyle \Lambda \leq t + \frac{1}{2} \sigma_{max}(t)^2</math>
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for any <math>t</math>.
 
for any <math>t</math>.
  
The zeroes <math>z_j(t)</math> of <math>H_t</math> (formally, at least) obey the system of ODE
+
The zeroes <math>z_j(t)</math> of <math>H_t</math> obey the system of ODE
  
 
:<math>\partial_t z_j(t) = - \sum_{k \neq j} \frac{2}{z_k(t) - z_j(t)}</math>
 
:<math>\partial_t z_j(t) = - \sum_{k \neq j} \frac{2}{z_k(t) - z_j(t)}</math>
  
where the sum may have to be interpreted in a principal value sense.  (See for instance [CSV1994, Lemma 2.4].  This lemma assumes that <math>t > \Lambda</math>, but it is likely that one can extend to other <math>t \geq 0</math> as well.)
+
where the sum is interpreted in a principal value sense, and excluding those times in which <math>z_j(t)</math> is a repeated zero.  See [[dynamics of zeros]] for more detailsWriting <math>z_j(t) = x_j(t) + i y_j(t)</math>, we can write the dynamics as
 +
 
 +
:<math> \partial_t x_j = - \sum_{k \neq j} \frac{2 (x_k - x_j)}{(x_k-x_j)^2 + (y_k-y_j)^2} </math>
 +
:<math> \partial_t y_j = \sum_{k \neq j} \frac{2 (y_k - y_j)}{(x_k-x_j)^2 + (y_k-y_j)^2} </math>
 +
 
 +
where the dependence on <math>t</math> has been omitted for brevity.
  
 
In [KKL2009, Theorem 1.4], it is shown that for any fixed <math>t>0</math>, the number <math>N_t(T)</math> of zeroes of <math>H_t</math> with real part between 0 and T obeys the asymptotic
 
In [KKL2009, Theorem 1.4], it is shown that for any fixed <math>t>0</math>, the number <math>N_t(T)</math> of zeroes of <math>H_t</math> with real part between 0 and T obeys the asymptotic
  
:<math>N_t(T) = \frac{T}{4\pi} \log \frac{T}{4\pi} - \frac{T}{4\pi} + t \log T + O(1) </math>
+
:<math>N_t(T) = \frac{T}{4\pi} \log \frac{T}{4\pi} - \frac{T}{4\pi} + \frac{t}{16} \log T + O(1) </math>
  
 
as <math>T \to \infty</math> (caution: the error term here is not uniform in t).  Also, the zeroes behave like an arithmetic progression in the sense that
 
as <math>T \to \infty</math> (caution: the error term here is not uniform in t).  Also, the zeroes behave like an arithmetic progression in the sense that
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as <math>k \to +\infty</math>.
 
as <math>k \to +\infty</math>.
  
 +
See [[asymptotics of H_t]] for asymptotics of the function <math>H_t</math>, and [[Effective bounds on H_t]] and [[Effective bounds on H_t - second approach]] for explicit bounds.
  
 
== Threads ==
 
== Threads ==
  
 
* [https://terrytao.wordpress.com/2018/01/24/polymath-proposal-upper-bounding-the-de-bruijn-newman-constant/ Polymath proposal: upper bounding the de Bruijn-Newman constant], Terence Tao, Jan 24, 2018.
 
* [https://terrytao.wordpress.com/2018/01/24/polymath-proposal-upper-bounding-the-de-bruijn-newman-constant/ Polymath proposal: upper bounding the de Bruijn-Newman constant], Terence Tao, Jan 24, 2018.
 +
* [https://terrytao.wordpress.com/2018/01/27/polymath15-first-thread-computing-h_t-asymptotics-and-dynamics-of-zeroes/ Polymath15, first thread: computing H_t, asymptotics, and dynamics of zeroes], Terence Tao, Jan 27, 2018.
 +
* [https://terrytao.wordpress.com/2018/02/02/polymath15-second-thread-generalising-the-riemann-siegel-approximate-functional-equation/ Polymath15, second thread: generalising the Riemann-Siegel approximate functional equation], Terence Tao and Sujit Nair, Feb 2, 2018.
 +
* [https://terrytao.wordpress.com/2018/02/12/polymath15-third-thread-computing-and-approximating-h_t/ Polymath15, third thread: computing and approximating H_t], Terence Tao and Sujit Nair, Feb 12, 2018.
 +
* [https://terrytao.wordpress.com/2018/02/24/polymath15-fourth-thread-closing-in-on-the-test-problem/ Polymath 15, fourth thread: closing in on the test problem], Terence Tao, Feb 24, 2018.
 +
* [https://terrytao.wordpress.com/2018/03/02/polymath15-fifth-thread-finishing-off-the-test-problem/ Polymath15, fifth thread: finishing off the test problem?], Terence Tao, Mar 2, 2018.
 +
* [https://terrytao.wordpress.com/2018/03/18/polymath15-sixth-thread-the-test-problem-and-beyond/ Polymath15, sixth thread: the test problem and beyond], Terence Tao, Mar 18, 2018.
 +
* [https://terrytao.wordpress.com/2018/03/28/polymath15-seventh-thread-going-below-0-48/ Polymath15, seventh thread: going below 0.48], Terence Tao, Mar 28, 2018.
 +
* [https://terrytao.wordpress.com/2018/04/17/polymath15-eighth-thread-going-below-0-28/ Polymath15, eighth thread: going below 0.28], Terence Tao, Apr 17, 2018.
 +
* [https://terrytao.wordpress.com/2018/05/04/polymath15-ninth-thread-going-below-0-22/ Polymath15, ninth thread: going below 0.22?], Terence Tao, May 4, 2018.
 +
* [https://terrytao.wordpress.com/10725 Polymath15, tenth thread: numerics update], Rudolph Dwars and Kalpesh Muchhal, Sep 6, 2018.
 +
* [https://terrytao.wordpress.com/2018/12/28/polymath-15-eleventh-thread-writing-up-the-results-and-exploring-negative-t/ Polymath15, eleventh thread: Writing up the results, and exploring negative t], Terence Tao, Dec 28, 2018.
 +
* [https://terrytao.wordpress.com/2019/04/30/11075/ Effective approximation of heat flow evolution of the Riemann xi function, and a new upper bound for the de Bruijn-Newman constant], Terence Tao, Apr 30, 2019.
  
 
== Other blog posts and online discussion ==
 
== Other blog posts and online discussion ==
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* [https://github.com/km-git-acc/dbn_upper_bound Github repository]
 
* [https://github.com/km-git-acc/dbn_upper_bound Github repository]
 +
 +
== Writeup ==
 +
 +
* [https://github.com/km-git-acc/dbn_upper_bound/tree/master/Writeup Writeup subdirectory of Github repository]
 +
 +
Here are the [[Polymath15 grant acknowledgments]].
 +
 +
== Test problem ==
 +
 +
See [[Polymath15 test problem]].
 +
 +
== Zero-free regions ==
 +
 +
See [[Zero-free regions]].
  
 
== Wikipedia and other references ==
 
== Wikipedia and other references ==
  
 
* [https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Newman_constant de Bruijn-Newman constant]
 
* [https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Newman_constant de Bruijn-Newman constant]
 +
* [https://en.wikipedia.org/wiki/Odlyzko%E2%80%93Sch%C3%B6nhage_algorithm Odlyzko–Schönhage algorithm]
 +
* [https://en.wikipedia.org/wiki/Riemann%E2%80%93Siegel_formula Riemann–Siegel formula]
 
* [https://en.wikipedia.org/wiki/Riemann_Xi_function Riemann xi function]
 
* [https://en.wikipedia.org/wiki/Riemann_Xi_function Riemann xi function]
  
 
== Bibliography ==
 
== Bibliography ==
  
* [B1950] N. C. de Bruijn, The roots of trigonometric integrals, Duke J. Math. 17 (1950), 197–226.
+
* [A2011] J. Arias de Reyna, [https://pdfs.semanticscholar.org/7964/fbdc0caeec0a41304deb8d2d8b2e2be639ee.pdf High-precision computation of Riemann's zeta function by the Riemann-Siegel asymptotic formula, I], Mathematics of Computation, Volume 80, Number 274, April 2011, Pages 995–1009.
 +
* [B1994] W. G. C. Boyd, [http://www.jstor.org/stable/52450 Gamma Function Asymptotics by an Extension of the Method of Steepest Descents], Proceedings: Mathematical and Physical Sciences, Vol. 447, No. 1931 (Dec. 8, 1994),pp. 609-630.
 +
* [B1950] N. C. de Bruijn, [https://pure.tue.nl/ws/files/1769368/597490.pdf The roots of trigonometric integrals], Duke J. Math. 17 (1950), 197–226.
 
* [CSV1994] G. Csordas, W. Smith, R. S. Varga, [https://link.springer.com/article/10.1007/BF01205170 Lehmer pairs of zeros, the de Bruijn-Newman constant Λ, and the Riemann hypothesis], Constr. Approx. 10 (1994), no. 1, 107–129.
 
* [CSV1994] G. Csordas, W. Smith, R. S. Varga, [https://link.springer.com/article/10.1007/BF01205170 Lehmer pairs of zeros, the de Bruijn-Newman constant Λ, and the Riemann hypothesis], Constr. Approx. 10 (1994), no. 1, 107–129.
* [G2004] Gourdon, Xavier (2004), [http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeros1e13-1e24.pdf The <math>10^13</math> first zeros of the Riemann Zeta function, and zeros computation at very large height]
+
* [G2004] Gourdon, Xavier (2004), [http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeros1e13-1e24.pdf The <math>10^{13}</math> first zeros of the Riemann Zeta function, and zeros computation at very large height]
 
* [KKL2009] H. Ki, Y. O. Kim, and J. Lee, On the de Bruijn-Newman constant, Advances in Mathematics, 22 (2009), 281–306. [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.164.5595&rep=rep1&type=pdf Citeseer]
 
* [KKL2009] H. Ki, Y. O. Kim, and J. Lee, On the de Bruijn-Newman constant, Advances in Mathematics, 22 (2009), 281–306. [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.164.5595&rep=rep1&type=pdf Citeseer]
* [N1976] C. M. Newman, Fourier transforms with only real zeroes, Proc. Amer. Math. Soc. 61 (1976), 246–251.
+
* [N1976] C. M. Newman, [http://www.ams.org/journals/proc/1976-061-02/S0002-9939-1976-0434982-5/S0002-9939-1976-0434982-5.pdf Fourier transforms with only real zeroes], Proc. Amer. Math. Soc. 61 (1976), 246–251.
* [RT2018] B. Rodgers, T. Tao, The de Bruijn-Newman constant is negative, preprint. [https://arxiv.org/abs/1801.05914 arXiv:1801.05914]
+
* [P2017] D. J. Platt, [http://www.ams.org/journals/mcom/2017-86-307/S0025-5718-2017-03198-7/ Isolating some non-trivial zeros of zeta], Math. Comp. 86 (2017), 2449-2467.
 +
* [P1992] G. Pugh, [https://web.viu.ca/pughg/thesis.d/masters.thesis.pdf The Riemann-Siegel formula and large scale computations of the Riemann zeta function], M.Sc. Thesis, U. British Columbia, 1992.
 +
* [RT2018] B. Rodgers, T. Tao, The de Bruijn-Newman constant is non-negative, preprint. [https://arxiv.org/abs/1801.05914 arXiv:1801.05914]
 +
* [T1986] E. C. Titchmarsh, The theory of the Riemann zeta-function. Second edition. Edited and with a preface by D. R. Heath-Brown. The Clarendon Press, Oxford University Press, New York, 1986. [http://plouffe.fr/simon/math/The%20Theory%20Of%20The%20Riemann%20Zeta-Function%20-Titshmarch.pdf pdf]
 +
 
 +
[[Category:Polymath15]]

Latest revision as of 16:37, 30 April 2019

For each real number [math]t[/math], define the entire function [math]H_t: {\mathbf C} \to {\mathbf C}[/math] by the formula

[math]\displaystyle H_t(z) := \int_0^\infty e^{tu^2} \Phi(u) \cos(zu)\ du[/math]

where [math]\Phi[/math] is the super-exponentially decaying function

[math]\displaystyle \Phi(u) := \sum_{n=1}^\infty (2\pi^2 n^4 e^{9u} - 3 \pi n^2 e^{5u}) \exp(-\pi n^2 e^{4u}).[/math]

It is known that [math]\Phi[/math] is even, and that [math]H_t[/math] is even, real on the real axis, and obeys the functional equation [math]H_t(\overline{z}) = \overline{H_t(z)}[/math]. In particular, the zeroes of [math]H_t[/math] are symmetric about both the real and imaginary axes. One can also express [math]H_t[/math] in a number of different forms, such as

[math]\displaystyle H_t(z) = \frac{1}{2} \int_{\bf R} e^{tu^2} \Phi(u) e^{izu}\ du[/math]

or

[math]\displaystyle H_t(z) = \frac{1}{2} \int_0^\infty e^{t\log^2 x} \Phi(\log x) e^{iz \log x}\ \frac{dx}{x}.[/math]

In the notation of [KKL2009], one has

[math]\displaystyle H_t(z) = \frac{1}{8} \Xi_{t/4}(z/2).[/math]

De Bruijn [B1950] and Newman [N1976] showed that there existed a constant, the de Bruijn-Newman constant [math]\Lambda[/math], such that [math]H_t[/math] has all zeroes real precisely when [math]t \geq \Lambda[/math]. The Riemann hypothesis is equivalent to the claim that [math]\Lambda \leq 0[/math]. Currently it is known that [math]0 \leq \Lambda \lt 1/2[/math] (lower bound in [RT2018], upper bound in [KKL2009]).

The Polymath15 project seeks to improve the upper bound on [math]\Lambda[/math]. The current strategy is to combine the following three ingredients:

  1. Numerical zero-free regions for [math]H_t(x+iy)[/math] of the form [math]\{ x+iy: 0 \leq x \leq T; y \geq \varepsilon \}[/math] for explicit [math]T, \varepsilon, t \gt 0[/math].
  2. Rigorous asymptotics that show that [math]H_t(x+iy)[/math] whenever [math]y \geq \varepsilon[/math] and [math]x \geq T[/math] for a sufficiently large [math]T[/math].
  3. Dynamics of zeroes results that control [math]\Lambda[/math] in terms of the maximum imaginary part of a zero of [math]H_t[/math].

[math]t=0[/math]

When [math]t=0[/math], one has

[math]\displaystyle H_0(z) = \frac{1}{8} \xi( \frac{1}{2} + \frac{iz}{2} ) [/math]

where

[math]\displaystyle \xi(s) := \frac{s(s-1)}{2} \pi^{-s/2} \Gamma(s/2) \zeta(s)[/math]

is the Riemann xi function. In particular, [math]z[/math] is a zero of [math]H_0[/math] if and only if [math]\frac{1}{2} + \frac{iz}{2}[/math] is a non-trivial zero of the Riemann zeta function. Thus, for instance, the Riemann hypothesis is equivalent to all the zeroes of [math]H_0[/math] being real, and Riemann-von Mangoldt formula (in the explicit form given by Backlund) gives

[math]\displaystyle \left|N_0(T) - (\frac{T}{4\pi} \log \frac{T}{4\pi} - \frac{T}{4\pi} - \frac{7}{8})\right| \lt 0.137 \log (T/2) + 0.443 \log\log(T/2) + 4.350 [/math]

for any [math]T \gt 4[/math], where [math]N_0(T)[/math] denotes the number of zeroes of [math]H_0[/math] with real part between 0 and T.

The first [math]10^{13}[/math] zeroes of [math]H_0[/math] (to the right of the origin) are real [G2004]. This numerical computation uses the Odlyzko-Schonhage algorithm. In [P2017] it was independently verified that all zeroes of [math]H_0[/math] between 0 and 61,220,092,000 were real.

[math]t\gt0[/math]

For any [math]t\gt0[/math], it is known that all but finitely many of the zeroes of [math]H_t[/math] are real and simple [KKL2009, Theorem 1.3]. In fact, assuming the Riemann hypothesis, all of the zeroes of [math]H_t[/math] are real and simple [CSV1994, Corollary 2].

It is known that [math]\xi[/math] is an entire function of order one ([T1986, Theorem 2.12]). Hence by the fundamental solution for the heat equation, the [math]H_t[/math] are also entire functions of order one for any [math]t[/math].

Because [math]\Phi[/math] is positive, [math]H_t(iy)[/math] is positive for any [math]y[/math], and hence there are no zeroes on the imaginary axis.

Let [math]\sigma_{max}(t)[/math] denote the largest imaginary part of a zero of [math]H_t[/math], thus [math]\sigma_{max}(t)=0[/math] if and only if [math]t \geq \Lambda[/math]. It is known that the quantity [math]\frac{1}{2} \sigma_{max}(t)^2 + t[/math] is non-increasing in time whenever [math]\sigma_{max}(t)\gt0[/math] (see [KKL2009, Proposition A]. In particular we have

[math]\displaystyle \Lambda \leq t + \frac{1}{2} \sigma_{max}(t)^2[/math]

for any [math]t[/math].

The zeroes [math]z_j(t)[/math] of [math]H_t[/math] obey the system of ODE

[math]\partial_t z_j(t) = - \sum_{k \neq j} \frac{2}{z_k(t) - z_j(t)}[/math]

where the sum is interpreted in a principal value sense, and excluding those times in which [math]z_j(t)[/math] is a repeated zero. See dynamics of zeros for more details. Writing [math]z_j(t) = x_j(t) + i y_j(t)[/math], we can write the dynamics as

[math] \partial_t x_j = - \sum_{k \neq j} \frac{2 (x_k - x_j)}{(x_k-x_j)^2 + (y_k-y_j)^2} [/math]
[math] \partial_t y_j = \sum_{k \neq j} \frac{2 (y_k - y_j)}{(x_k-x_j)^2 + (y_k-y_j)^2} [/math]

where the dependence on [math]t[/math] has been omitted for brevity.

In [KKL2009, Theorem 1.4], it is shown that for any fixed [math]t\gt0[/math], the number [math]N_t(T)[/math] of zeroes of [math]H_t[/math] with real part between 0 and T obeys the asymptotic

[math]N_t(T) = \frac{T}{4\pi} \log \frac{T}{4\pi} - \frac{T}{4\pi} + \frac{t}{16} \log T + O(1) [/math]

as [math]T \to \infty[/math] (caution: the error term here is not uniform in t). Also, the zeroes behave like an arithmetic progression in the sense that

[math] z_{k+1}(t) - z_k(t) = (1+o(1)) \frac{4\pi}{\log |z_k|(t)} = (1+o(1)) \frac{4\pi}{\log k} [/math]

as [math]k \to +\infty[/math].

See asymptotics of H_t for asymptotics of the function [math]H_t[/math], and Effective bounds on H_t and Effective bounds on H_t - second approach for explicit bounds.

Threads

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Test problem

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Zero-free regions

See Zero-free regions.

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