The Riemann zeta function and tuning: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-04-07 02:28:56 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-04-07 04:14:12 UTC</tt>.<br>

: The original revision id was <tt>~~217990126~~</tt>.<br>

: The original revision id was <tt>218005584</tt>.<br>

: The revision comment was: <tt></tt><br>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>

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[[math]]

If s is greater than one, this does converge. However, we might want to make a few adjustments. For one thing, if the error is low enough that the tuning is consistent, then the error of the square of a prime is twice that of the prime, of the cube tripled, and so forth until the error becomes inconsistent. When the weighting ~~used~~ logarithms and error measures are consistent, then the logarithmic weighting ~~canceled~~ this effect out, so we might consider that prime powers were implicitly included in the Tenney-Euclidean measure. We can go ahead and include them by adding a factor of 1/n for each prime power p^n. A somewhat peculiar but useful way to write the result of doing this is in terms of the [[http://en.wikipedia.org/wiki/Von_Mangoldt_function|Von Mangoldt function]], an [[http://en.wikipedia.org/wiki/Arithmetic_function|arithmetic function]] on positive integers which is equal to ln p on prime powers p^n, and is zero elsewhere. This is written using a capital lambda, and in terms of it we can include prime powers in our error function as

If s is greater than one, this does converge. However, we might want to make a few adjustments. For one thing, if the error is low enough that the tuning is consistent, then the error of the square of a prime is twice that of the prime, of the cube tripled, and so forth until the error becomes inconsistent. When the weighting uses logarithms and error measures are consistent, then the logarithmic weighting canceles this effect out, so we might consider that prime powers were implicitly included in the Tenney-Euclidean measure. We can go ahead and include them by adding a factor of 1/n for each prime power p^n. A somewhat peculiar but useful way to write the result of doing this is in terms of the [[http://en.wikipedia.org/wiki/Von_Mangoldt_function|Von Mangoldt function]], an [[http://en.wikipedia.org/wiki/Arithmetic_function|arithmetic function]] on positive integers which is equal to ln p on prime powers p^n, and is zero elsewhere. This is written using a capital lambda, and in terms of it we can include prime powers in our error function as

[[math]]

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where the summation is taken formally over all positive integers, though only the primes and prime powers make a nonzero contribution.

Another consequence of the above definition which might be objected to is that it results in a [[http://en.wikipedia.org/wiki/Continuous_function|discontinuous function]], whereas ~~a continuous and in fact~~ a smooth function ~~might~~ be preferred. The function ||x||^2 is quadratically increasing near integer values of x, and is periodic with period 1. Another function with these same properties is 1 - cos(2 pi x), which is a smooth and in fact an [[http://en.wikipedia.org/wiki/Entire_function|entire]] function. Let us therefore now define for any s > 1

Another consequence of the above definition which might be objected to is that it results in a function with [[http://en.wikipedia.org/wiki/Continuous_function|discontinuous function]] derivative, whereas a smooth function be preferred. The function ||x||^2 is quadratically increasing near integer values of x, and is periodic with period 1. Another function with these same properties is 1 - cos(2 pi x), which is a smooth and in fact an [[http://en.wikipedia.org/wiki/Entire_function|entire]] function. Let us therefore now define for any s > 1

[[math]]

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[[math]]

~~This is~~ a [[http://en.wikipedia.org/wiki/~~Uniform_convergence~~|~~uniformly convergent~~]] ~~series of continuous functions~~, and hence ~~is continuous~~. We can define essentially the same function by subtracting it from E_s(1/2)/2:

For any fixed s > 1 this gives a [[http://en.wikipedia.org/wiki/Analytic_function|real analytic function]] defined for all x, and hence with all the smoothness properties we could desire. We can define essentially the same function by subtracting it from E_s(1/2)/2:

[[math]]

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--><script type="math/tex"> \sum_2^\infty \frac{\|x \log_2 q\|^2}{q^s}</script> <br />

<br />

If s is greater than one, this does converge. However, we might want to make a few adjustments. For one thing, if the error is low enough that the tuning is consistent, then the error of the square of a prime is twice that of the prime, of the cube tripled, and so forth until the error becomes inconsistent. When the weighting ~~used~~ logarithms and error measures are consistent, then the logarithmic weighting ~~canceled~~ this effect out, so we might consider that prime powers were implicitly included in the Tenney-Euclidean measure. We can go ahead and include them by adding a factor of 1/n for each prime power p^n. A somewhat peculiar but useful way to write the result of doing this is in terms of the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Von_Mangoldt_function" rel="nofollow">Von Mangoldt function</a>, an <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Arithmetic_function" rel="nofollow">arithmetic function</a> on positive integers which is equal to ln p on prime powers p^n, and is zero elsewhere. This is written using a capital lambda, and in terms of it we can include prime powers in our error function as<br />

If s is greater than one, this does converge. However, we might want to make a few adjustments. For one thing, if the error is low enough that the tuning is consistent, then the error of the square of a prime is twice that of the prime, of the cube tripled, and so forth until the error becomes inconsistent. When the weighting uses logarithms and error measures are consistent, then the logarithmic weighting canceles this effect out, so we might consider that prime powers were implicitly included in the Tenney-Euclidean measure. We can go ahead and include them by adding a factor of 1/n for each prime power p^n. A somewhat peculiar but useful way to write the result of doing this is in terms of the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Von_Mangoldt_function" rel="nofollow">Von Mangoldt function</a>, an <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Arithmetic_function" rel="nofollow">arithmetic function</a> on positive integers which is equal to ln p on prime powers p^n, and is zero elsewhere. This is written using a capital lambda, and in terms of it we can include prime powers in our error function as<br />

<br />

<!-- ws:start:WikiTextMathRule:2:

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where the summation is taken formally over all positive integers, though only the primes and prime powers make a nonzero contribution.<br />

<br />

Another consequence of the above definition which might be objected to is that it results in a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Continuous_function" rel="nofollow">discontinuous function</a>, whereas ~~a continuous and in fact~~ a smooth function ~~might~~ be preferred. The function ||x||^2 is quadratically increasing near integer values of x, and is periodic with period 1. Another function with these same properties is 1 - cos(2 pi x), which is a smooth and in fact an <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Entire_function" rel="nofollow">entire</a> function. Let us therefore now define for any s &gt; 1<br />

Another consequence of the above definition which might be objected to is that it results in a function with <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Continuous_function" rel="nofollow">discontinuous function</a> derivative, whereas a smooth function be preferred. The function ||x||^2 is quadratically increasing near integer values of x, and is periodic with period 1. Another function with these same properties is 1 - cos(2 pi x), which is a smooth and in fact an <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Entire_function" rel="nofollow">entire</a> function. Let us therefore now define for any s &gt; 1<br />

<br />

<!-- ws:start:WikiTextMathRule:3:

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--><script type="math/tex">E_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{1 - \cos(2 \pi x)}{n^s}</script><br />

<br />

~~This is~~ a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/~~Uniform_convergence~~" rel="nofollow">~~uniformly convergent~~</a> ~~series of continuous functions~~, and hence ~~is continuous~~. We can define essentially the same function by subtracting it from E_s(1/2)/2:<br />

For any fixed s &gt; 1 this gives a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Analytic_function" rel="nofollow">real analytic function</a> defined for all x, and hence with all the smoothness properties we could desire. We can define essentially the same function by subtracting it from E_s(1/2)/2:<br />

<br />

<!-- ws:start:WikiTextMathRule:4:

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-04-07 02:28:56 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-04-07 04:14:12 UTC</tt>.<br>
-: The original revision id was <tt>217990126</tt>.<br>
+: The original revision id was <tt>218005584</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
@@ Line 20: / Line 20: @@
 [[math]]
-If s is greater than one, this does converge. However, we might want to make a few adjustments. For one thing, if the error is low enough that the tuning is consistent, then the error of the square of a prime is twice that of the prime, of the cube tripled, and so forth until the error becomes inconsistent. When the weighting used logarithms and error measures are consistent, then the logarithmic weighting canceled this effect out, so we might consider that prime powers were implicitly included in the Tenney-Euclidean measure. We can go ahead and include them by adding a factor of 1/n for each prime power p^n. A somewhat peculiar but useful way to write the result of doing this is in terms of the [[http://en.wikipedia.org/wiki/Von_Mangoldt_function|Von Mangoldt function]], an [[http://en.wikipedia.org/wiki/Arithmetic_function|arithmetic function]] on positive integers which is equal to ln p on prime powers p^n, and is zero elsewhere. This is written using a capital lambda, and in terms of it we can include prime powers in our error function as
+If s is greater than one, this does converge. However, we might want to make a few adjustments. For one thing, if the error is low enough that the tuning is consistent, then the error of the square of a prime is twice that of the prime, of the cube tripled, and so forth until the error becomes inconsistent. When the weighting uses logarithms and error measures are consistent, then the logarithmic weighting canceles this effect out, so we might consider that prime powers were implicitly included in the Tenney-Euclidean measure. We can go ahead and include them by adding a factor of 1/n for each prime power p^n. A somewhat peculiar but useful way to write the result of doing this is in terms of the [[http://en.wikipedia.org/wiki/Von_Mangoldt_function|Von Mangoldt function]], an [[http://en.wikipedia.org/wiki/Arithmetic_function|arithmetic function]] on positive integers which is equal to ln p on prime powers p^n, and is zero elsewhere. This is written using a capital lambda, and in terms of it we can include prime powers in our error function as
 [[math]]
@@ Line 28: / Line 28: @@
 where the summation is taken formally over all positive integers, though only the primes and prime powers make a nonzero contribution.
-Another consequence of the above definition which might be objected to is that it results in a [[http://en.wikipedia.org/wiki/Continuous_function|discontinuous function]], whereas a continuous and in fact a smooth function might be preferred. The function ||x||^2 is quadratically increasing near integer values of x, and is periodic with period 1. Another function with these same properties is 1 - cos(2 pi x), which is a smooth and in fact an [[http://en.wikipedia.org/wiki/Entire_function|entire]] function. Let us therefore now define for any s &gt; 1
+Another consequence of the above definition which might be objected to is that it results in a function with [[http://en.wikipedia.org/wiki/Continuous_function|discontinuous function]] derivative, whereas a smooth function be preferred. The function ||x||^2 is quadratically increasing near integer values of x, and is periodic with period 1. Another function with these same properties is 1 - cos(2 pi x), which is a smooth and in fact an [[http://en.wikipedia.org/wiki/Entire_function|entire]] function. Let us therefore now define for any s &gt; 1
 [[math]]
@@ Line 34: / Line 34: @@
 [[math]]
-This is a [[http://en.wikipedia.org/wiki/Uniform_convergence|uniformly convergent]] series of continuous functions, and hence is continuous. We can define essentially the same function by subtracting it from E_s(1/2)/2:
+For any fixed s &gt; 1 this gives a [[http://en.wikipedia.org/wiki/Analytic_function|real analytic function]] defined for all x, and hence with all the smoothness properties we could desire. We can define essentially the same function by subtracting it from E_s(1/2)/2:
 [[math]]
@@ Line 72: / Line 72: @@
   --&gt;&lt;script type="math/tex"&gt; \sum_2^\infty \frac{\|x \log_2 q\|^2}{q^s}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:1 --&gt; &lt;br /&gt;
 &lt;br /&gt;
-If s is greater than one, this does converge. However, we might want to make a few adjustments. For one thing, if the error is low enough that the tuning is consistent, then the error of the square of a prime is twice that of the prime, of the cube tripled, and so forth until the error becomes inconsistent. When the weighting used logarithms and error measures are consistent, then the logarithmic weighting canceled this effect out, so we might consider that prime powers were implicitly included in the Tenney-Euclidean measure. We can go ahead and include them by adding a factor of 1/n for each prime power p^n. A somewhat peculiar but useful way to write the result of doing this is in terms of the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Von_Mangoldt_function" rel="nofollow"&gt;Von Mangoldt function&lt;/a&gt;, an &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Arithmetic_function" rel="nofollow"&gt;arithmetic function&lt;/a&gt; on positive integers which is equal to ln p on prime powers p^n, and is zero elsewhere. This is written using a capital lambda, and in terms of it we can include prime powers in our error function as&lt;br /&gt;
+If s is greater than one, this does converge. However, we might want to make a few adjustments. For one thing, if the error is low enough that the tuning is consistent, then the error of the square of a prime is twice that of the prime, of the cube tripled, and so forth until the error becomes inconsistent. When the weighting uses logarithms and error measures are consistent, then the logarithmic weighting canceles this effect out, so we might consider that prime powers were implicitly included in the Tenney-Euclidean measure. We can go ahead and include them by adding a factor of 1/n for each prime power p^n. A somewhat peculiar but useful way to write the result of doing this is in terms of the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Von_Mangoldt_function" rel="nofollow"&gt;Von Mangoldt function&lt;/a&gt;, an &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Arithmetic_function" rel="nofollow"&gt;arithmetic function&lt;/a&gt; on positive integers which is equal to ln p on prime powers p^n, and is zero elsewhere. This is written using a capital lambda, and in terms of it we can include prime powers in our error function as&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:2:
@@ Line 81: / Line 81: @@
 where the summation is taken formally over all positive integers, though only the primes and prime powers make a nonzero contribution.&lt;br /&gt;
 &lt;br /&gt;
-Another consequence of the above definition which might be objected to is that it results in a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Continuous_function" rel="nofollow"&gt;discontinuous function&lt;/a&gt;, whereas a continuous and in fact a smooth function might be preferred. The function ||x||^2 is quadratically increasing near integer values of x, and is periodic with period 1. Another function with these same properties is 1 - cos(2 pi x), which is a smooth and in fact an &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Entire_function" rel="nofollow"&gt;entire&lt;/a&gt; function. Let us therefore now define for any s &amp;gt; 1&lt;br /&gt;
+Another consequence of the above definition which might be objected to is that it results in a function with &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Continuous_function" rel="nofollow"&gt;discontinuous function&lt;/a&gt; derivative, whereas a smooth function be preferred. The function ||x||^2 is quadratically increasing near integer values of x, and is periodic with period 1. Another function with these same properties is 1 - cos(2 pi x), which is a smooth and in fact an &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Entire_function" rel="nofollow"&gt;entire&lt;/a&gt; function. Let us therefore now define for any s &amp;gt; 1&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:3:
@@ Line 88: / Line 88: @@
   --&gt;&lt;script type="math/tex"&gt;E_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{1 - \cos(2 \pi x)}{n^s}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:3 --&gt;&lt;br /&gt;
 &lt;br /&gt;
-This is a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Uniform_convergence" rel="nofollow"&gt;uniformly convergent&lt;/a&gt; series of continuous functions, and hence is continuous. We can define essentially the same function by subtracting it from E_s(1/2)/2:&lt;br /&gt;
+For any fixed s &amp;gt; 1 this gives a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Analytic_function" rel="nofollow"&gt;real analytic function&lt;/a&gt; defined for all x, and hence with all the smoothness properties we could desire. We can define essentially the same function by subtracting it from E_s(1/2)/2:&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:4: