The Riemann zeta function and tuning

IMPORTED REVISION FROM WIKISPACES

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

This revision was by author genewardsmith and made on 2011-04-10 15:43:37 UTC.

The original revision id was 218897896.

The revision comment was:

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

Original Wikitext content:

[[toc|flat]]
=Preliminaries= 
Suppose x is a continuous variable equal to the reciprocal of the step size of an equal division of the octave in fractions of an octave. For example, if the step size was 15 cents, then x = 1200/15 = 80, and we would be considering 80edo in pure octave tuning. If ||x|| denotes x minus x rounded to the nearest integer, or in other words the x - floor(x+1/2), then the [[Tenney-Euclidean metrics|Tenney-Euclidean error]] for the [[p-limit]] [[val]] obtained by rounding log2(q) x to the nearest integer for each prime q up to p will be

[[math]]
 \sum_2^p (\frac{\|x \log_2 q\|}{\log_2 q})^2
[[math]] 

Now suppose we don't want a formula for any specific prime limit, but which applies to all primes. We can't take the above sum to infinity, since it doesn't converge. However, we could change the weighting factor to a power so that it does converge:

[[math]]
 \sum_2^\infty \frac{\|x \log_2 q\|^2}{q^s}
[[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 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]]
\sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{\|x \log_2 n\|^2}{n^s}
[[math]]

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 function with a [[http://en.wikipedia.org/wiki/Continuous_function|discontinuous 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]]
E_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{1 - \cos(2 \pi x)}{n^s}
[[math]]

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]]
F_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{\cos(2 \pi x)}{n^s}
[[math]]

This now increases to a maximum value for low errors, rather than declining to a minimum. Of more interest is the fact that it is a known mathematical function, which can be expressed in terms of the real part of the logarithm of the [[http://en.wikipedia.org/wiki/Riemann_zeta_function|Riemann zeta function]]:

[[math]]
F_s(x) = \Re \ln \zeta(s + 2 \pi i x/\ln 2)
[[math]]

If we take exponentials of both sides, then

[[math]]
\exp(F_s(x)) = |\zeta(s + 2 \pi i x/\ln 2)|
[[math]]

so that we see that the absolute value of the zeta function serves to measure the error of an equal division.

=Into the critical strip=
So long as s is greater than or equal to one, the absolute value of the zeta function can be seen as an error measurement. However, the rationale for that view of things departs when s is less than one, particularly in the [[http://mathworld.wolfram.com/CriticalStrip.html|critical strip]], when s lies between zero and one. As s approaches the value s=1/2 of the [[http://mathworld.wolfram.com/CriticalLine.html|critical line]], the information content, so to speak, of the zeta function concerning higher primes increases and it behaves increasingly like a badness measure (or more correctly, since we have inverted it, like a goodness measure.) The quasi-symmetric [[http://planetmath.org/encyclopedia/FunctionalEquationOfTheRiemannZetaFunction.html|functional equation]] of the zeta function tells us that past the critical line the information content starts to decrease again, with 1-s and s having the same information content. Hence it is the zeta function between s=1/2 and s=1, and especially the zeta function along the critical line s=1/2, which is of the most interest.

As s>0 gets larger, the Dirichlet series for the zeta function is increasingly dominated by the 2 term, getting ever closer to simply 1 + 2^(-z), which approaches 1 as s = Re(z) becomes larger. When s >> 1 and x is an integer, the real part of zeta is approximately 1 + 2^(-s), and the imaginary part is approximately zero; that is, zeta is real. Starting from +infinity with x an integer, we can trace a line back towards the critical strip on which zeta is real. Since when s >> 1 the derivative is approximately -ln(2)/2^s, it is negative, meaning that this real value for zeta increases as we go back. The zeta function approaches 1 uniformly as s increases to infinity, so going back, the real-valued zeta function along this line of real values continues to increase, though all real values from 1 to infinity monotonically. When it crosses the critical line where s=1/2, it produces a real value of zeta on the critical line. Points on the critical line where zeta(1/2 + i g) are real are called "Gram points", after [[http://en.wikipedia.org/wiki/J%C3%B8rgen_Pedersen_Gram|Jørgen Pedersen Gram]]. We thus have associated pure-octave edos, where x is an integer, to a value near to the pure octave, at the special sort of Gram point which corresponds to an edo.

Because the value of zeta increased continuously as it made its way from +infinity to the critical line, we might expect the values of zeta at these special Gram points to be relatively large. This would be especially true if -zeta'(z) is getting a boost from other small primes as it travels toward the Gram point. A complex formula due to [[http://en.wikipedia.org/wiki/Riemann|Bernhard Riemann]] which he failed to publish because it was so nasty becomes a bit simpler when used at a Gram point. It is named the [[Riemann-Siegel formula]] since [[http://en.wikipedia.org/wiki/Carl_Ludwig_Siegel|Carl Ludwig Siegel]] went looking for it and was able to reconstruct it after rooting industriously around in Riemann's unpublished papers. From this formula, it is apparent that when x corresponds to a good edo, the value of zeta(1/2 + i g) at the corresponding Gram point should be especially large.

=The Z function=
The absolute value zeta(1/2 + i g) at a Gram point corresponding to an edo is near to a local maximum, but not actually at one. At the local maximum, of course, the partial derivative of zeta(1/2 + i t) with respect to t will be zero; however this does not mean its derivative there will be zero. In fact, the [[http://en.wikipedia.org/wiki/Riemann_hypothesis|Riemann hypothesis]] is equivalent to the claim that all zeros of zeta'(s + i t) occur when s > 1/2, which is where all known zeros lie. These do not have values of t corresponding to good edos. For this and other reasons, it is helpful to have a function which is real for values on the critical line but whose absolute value is the same as that of zeta. This is provided by the [[http://en.wikipedia.org/wiki/Z_function|Z function]].



=Computing zeta=
There are various approaches to the question of computing the zeta function, but perhaps the simplest is the use of the [[http://en.wikipedia.org/wiki/Dirichlet_eta_function|Dirichlet eta function]] which was introduced to mathematics by [[http://en.wikipedia.org/wiki/Johann_Peter_Gustav_Lejeune_Dirichlet|Johann Peter Gustav Lejeune Dirichlet]], who despite his name was a German and the brother-in-law of [[http://en.wikipedia.org/wiki/Felix_Mendelssohn_Bartholdy|Felix Mendelssohn]].

The zeta function has a [[http://mathworld.wolfram.com/SimplePole.html|simple pole]] at z=1 which forms a barrier against continuing it with its [[http://en.wikipedia.org/wiki/Euler_product|Euler product]] or [[http://en.wikipedia.org/wiki/Dirichlet_series|Dirichlet series]] representation. We could subtract off the pole, or multiply by a factor of (z-1), but at the expense of losing the character of a Dirichlet series or Euler product. A better method is to multiply by a factor of (1-2^(1-z)), leading to the eta function:

[[math]]
\eta(z) = (1-2^{1-z})\zeta(z) = \sum_{n=1}^\infty (-1)^{n-1} n^{-z}
= \frac{1}{1^z} - \frac{1}{2^z} + \frac{1}{3^z} - \frac{1}{4^z} + \cdots
[[math]]

The Dirichlet series for the zeta function is absolutely convergent when s>1, justifying the rearrangement of terms leading to the alternating series for eta, which converges conditionally in the critical strip. The extra factor introduces zeros of the eta function at the points 1 + 2pi i x corresponding to pure octave divisions along the line s=1, but no other zeros, and in particular none in the critical strip and along the critical line. The convergence of the alternating series can be greatly accelerated by applying [[http://en.wikipedia.org/wiki/Euler_summation|Euler summation]].

=Links=
[[http://front.math.ucdavis.edu/0309.5433|X-Ray of Riemann zeta-function]] by Juan Arias-de-Reyna

Original HTML content:

<html><head><title>The Riemann Zeta Function and Tuning</title></head><body><!-- ws:start:WikiTextTocRule:18:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --><a href="#Preliminaries">Preliminaries</a><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --> | <a href="#Into the critical strip">Into the critical strip</a><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --> | <a href="#The Z function">The Z function</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --> | <a href="#Computing zeta">Computing zeta</a><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --> | <a href="#Links">Links</a><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: -->
<!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc0"><a name="Preliminaries"></a><!-- ws:end:WikiTextHeadingRule:8 -->Preliminaries</h1>
 Suppose x is a continuous variable equal to the reciprocal of the step size of an equal division of the octave in fractions of an octave. For example, if the step size was 15 cents, then x = 1200/15 = 80, and we would be considering 80edo in pure octave tuning. If ||x|| denotes x minus x rounded to the nearest integer, or in other words the x - floor(x+1/2), then the <a class="wiki_link" href="/Tenney-Euclidean%20metrics">Tenney-Euclidean error</a> for the <a class="wiki_link" href="/p-limit">p-limit</a> <a class="wiki_link" href="/val">val</a> obtained by rounding log2(q) x to the nearest integer for each prime q up to p will be<br />
<br />
<!-- ws:start:WikiTextMathRule:0:
[[math]]&lt;br/&gt;
 \sum_2^p (\frac{\|x \log_2 q\|}{\log_2 q})^2&lt;br/&gt;[[math]]
 --><script type="math/tex"> \sum_2^p (\frac{\|x \log_2 q\|}{\log_2 q})^2</script><!-- ws:end:WikiTextMathRule:0 --> <br />
<br />
Now suppose we don't want a formula for any specific prime limit, but which applies to all primes. We can't take the above sum to infinity, since it doesn't converge. However, we could change the weighting factor to a power so that it does converge:<br />
<br />
<!-- ws:start:WikiTextMathRule:1:
[[math]]&lt;br/&gt;
 \sum_2^\infty \frac{\|x \log_2 q\|^2}{q^s}&lt;br/&gt;[[math]]
 --><script type="math/tex"> \sum_2^\infty \frac{\|x \log_2 q\|^2}{q^s}</script><!-- ws:end:WikiTextMathRule:1 --> <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 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:
[[math]]&lt;br/&gt;
\sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{\|x \log_2 n\|^2}{n^s}&lt;br/&gt;[[math]]
 --><script type="math/tex">\sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{\|x \log_2 n\|^2}{n^s}</script><!-- ws:end:WikiTextMathRule:2 --><br />
<br />
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 function with a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Continuous_function" rel="nofollow">discontinuous derivative</a>, 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:
[[math]]&lt;br/&gt;
E_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{1 - \cos(2 \pi x)}{n^s}&lt;br/&gt;[[math]]
 --><script type="math/tex">E_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{1 - \cos(2 \pi x)}{n^s}</script><!-- ws:end:WikiTextMathRule:3 --><br />
<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:
[[math]]&lt;br/&gt;
F_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{\cos(2 \pi x)}{n^s}&lt;br/&gt;[[math]]
 --><script type="math/tex">F_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{\cos(2 \pi x)}{n^s}</script><!-- ws:end:WikiTextMathRule:4 --><br />
<br />
This now increases to a maximum value for low errors, rather than declining to a minimum. Of more interest is the fact that it is a known mathematical function, which can be expressed in terms of the real part of the logarithm of the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Riemann_zeta_function" rel="nofollow">Riemann zeta function</a>:<br />
<br />
<!-- ws:start:WikiTextMathRule:5:
[[math]]&lt;br/&gt;
F_s(x) = \Re \ln \zeta(s + 2 \pi i x/\ln 2)&lt;br/&gt;[[math]]
 --><script type="math/tex">F_s(x) = \Re \ln \zeta(s + 2 \pi i x/\ln 2)</script><!-- ws:end:WikiTextMathRule:5 --><br />
<br />
If we take exponentials of both sides, then<br />
<br />
<!-- ws:start:WikiTextMathRule:6:
[[math]]&lt;br/&gt;
\exp(F_s(x)) = |\zeta(s + 2 \pi i x/\ln 2)|&lt;br/&gt;[[math]]
 --><script type="math/tex">\exp(F_s(x)) = |\zeta(s + 2 \pi i x/\ln 2)|</script><!-- ws:end:WikiTextMathRule:6 --><br />
<br />
so that we see that the absolute value of the zeta function serves to measure the error of an equal division.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc1"><a name="Into the critical strip"></a><!-- ws:end:WikiTextHeadingRule:10 -->Into the critical strip</h1>
So long as s is greater than or equal to one, the absolute value of the zeta function can be seen as an error measurement. However, the rationale for that view of things departs when s is less than one, particularly in the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/CriticalStrip.html" rel="nofollow">critical strip</a>, when s lies between zero and one. As s approaches the value s=1/2 of the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/CriticalLine.html" rel="nofollow">critical line</a>, the information content, so to speak, of the zeta function concerning higher primes increases and it behaves increasingly like a badness measure (or more correctly, since we have inverted it, like a goodness measure.) The quasi-symmetric <a class="wiki_link_ext" href="http://planetmath.org/encyclopedia/FunctionalEquationOfTheRiemannZetaFunction.html" rel="nofollow">functional equation</a> of the zeta function tells us that past the critical line the information content starts to decrease again, with 1-s and s having the same information content. Hence it is the zeta function between s=1/2 and s=1, and especially the zeta function along the critical line s=1/2, which is of the most interest.<br />
<br />
As s&gt;0 gets larger, the Dirichlet series for the zeta function is increasingly dominated by the 2 term, getting ever closer to simply 1 + 2^(-z), which approaches 1 as s = Re(z) becomes larger. When s &gt;&gt; 1 and x is an integer, the real part of zeta is approximately 1 + 2^(-s), and the imaginary part is approximately zero; that is, zeta is real. Starting from +infinity with x an integer, we can trace a line back towards the critical strip on which zeta is real. Since when s &gt;&gt; 1 the derivative is approximately -ln(2)/2^s, it is negative, meaning that this real value for zeta increases as we go back. The zeta function approaches 1 uniformly as s increases to infinity, so going back, the real-valued zeta function along this line of real values continues to increase, though all real values from 1 to infinity monotonically. When it crosses the critical line where s=1/2, it produces a real value of zeta on the critical line. Points on the critical line where zeta(1/2 + i g) are real are called &quot;Gram points&quot;, after <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/J%C3%B8rgen_Pedersen_Gram" rel="nofollow">Jørgen Pedersen Gram</a>. We thus have associated pure-octave edos, where x is an integer, to a value near to the pure octave, at the special sort of Gram point which corresponds to an edo.<br />
<br />
Because the value of zeta increased continuously as it made its way from +infinity to the critical line, we might expect the values of zeta at these special Gram points to be relatively large. This would be especially true if -zeta'(z) is getting a boost from other small primes as it travels toward the Gram point. A complex formula due to <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Riemann" rel="nofollow">Bernhard Riemann</a> which he failed to publish because it was so nasty becomes a bit simpler when used at a Gram point. It is named the <a class="wiki_link" href="/Riemann-Siegel%20formula">Riemann-Siegel formula</a> since <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Carl_Ludwig_Siegel" rel="nofollow">Carl Ludwig Siegel</a> went looking for it and was able to reconstruct it after rooting industriously around in Riemann's unpublished papers. From this formula, it is apparent that when x corresponds to a good edo, the value of zeta(1/2 + i g) at the corresponding Gram point should be especially large.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc2"><a name="The Z function"></a><!-- ws:end:WikiTextHeadingRule:12 -->The Z function</h1>
The absolute value zeta(1/2 + i g) at a Gram point corresponding to an edo is near to a local maximum, but not actually at one. At the local maximum, of course, the partial derivative of zeta(1/2 + i t) with respect to t will be zero; however this does not mean its derivative there will be zero. In fact, the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Riemann_hypothesis" rel="nofollow">Riemann hypothesis</a> is equivalent to the claim that all zeros of zeta'(s + i t) occur when s &gt; 1/2, which is where all known zeros lie. These do not have values of t corresponding to good edos. For this and other reasons, it is helpful to have a function which is real for values on the critical line but whose absolute value is the same as that of zeta. This is provided by the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Z_function" rel="nofollow">Z function</a>.<br />
<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc3"><a name="Computing zeta"></a><!-- ws:end:WikiTextHeadingRule:14 -->Computing zeta</h1>
There are various approaches to the question of computing the zeta function, but perhaps the simplest is the use of the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Dirichlet_eta_function" rel="nofollow">Dirichlet eta function</a> which was introduced to mathematics by <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Johann_Peter_Gustav_Lejeune_Dirichlet" rel="nofollow">Johann Peter Gustav Lejeune Dirichlet</a>, who despite his name was a German and the brother-in-law of <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Felix_Mendelssohn_Bartholdy" rel="nofollow">Felix Mendelssohn</a>.<br />
<br />
The zeta function has a <a class="wiki_link_ext" href="http://mathworld.wolfram.com/SimplePole.html" rel="nofollow">simple pole</a> at z=1 which forms a barrier against continuing it with its <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Euler_product" rel="nofollow">Euler product</a> or <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Dirichlet_series" rel="nofollow">Dirichlet series</a> representation. We could subtract off the pole, or multiply by a factor of (z-1), but at the expense of losing the character of a Dirichlet series or Euler product. A better method is to multiply by a factor of (1-2^(1-z)), leading to the eta function:<br />
<br />
<!-- ws:start:WikiTextMathRule:7:
[[math]]&lt;br/&gt;
\eta(z) = (1-2^{1-z})\zeta(z) = \sum_{n=1}^\infty (-1)^{n-1} n^{-z}&lt;br /&gt;
= \frac{1}{1^z} - \frac{1}{2^z} + \frac{1}{3^z} - \frac{1}{4^z} + \cdots&lt;br/&gt;[[math]]
 --><script type="math/tex">\eta(z) = (1-2^{1-z})\zeta(z) = \sum_{n=1}^\infty (-1)^{n-1} n^{-z}
= \frac{1}{1^z} - \frac{1}{2^z} + \frac{1}{3^z} - \frac{1}{4^z} + \cdots</script><!-- ws:end:WikiTextMathRule:7 --><br />
<br />
The Dirichlet series for the zeta function is absolutely convergent when s&gt;1, justifying the rearrangement of terms leading to the alternating series for eta, which converges conditionally in the critical strip. The extra factor introduces zeros of the eta function at the points 1 + 2pi i x corresponding to pure octave divisions along the line s=1, but no other zeros, and in particular none in the critical strip and along the critical line. The convergence of the alternating series can be greatly accelerated by applying <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Euler_summation" rel="nofollow">Euler summation</a>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc4"><a name="Links"></a><!-- ws:end:WikiTextHeadingRule:16 -->Links</h1>
<a class="wiki_link_ext" href="http://front.math.ucdavis.edu/0309.5433" rel="nofollow">X-Ray of Riemann zeta-function</a> by Juan Arias-de-Reyna</body></html>