Omega(N) - Revision history

OBryant: Added "inf" to final equation

2010-05-13T18:01:08Z

Added "inf" to final equation

← Older revision		Revision as of 11:01, 13 May 2010
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	<math> \omega(N,g) = \frac 1N \left( \sum_{1\leq i , j \leq n} (n+1-\max\{i,j\}) \cos(2\pi (x_i - x_j)) \right)</math>		<math> \omega(N,g) = \frac 1N \left( \sum_{1\leq i , j \leq n} (n+1-\max\{i,j\}) \cos(2\pi (x_i - x_j)) \right)</math>

	Setting <math> y_k = \tan(\pi x_k)</math> (and not being concerned about the possibility </math> y_k=\infty </math>), we want to handle		Setting <math> y_k = \tan(\pi x_k)</math> (and not being concerned about the possibility <math> y_k=\infty </math>), we want to handle

	<math> \omega(N,g) =\frac1N \sum_{i,j}^N (n+1-\max\{i,j\}) \frac{ \left(y_i y_j+y_i-y_j+1\right) \left(y_i y_j-y_i+y_j+1\right)}{\left(y_i^2+1\right) \left(y_j^2+1\right)}</math>		<math> \omega(N,g) =\frac1N \sum_{i,j}^N (n+1-\max\{i,j\}) \frac{ \left(y_i y_j+y_i-y_j+1\right) \left(y_i y_j-y_i+y_j+1\right)}{\left(y_i^2+1\right) \left(y_j^2+1\right)}</math>
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	A variant of the above expression for <math>\omega(N)</math> is		A variant of the above expression for <math>\omega(N)</math> is

	:<math> \omega(N) = \frac1N \left( \binom{N+1}{2} +2 \sum_{i=1}^{n-1} \sum_{j=i+1}^N (n+1-j) \frac{ \left(y_i y_j+y_i-y_j+1\right) \left(y_i y_j-y_i+y_j+1\right)}{\left(y_i^2+1\right) \left(y_j^2+1\right)}\right).</math>		:<math> \omega(N) = \inf_{y_i\in {\mathbb R}} \frac1N \left( \binom{N+1}{2} +2 \sum_{i=1}^{n-1} \sum_{j=i+1}^N (n+1-j) \frac{ \left(y_i y_j+y_i-y_j+1\right) \left(y_i y_j-y_i+y_j+1\right)}{\left(y_i^2+1\right) \left(y_j^2+1\right)}\right).</math>


	== Exact values ==		== Exact values ==

Klasm at 13:11, 2 February 2010

2010-02-02T13:11:10Z

Klasm at 09:30, 2 February 2010

2010-02-02T09:30:04Z

← Older revision		Revision as of 02:30, 2 February 2010
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	<math>\Omega(3)=\frac{1}{2}</math> at <math>y_1=0, y_2 = -\sqrt{15}, y_3 = \sqrt{\frac{5}{3}}</math>		<math>\Omega(3)=\frac{1}{2}</math> at <math>y_1=0, y_2 = -\sqrt{15}, y_3 = \sqrt{\frac{5}{3}}</math>

	<math>\Omega(4)=\frac{1}{2}</math>		<math>\Omega(4)=\frac{1}{2}</math> at <math>y_1=0, y_2=\infty, y_3=\infty</math>

Klasm at 09:11, 2 February 2010

2010-02-02T09:11:22Z

← Older revision		Revision as of 02:11, 2 February 2010
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	:<math> \omega(N) = \frac1N \left( \binom{N+1}{2} +2 \sum_{i=1}^{n-1} \sum_{j=i+1}^N (n+1-j) \frac{ \left(y_i y_j+y_i-y_j+1\right) \left(y_i y_j-y_i+y_j+1\right)}{\left(y_i^2+1\right) \left(y_j^2+1\right)}\right).</math>		:<math> \omega(N) = \frac1N \left( \binom{N+1}{2} +2 \sum_{i=1}^{n-1} \sum_{j=i+1}^N (n+1-j) \frac{ \left(y_i y_j+y_i-y_j+1\right) \left(y_i y_j-y_i+y_j+1\right)}{\left(y_i^2+1\right) \left(y_j^2+1\right)}\right).</math>


			== Exact values ==

			These are the currently known values of <math>\Omega(N)</math>

			<math>\Omega(1)=1</math> at <math>y_1=0</math>

			<math>\Omega(2)=\frac{1}{2}</math> at <math>y_1=0, y_2=-\infty</math>

			<math>\Omega(3)=\frac{1}{2}</math> at <math>y_1=0, y_2 = -\sqrt{15}, y_3 = \sqrt{\frac{5}{3}}</math>

			<math>\Omega(4)=\frac{1}{2}</math>


	== Upper bounds for <math> \omega(N)</math> ==		== Upper bounds for <math> \omega(N)</math> ==

Klasm at 08:55, 2 February 2010

2010-02-02T08:55:07Z

← Older revision		Revision as of 01:55, 2 February 2010
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	Using a numerical optimization routine the following upper bounds on <math> \omega(N)</math> were found.		Using a numerical optimization routine the following upper bounds on <math> \omega(N)</math> were found.

	For <math> N=1,...,71</math> bound given below were found. the bounds are given as a Mathematica formatted list with each item of the form {upper bound,{ list of values for the y[i]}}		For <math> N=1,...,84</math> bound given below were found. the bounds are given as a Mathematica formatted list with each item of the form {upper bound,{ list of values for the y[i]}}

			A plot of the bounds for <math> N=1,...,84</math>
			[http://abel.math.umu.se/~klasm/Data/EDP/omega-bound-plot.pdf]

	<nowiki>		<nowiki>

Klasm at 08:19, 2 February 2010

2010-02-02T08:19:11Z

@@ Line 612: / Line 612: @@
         y[59] -> 4.563568980061795, y[61] -> -0.9119451367851588,
         y[67] -> -0.33255497165614445, y[71] -> 4.939650926563234,
-        y[73] -> -0.08134651616351249}}
+        y[73] -> -0.08134651616351249}},
+}

OBryant: /* Upper bounds for \omega(N) */

2010-02-01T20:06:04Z

Upper bounds for \omega(N)

← Older revision		Revision as of 13:06, 1 February 2010
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	{1,		{1,
	{y[1]->0}},		{y[1]->0}},
	{~~0.5~~,		{1/2,
	{y[2] -> -~~1.462885206912493*10^8~~}},		{y[2] -> -Infinity}},
	{0.4999999999999996,		{0.4999999999999996,
	{y[2] -> 3.8729833462074157, y[3] -> -1.2909944487358054}},		{y[2] -> 3.8729833462074157, y[3] -> -1.2909944487358054}},
	{0.4999999999999998,		{0.4999999999999998,
	{y[2] -> -~~3.91064817038239310^8~~, y[3] -> ~~7.31001135706977610^8~~}},		{y[2] -> -Infinity, y[3] -> Infinity}},
	{0.4801338227148131,		{0.4801338227148131,
	{y[2] -> 9.463147074619814, y[3] -> -4.8114722029865895, y[5] -> 1.0297311434019392}},		{y[2] -> 9.463147074619814, y[3] -> -4.8114722029865895, y[5] -> 1.0297311434019392}},

OBryant: /* Definitions */ fixed (n+1-j) to (n+1-max{i,j})

2010-02-01T19:54:15Z

Definitions: fixed (n+1-j) to (n+1-max{i,j})

← Older revision		Revision as of 12:54, 1 February 2010
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	== Definitions ==		== Definitions ==

	In this [http://gowers.wordpress.com/2010/01/26/edp3-a-very-brief-report-on-where-we-are/#comment-5566] comment Terrence Tao defined a function <math> \omega(N)</math> which is a lower bound for the discrepancy of a completely multiplicative function. If <math> \omega(N)~~</math>~~</math> is unbounded as a function of <math> N</math> then all compeltely mutplicative functions will have unbounded discrepancy.		In this [http://gowers.wordpress.com/2010/01/26/edp3-a-very-brief-report-on-where-we-are/#comment-5566] comment Terrence Tao defined a function <math> \omega(N)</math> which is a lower bound for the discrepancy of a completely multiplicative function. If <math> \omega(N)</math> is unbounded as a function of <math> N</math> then all compeltely mutplicative functions will have unbounded discrepancy.


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	Setting <math> g(k)=e^{2\pi i x_k}</math>, the multiplicativity of <math> g</math> becomes <math> x_{mn}=x_m+x_n</math>. We wish to handle:		Setting <math> g(k)=e^{2\pi i x_k}</math>, the multiplicativity of <math> g</math> becomes <math> x_{mn}=x_m+x_n</math>. We wish to handle:

	<math> \omega(N,g) = \frac 1N \left( \sum_{1\leq i , j \leq n} (n+1-j) \cos(2\pi (x_i - x_j)) \right)</math>		<math> \omega(N,g) = \frac 1N \left( \sum_{1\leq i , j \leq n} (n+1-\max\{i,j\}) \cos(2\pi (x_i - x_j)) \right)</math>

	Setting <math> y_k = \tan(\pi x_k)</math> (and not being concerned about the possibility </math> y_k=\infty </math>), we want to handle		Setting <math> y_k = \tan(\pi x_k)</math> (and not being concerned about the possibility </math> y_k=\infty </math>), we want to handle

	<math> \omega(N,g) =\frac1N \sum_{i,j}^N ~~\frac{~~(n-j+1) \left(y_i y_j+y_i-y_j+1\right) \left(y_i y_j-y_i+y_j+1\right)}{\left(y_i^2+1\right) \left(y_j^2+1\right)}</math>		<math> \omega(N,g) =\frac1N \sum_{i,j}^N (n+1-\max\{i,j\}) \frac{ \left(y_i y_j+y_i-y_j+1\right) \left(y_i y_j-y_i+y_j+1\right)}{\left(y_i^2+1\right) \left(y_j^2+1\right)}</math>

	where the multiplicativity of <math> g</math> is expressed through		where the multiplicativity of <math> g</math> is expressed through

	<math> y_{mn} = \frac{y_m+y_n}{1-y_my_n}</math>		<math> y_{mn} = \frac{y_m+y_n}{1-y_my_n}.</math>

			A variant of the above expression for <math>\omega(N)</math> is

			:<math> \omega(N) = \frac1N \left( \binom{N+1}{2} +2 \sum_{i=1}^{n-1} \sum_{j=i+1}^N (n+1-j) \frac{ \left(y_i y_j+y_i-y_j+1\right) \left(y_i y_j-y_i+y_j+1\right)}{\left(y_i^2+1\right) \left(y_j^2+1\right)}\right).</math>

	== Upper bounds for <math> \omega(N)</math> ==		== Upper bounds for <math> \omega(N)</math> ==

Klasm at 19:22, 1 February 2010

2010-02-01T19:22:03Z

← Older revision		Revision as of 12:22, 1 February 2010
Line 38:		Line 38:

	{		{
			{1,
			{y[1]->0}},
	{0.5,		{0.5,
	{y[2] -> -1.462885206912493*10^8}},		{y[2] -> -1.462885206912493*10^8}},

Klasm at 19:20, 1 February 2010

2010-02-01T19:20:42Z

Show changes

← Older revision		Revision as of 06:11, 2 February 2010
Line 632:		Line 632:
	{0.5347953853151964,		{0.5347953853151964,
	{y[2] -> 826616.6264389796,		{y[2] -> 826616.6264389796,
	y[3] -> -3.2445194137959525, y[5] -> 1.1337838988884934*^6,		y[3] -> -3.2445194137959525, y[5] -> 1.1337838988884934*10^6,
	y[7] -> -0.053183445803101505, y[11] -> 813021.9382822418,		y[7] -> -0.053183445803101505, y[11] -> 813021.9382822418,
	y[13] -> -0.35844167042277564, y[17] -> 5.339307410851948,		y[13] -> -0.35844167042277564, y[17] -> 5.339307410851948,
	y[19] -> 0.219050505622947, y[23] -> 499949.5722852526,		y[19] -> 0.219050505622947, y[23] -> 499949.5722852526,
	y[29] -> -603285.5606499284, y[31] -> 0.2483912347244814,		y[29] -> -603285.5606499284, y[31] -> 0.2483912347244814,
	y[37] -> -0.5099629142111536, y[41] -> 1.0913022635794224*^6,		y[37] -> -0.5099629142111536, y[41] -> 1.0913022635794224*10^6,
	y[43] -> 0.16012687574803594, y[47] -> 1.4162187543877018,		y[43] -> 0.16012687574803594, y[47] -> 1.4162187543877018,
	y[53] -> -708771.0434949356, y[59] -> 4.926882173473444,		y[53] -> -708771.0434949356, y[59] -> 4.926882173473444,
Line 669:		Line 669:
	{0.5446127381623379,		{0.5446127381623379,
	{y[2] -> -5.014708963920774,		{y[2] -> -5.014708963920774,
	y[3] -> 1.6480739954794408*^6, y[5] -> -1.8054886612263317,		y[3] -> 1.6480739954794408*10^6, y[5] -> -1.8054886612263317,
	y[7] -> 0.8906342291461616, y[11] -> -1.707073396117987,		y[7] -> 0.8906342291461616, y[11] -> -1.707073396117987,
	y[13] -> 1.5628224634363812, y[17] -> -0.6611721077818026,		y[13] -> 1.5628224634363812, y[17] -> -0.6611721077818026,
Line 678:		Line 678:
	y[53] -> -0.7210168316402203, y[59] -> -0.11958679378827772,		y[53] -> -0.7210168316402203, y[59] -> -0.11958679378827772,
	y[61] -> -2.922500109233194, y[67] -> -288350.9761325468,		y[61] -> -2.922500109233194, y[67] -> -288350.9761325468,
	y[71] -> 0.18284299894759842, y[73] -> -4.0197907085561515*^6,		y[71] -> 0.18284299894759842, y[73] -> -4.0197907085561515*10^6,
	y[79] -> 16.793919038925115}}, {0.547862473559509,		y[79] -> 16.793919038925115}}, {0.547862473559509,
	{y[2] -> -6.547748625076605, y[3] -> -1.6611578546016958,		{y[2] -> -6.547748625076605, y[3] -> -1.6611578546016958,