Dynamics of zeros - Revision history

Teorth: /* Bounding \Lambda */

2018-04-05T16:39:00Z

Bounding \Lambda

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	'''Proof''' (Sketch) By the previous lemma, all the zeroes <math>x_j+iy_j</math> of <math>H_{t}</math> for <math>0 < t < t_0</math>, <math>0 \leq x \leq T</math> and <math>y > \varepsilon</math> obey <math>\partial_t y_j < 0</math>. By a continuity argument this shows in fact that there are no such zeroes. In particular all zeroes of <math>H_{t_0}</math> have imaginary part at most <math>\varepsilon</math>, and the claim then follows from Theorem 1. <math>\Box</math>		'''Proof''' (Sketch) By the previous lemma, all the zeroes <math>x_j+iy_j</math> of <math>H_{t}</math> for <math>0 < t < t_0</math>, <math>0 \leq x \leq T</math> and <math>y > \varepsilon</math> obey <math>\partial_t y_j < 0</math>. By a continuity argument this shows in fact that there are no such zeroes. In particular all zeroes of <math>H_{t_0}</math> have imaginary part at most <math>\varepsilon</math>, and the claim then follows from Theorem 1. <math>\Box</math>

			Corollary 3 establishes a large zero free region at positive times by exploiting a similarly large zero free region at time zero, combined with a "barrier" to keep zeroes from entering the region. See <A HREF="https://drive.google.com/file/d/1KtD0vUc6WATXWadsCve8mWA8MMAev-qI/view">this graphic</A> for a visual.

	== Derivative analysis ==		== Derivative analysis ==

Teorth: /* Dynamics of a simple zero */

2018-04-04T19:56:40Z

Dynamics of a simple zero

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	and thus		and thus

	:<math>\displaystyle \partial_t z_j(t) = 2 \lim_{z \to z_j(t)} \frac{\partial_z H_t(z)}{H_t(z)} - \frac{1}{z-z_j(t)}.</math>		:<math>\displaystyle \partial_t z_j(t) = 2 \lim_{z \to z_j(t)} \left( \frac{\partial_z H_t(z)}{H_t(z)} - \frac{1}{z-z_j(t)} \right).</math>

	As <math>H_t</math> is even, of order 1, and has no zero at the origin, we see from the [https://en.wikipedia.org/wiki/Weierstrass_factorization_theorem Hadamard factorisation theorem] that		As <math>H_t</math> is even, of order 1, and has no zero at the origin, we see from the [https://en.wikipedia.org/wiki/Weierstrass_factorization_theorem Hadamard factorisation theorem] that

Teorth: /* Dynamics of a repeated zero */

2018-04-04T19:56:05Z

Dynamics of a repeated zero

← Older revision		Revision as of 12:56, 4 April 2018
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	If <math>z_j</math> is a zero with maximal imaginary part, and this imaginary part is positive, then the left-hand side of this equation has positive imaginary part, which is absurd. Thus there are no zeroes with positive imaginary part, and similarly none with negative imaginary part, hence all zeroes are real. (One can also establish these claims using Sturm's theorem; see [https://terrytao.wordpress.com/2018/01/27/polymath15-first-thread-computing-h_t-asymptotics-and-dynamics-of-zeroes/#comment-491899 this comment].)		If <math>z_j</math> is a zero with maximal imaginary part, and this imaginary part is positive, then the left-hand side of this equation has positive imaginary part, which is absurd. Thus there are no zeroes with positive imaginary part, and similarly none with negative imaginary part, hence all zeroes are real. (One can also establish these claims using Sturm's theorem; see [https://terrytao.wordpress.com/2018/01/27/polymath15-first-thread-computing-h_t-asymptotics-and-dynamics-of-zeroes/#comment-491899 this comment].)

	'''Remark''' The <math>z_j</math> minimize the Hamiltonian <math>\sum_{j \neq l} \log \frac{1}{\|z_j-z_l\|} + \sum_j \frac{1}{2} \|z_j\|^2</math> and are also known as ''Fekete points'' in random matrix theory. For large values of <math>k</math>, the distribution of these Fekete points approaches a semicircular distribution.		'''Remark''' The <math>z_j</math> minimize the Hamiltonian <math>\sum_{j \neq l} \log \frac{1}{\|z_j-z_l\|} + \sum_j \frac{1}{2} \|z_j\|^2</math> and are also known as ''Fekete points'' in random matrix theory. For large values of <math>k</math>, the distribution of these Fekete points approaches a semicircular distribution. Up to a scaling, the <math>z_j</math> are also the zeroes of the degree <math>k</math> Hermite polynomial.

	== Bounding <math>\Lambda</math> ==		== Bounding <math>\Lambda</math> ==

Teorth: /* Derivative analysis */

2018-03-29T16:53:26Z

Derivative analysis

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	:<math> \partial_t S(t) \leq O( \frac{\log^2 X}{X} S(t) + \frac{\log^2 X}{X} )</math>		:<math> \partial_t S(t) \leq O( \frac{\log^2 X}{X} S(t) + \frac{\log^2 X}{X} )</math>
	and in particular by Gronwall's inequality		and in particular by Gronwall's inequality
	:<math> S(t) \leq \exp( O( \frac{\log^2 X}{X} ) ) S(t) + O( \frac{\log^2 X}{X} )</math>		:<math> S(t) \leq \exp( O( \frac{\log^2 X}{X} ) ) S(0) + O( \frac{\log^2 X}{X} )</math>
	for <math>0 \leq t \leq 1/2</math> (say), which should be good enough to create non-trivial zero-free regions for <math>H_t</math>.		for <math>0 \leq t \leq 1/2</math> (say), which should be good enough to create non-trivial zero-free regions for <math>H_t</math>.

Teorth: /* Bounding \Lambda */

2018-03-28T22:41:05Z

Bounding \Lambda

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	:<math> \frac{y_k-y_j}{(x_k-x_j)^2 + (y_k-y_j)^2} + \frac{-y_k-y_j}{(x_k-x_j)^2 + (-y_k-y_j)^2} \leq 0.</math>		:<math> \frac{y_k-y_j}{(x_k-x_j)^2 + (y_k-y_j)^2} + \frac{-y_k-y_j}{(x_k-x_j)^2 + (-y_k-y_j)^2} \leq 0.</math>
	Cross-multiplying and canceling, the above inequality eventually simplifies to		Cross-multiplying and canceling, the above inequality eventually simplifies to
	:<math> y_k^2 \leq (x_k -x_j)^2 + y_j^2)</math>		:<math> y_k^2 \leq (x_k -x_j)^2 + y_j^2</math>
	which is automatic due to the hypothesis. <math>\Box</math>		which is automatic due to the hypothesis. <math>\Box</math>

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	'''Proof''' (Sketch) By the previous lemma, all the zeroes <math>x_j+iy_j</math> of <math>H_{t}</math> for <math>0 < t < t_0</math>, <math>0 \leq x \leq T</math> and <math>y > \varepsilon</math> obey <math>\partial_t y_j < 0</math>. By a continuity argument this shows in fact that there are no such zeroes. In particular all zeroes of <math>H_{t_0}</math> have imaginary part at most <math>\varepsilon</math>, and the claim then follows from Theorem 1. <math>\Box</math>		'''Proof''' (Sketch) By the previous lemma, all the zeroes <math>x_j+iy_j</math> of <math>H_{t}</math> for <math>0 < t < t_0</math>, <math>0 \leq x \leq T</math> and <math>y > \varepsilon</math> obey <math>\partial_t y_j < 0</math>. By a continuity argument this shows in fact that there are no such zeroes. In particular all zeroes of <math>H_{t_0}</math> have imaginary part at most <math>\varepsilon</math>, and the claim then follows from Theorem 1. <math>\Box</math>


	== Derivative analysis ==		== Derivative analysis ==

Teorth: /* Derivative analysis */

2018-03-26T23:26:52Z

Derivative analysis

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	Next we consider the contribution of those terms with <math>x_j > 0, y_j \neq 0</math> and <math>\|x_k-x_j\| \geq 1</math>. The <math>(x_j-x_k)</math> terms can be summed using the Riemann von Mangoldt formula to be at most <math>O( \frac{\log^2 X}{X} S(t))</math>. The <math>y_j-y_k</math> terms, when paired with their complex conjugates, are negative thanks to Lemma 2.		Next we consider the contribution of those terms with <math>x_j > 0, y_j \neq 0</math> and <math>\|x_k-x_j\| \geq 1</math>. The <math>(x_j-x_k)</math> terms can be summed using the Riemann von Mangoldt formula to be at most <math>O( \frac{\log^2 X}{X} S(t))</math>. The <math>y_j-y_k</math> terms, when paired with their complex conjugates, are negative thanks to Lemma 2.

	Next consider the contribution of those terms with <math>x_j > 0</math> and <math>y_j \leq 0</math> that have not previously been treated. The <math>(y_j-y_k)</math> term is now negative (and equal to <math>\frac{-1}{y_k} e^{-x_k/X}$ when looking at the complex conjugate), and the <math>(x_j-x_k)</math> term is negative unless <math>x_j \geq x_k</math>. The terms here sum to at most		Next consider the contribution of those terms with <math>x_j > 0</math> and <math>y_j \leq 0</math> that have not previously been treated. The <math>(y_j-y_k)</math> term is now negative (and equal to <math>\frac{-1}{y_k} e^{-x_k/X}</math> when looking at the complex conjugate), and the <math>(x_j-x_k)</math> term is negative unless <math>x_j \geq x_k</math>. The terms here sum to at most
	:<math> \frac{1}{X} \sum_{k: x_k, y_k > 0} y_k e^{-x_k/X} ( \sum_{j: x_j \geq x_k; y_j \leq 0} \frac{2(x_j-x_k)}{(x_k-x_j)^2 + y_k^2} - \frac{1}{y_k^2})</math>		:<math> \frac{1}{X} \sum_{k: x_k, y_k > 0} y_k e^{-x_k/X} ( \sum_{j: x_j \geq x_k; y_j \leq 0} \frac{2(x_j-x_k)}{(x_k-x_j)^2 + y_k^2} - \frac{1}{y_k^2})</math>
	which by the Riemann von Mangoldt formula is		which by the Riemann von Mangoldt formula is

Teorth at 23:26, 26 March 2018

2018-03-26T23:26:26Z

← Older revision		Revision as of 16:26, 26 March 2018
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	'''Proof''' (Sketch) By the previous lemma, all the zeroes <math>x_j+iy_j</math> of <math>H_{t}</math> for <math>0 < t < t_0</math>, <math>0 \leq x \leq T</math> and <math>y > \varepsilon</math> obey <math>\partial_t y_j < 0</math>. By a continuity argument this shows in fact that there are no such zeroes. In particular all zeroes of <math>H_{t_0}</math> have imaginary part at most <math>\varepsilon</math>, and the claim then follows from Theorem 1. <math>\Box</math>		'''Proof''' (Sketch) By the previous lemma, all the zeroes <math>x_j+iy_j</math> of <math>H_{t}</math> for <math>0 < t < t_0</math>, <math>0 \leq x \leq T</math> and <math>y > \varepsilon</math> obey <math>\partial_t y_j < 0</math>. By a continuity argument this shows in fact that there are no such zeroes. In particular all zeroes of <math>H_{t_0}</math> have imaginary part at most <math>\varepsilon</math>, and the claim then follows from Theorem 1. <math>\Box</math>


			== Derivative analysis ==

			Let <math>X</math> be a large quantity, and consider the expression

			:<math>S(t) := \sum_{k: x_k, y_k > 0} y_k e^{-x_k/X}</math>

			where <math>x_k+iy_k</math> ranges over the zeroes of <math>H_t</math>. The exponential decay factor should allow us to justify termwise differentiation etc., so we obtain

			:<math>\partial_t S(t) := \sum_{k: x_k, y_k > 0} \partial_t y_k e^{-x_k/X} - \frac{1}{X} y_k e^{-x_k/X} \partial_t x_k.</math>

			We formally have
			:<math> \partial_t y_k = \sum_{j \neq k} \frac{2 (y_j - y_k)}{(x_k-x_j)^2 + (y_k-y_j)^2} </math>
			and
			:<math> \partial_t x_k = -\sum_{j \neq k} \frac{2 (x_j - x_k)}{(x_k-x_j)^2 + (y_k-y_j)^2} </math>

			so the right-hand side is equal to

			:<math>\partial_t S(t) := \sum_{k: x_k, y_k > 0} \sum_{j \neq k} \frac{2}{(x_k-x_j)^2 + (y_k-y_j)^2} ( (y_j - y_k) e^{-x_k/X} + \frac{1}{X} y_k e^{-x_k/X} (x_j - x_k) ).</math>

			Consider first the contribution of those terms with <math>x_j \leq 0</math>. The <math>(x_j-x_k)</math> term is negative. The <math>y_j-y_k</math> term is also negative if <math>y_j \neq 0</math>, so suppose <math>y_j \neq 0</math>. By symmetry we get a contribution of
			:<math> e^{-x_k/X} ( \frac{y_j - y_k}{(x_k-x_j)^2 + (y_k-y_j)^2} + \frac{-y_j - y_k}{(x_k-x_j)^2 + (y_k+y_j)^2} </math>
			and this should also be negative by Lemma 2 (provided we can ensure that say <math>x_j \leq -1</math> and <math>x_k \geq 1</math>, but this should be easy to do.

			Next we consider the contribution of those terms with <math>x_j > 0, y_j \neq 0</math> and <math>\|x_k-x_j\| \geq 1</math>. The <math>(x_j-x_k)</math> terms can be summed using the Riemann von Mangoldt formula to be at most <math>O( \frac{\log^2 X}{X} S(t))</math>. The <math>y_j-y_k</math> terms, when paired with their complex conjugates, are negative thanks to Lemma 2.

			Next consider the contribution of those terms with <math>x_j > 0</math> and <math>y_j \leq 0</math> that have not previously been treated. The <math>(y_j-y_k)</math> term is now negative (and equal to <math>\frac{-1}{y_k} e^{-x_k/X}$ when looking at the complex conjugate), and the <math>(x_j-x_k)</math> term is negative unless <math>x_j \geq x_k</math>. The terms here sum to at most
			:<math> \frac{1}{X} \sum_{k: x_k, y_k > 0} y_k e^{-x_k/X} ( \sum_{j: x_j \geq x_k; y_j \leq 0} \frac{2(x_j-x_k)}{(x_k-x_j)^2 + y_k^2} - \frac{1}{y_k^2})</math>
			which by the Riemann von Mangoldt formula is
			:<math> \frac{1}{X} \sum_{k: x_k, y_k > 0} y_k e^{-x_k/X} ( O( \log^2 x_k) + \frac{O( \log x_k) }{y_k} - \frac{1}{y_k^2})</math>
			which by Young's inequality (and another appeal to Riemann von Mangoldt to handle the tail when say <math>x_k \geq X^{20}</math>) sums to
			:<math> \frac{1}{X} \sum_{k: x_k, y_k > 0} y_k e^{-x_k/X} O( \log^2 x_k) = O( \frac{\log^2 X}{X} S(t) + X^{-10} ).</math>

			Finally, we consider the contribution of those terms with <math>x_j,y_j > 0</math> and <math>\|x_k-x_j\| \leq 1</math>. Here we can symmetrise to get

			:<math> \sum_{j,k: x_k,y_k,x_j,y_k > 0: \|x_k-x_j\| \leq 1} \frac{1}{(x_k-x_j)^2 + (y_k-y_j)^2} ( (y_j - y_k) (e^{-x_k/X} - e^{-x_j/X}) + \frac{1}{X} (y_k e^{-x_k/X} - y_j e^{-x_j/X}) (x_j - x_k) ).</math>
			By Taylor expansion we have
			:<math>e^{-x_k/X} - e^{-x_j/X} = e^{-x_k/X} ((x_j-x_k)/X + O( \|x_j-x_k\|/X^2) )</math>
			and also
			:<math>y_k e^{-x_k/X} - y_j e^{-x_j/X} = (y_k - y_j) e^{-x_k/X} + O(\|x_j-x_k\|/X)</math>
			so we can cancel down to
			:<math> \sum_{j,k: x_k,y_k,x_j,y_k > 0: \|x_k-x_j\| \leq 1} \frac{1}{(x_k-x_j)^2 + (y_k-y_j)^2} (y_j - y_k) O( \|x_j-x_k\|/X^2 ) e^{-x_k/X}</math>
			which by the arithmetic mean geometric mean inequality followed by Riemann von Mangoldt is bounded by
			:<math> O( \frac{1}{X^2} \sum_{j,k: x_k,y_k,x_j,y_k > 0: \|x_k-x_j\| \leq 1} e^{-x_k/X} )</math>
			:<math> \leq O( \frac{1}{X^2} \sum_{x_k > 0} e^{-x_k/X} \log X )</math>
			:<math> \leq O( \frac{\log^2 X}{X} ).</math>

			Putting all this together we obtain
			:<math> \partial_t S(t) \leq O( \frac{\log^2 X}{X} S(t) + \frac{\log^2 X}{X} )</math>
			and in particular by Gronwall's inequality
			:<math> S(t) \leq \exp( O( \frac{\log^2 X}{X} ) ) S(t) + O( \frac{\log^2 X}{X} )</math>
			for <math>0 \leq t \leq 1/2</math> (say), which should be good enough to create non-trivial zero-free regions for <math>H_t</math>.

Teorth: /* Dynamics of a simple zero */

2018-02-11T05:32:32Z

Dynamics of a simple zero

← Older revision		Revision as of 22:32, 10 February 2018
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	where the sum is in a principal value sense. Thus we have		where the sum is in a principal value sense. Thus we have

	:<math>\displaystyle \partial_t z_j(t) = - 2 \sum_{k \neq j} \frac{1}{z_j(t) - z_k(t)} </math>		:<math>\displaystyle \partial_t z_j(t) = 2 \sum_{k \neq j} \frac{1}{z_j(t) - z_k(t)} </math>

	where the sum is again in a principal value sense (cf. [CSV1994, Lemma 2.4]).		where the sum is again in a principal value sense (cf. [CSV1994, Lemma 2.4]).

	== Dynamics of a repeated zero ==		== Dynamics of a repeated zero ==

Teorth: /* Dynamics of a repeated zero */

2018-02-03T03:10:25Z

Dynamics of a repeated zero

← Older revision		Revision as of 20:10, 2 February 2018
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	'''Remark''' The <math>z_j</math> minimize the Hamiltonian <math>\sum_{j \neq l} \log \frac{1}{\|z_j-z_l\|} + \sum_j \frac{1}{2} \|z_j\|^2</math> and are also known as ''Fekete points'' in random matrix theory. For large values of <math>k</math>, the distribution of these Fekete points approaches a semicircular distribution.		'''Remark''' The <math>z_j</math> minimize the Hamiltonian <math>\sum_{j \neq l} \log \frac{1}{\|z_j-z_l\|} + \sum_j \frac{1}{2} \|z_j\|^2</math> and are also known as ''Fekete points'' in random matrix theory. For large values of <math>k</math>, the distribution of these Fekete points approaches a semicircular distribution.

			== Bounding <math>\Lambda</math> ==

			'''Theorem 1''' Suppose that all zeroes of <math>H_t(x+iy)=0</math> have imaginary part at most <math>\varepsilon</math>. Then <math>\Lambda \leq t + \frac{1}{2} \varepsilon^2</math>.

			'''Proof''' See [B1950, Theorem 13]. Informally, the claim follows by using the imaginary part dynamics
			:<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>
			to conclude that for the highest zero (with the maximum value of <math>y_j</math>), one has
			:<math> \partial_t y_j \leq - \frac{1}{y_j}</math>
			if <math>y_j > 0</math>, due to the attraction that that zero <math>x_j+iy_j</math> feels to its conjugate <math>x_j - iy_j</math>. (One has to exclude the repeated zeroes from this analysis somehow.) <math>\Box</math>

			In a similar spirit, we have

			'''Lemma 2''' Suppose <math>x_j+y_j</math> is a simple zero of <math>H_t</math> with <math>y_j>0</math> and no zeroes in the hyperbola <math>\{ x+iy: y > \sqrt{(x-x_j)^2 + y_j^2}</math>. Then <math>\partial_t y_j</math> is negative.

			'''Proof''' The contribution of the conjugate <math>x_j-iy_j</math> is negative, as is the contribution of any real zero <math>x_k</math>. It remains to show that the contribution of any other pair <math>x_k+iy_k, x_k - iy_k</math> is non-positive, that is to say that
			:<math> \frac{y_k-y_j}{(x_k-x_j)^2 + (y_k-y_j)^2} + \frac{-y_k-y_j}{(x_k-x_j)^2 + (-y_k-y_j)^2} \leq 0.</math>
			Cross-multiplying and canceling, the above inequality eventually simplifies to
			:<math> y_k^2 \leq (x_k -x_j)^2 + y_j^2)</math>
			which is automatic due to the hypothesis. <math>\Box</math>

			'''Corollary 3''' Suppose that one has parameters <math>t_0, T, \varepsilon > 0</math> obeying the following properties:
			* All the zeroes of <math>H_0(x+iy)=0</math> with <math>0 \leq x \leq T</math> are real.
			* There are no zeroes <math>H_t(x+iy)=0</math> with <math>0 \leq t \leq t_0</math> in the region <math>\{ x+iy: x \geq T; 1-2t \geq y^2 \geq \varepsilon^2 + (T-x)^2 \}</math>.
			* There are no zeroes <math>H_{t_0}(x+iy)=0</math> with <math>x > T</math> and <math>y \geq \varepsilon</math>.
			Then one has <math>\Lambda \leq t_0 + \frac{1}{2} \varepsilon^2</math>.

			'''Proof''' (Sketch) By the previous lemma, all the zeroes <math>x_j+iy_j</math> of <math>H_{t}</math> for <math>0 < t < t_0</math>, <math>0 \leq x \leq T</math> and <math>y > \varepsilon</math> obey <math>\partial_t y_j < 0</math>. By a continuity argument this shows in fact that there are no such zeroes. In particular all zeroes of <math>H_{t_0}</math> have imaginary part at most <math>\varepsilon</math>, and the claim then follows from Theorem 1. <math>\Box</math>

Teorth: /* Dynamics of a repeated zero */

2018-01-31T01:51:56Z

Dynamics of a repeated zero

← Older revision		Revision as of 18:51, 30 January 2018
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	This already shows that <math>P_k</math> cannot have a repeated zero (because <math>\partial_{zz} P_k</math> would then vanish to lower order than either <math>z \partial_z P_k</math> or <math>k P_k</math>). Factoring <math>P_k(z) = (z-z_1) \dots (z-z_k)</math>, and evaluating the above at some zero <math>z_j</math>, we conclude that		This already shows that <math>P_k</math> cannot have a repeated zero (because <math>\partial_{zz} P_k</math> would then vanish to lower order than either <math>z \partial_z P_k</math> or <math>k P_k</math>). Factoring <math>P_k(z) = (z-z_1) \dots (z-z_k)</math>, and evaluating the above at some zero <math>z_j</math>, we conclude that

	:<math> -2 \sum_{j \neq k} \frac{1}{~~z_k~~-z_j} + z_j = 0.</math>		:<math> -2 \sum_{j \neq l} \frac{1}{z_l-z_j} + z_j = 0.</math>

	If <math>z_j</math> is a zero with maximal imaginary part, and this imaginary part is positive, then the left-hand side of this equation has positive imaginary part, which is absurd. Thus there are no zeroes with positive imaginary part, and similarly none with negative imaginary part, hence all zeroes are real. (One can also establish these claims using Sturm's theorem; see [https://terrytao.wordpress.com/2018/01/27/polymath15-first-thread-computing-h_t-asymptotics-and-dynamics-of-zeroes/#comment-491899 this comment].)		If <math>z_j</math> is a zero with maximal imaginary part, and this imaginary part is positive, then the left-hand side of this equation has positive imaginary part, which is absurd. Thus there are no zeroes with positive imaginary part, and similarly none with negative imaginary part, hence all zeroes are real. (One can also establish these claims using Sturm's theorem; see [https://terrytao.wordpress.com/2018/01/27/polymath15-first-thread-computing-h_t-asymptotics-and-dynamics-of-zeroes/#comment-491899 this comment].)

			'''Remark''' The <math>z_j</math> minimize the Hamiltonian <math>\sum_{j \neq l} \log \frac{1}{\|z_j-z_l\|} + \sum_j \frac{1}{2} \|z_j\|^2</math> and are also known as ''Fekete points'' in random matrix theory. For large values of <math>k</math>, the distribution of these Fekete points approaches a semicircular distribution.