The Riemann zeta function and tuning: Difference between revisions
Wikispaces>genewardsmith **Imported revision 353276314 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 353276468 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
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: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-07-15 22: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-07-15 22:21:04 UTC</tt>.<br> | ||
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Since theta is holomorphic on the strip with imaginary part between -1/2 and 1/2, so is Z. Since the exponential function has no zeros, the zeros of Z in this strip correspond one to one with the zeros of zeta in the critical strip. Since the exponential of an imaginary argument has absolute value 1, the absolute value of Z along the real axis is the same as the absolute value of zeta at the corresponding place on the critical line. And since theta was defined so as to give precisely this property, Z is a real even function of the real variable t. | Since theta is holomorphic on the strip with imaginary part between -1/2 and 1/2, so is Z. Since the exponential function has no zeros, the zeros of Z in this strip correspond one to one with the zeros of zeta in the critical strip. Since the exponential of an imaginary argument has absolute value 1, the absolute value of Z along the real axis is the same as the absolute value of zeta at the corresponding place on the critical line. And since theta was defined so as to give precisely this property, Z is a real even function of the real variable t. | ||
Using the [[http://functions.wolfram.com/webMathematica/FunctionPlotting.jsp?name=RiemannSiegelZ|online plotter]] we can plot Z in the regions corresponding to scale divisions, using the conversion factor t = | Using the [[http://functions.wolfram.com/webMathematica/FunctionPlotting.jsp?name=RiemannSiegelZ|online plotter]] we can plot Z in the regions corresponding to scale divisions, using the conversion factor t = 2π/ln(2) x, for x a number near or at an edo number. Hence, for instance, to plot 12 plot around 108.777, to plot 31 plot around 281.006, and so forth. An alternative plotter is the applet [[http://web.viu.ca/pughg/RiemannZeta/RiemannZetaLong.html|here]]. | ||
If you have access to [[http://en.wikipedia.org/wiki/Mathematica|Mathematica]], which has Z, zeta and theta as a part of its suite of initially defined functions, you can do even better. Below is a Mathematicia-generated plot of Z(2πx/ln(2)) in the region around 12edo: | If you have access to [[http://en.wikipedia.org/wiki/Mathematica|Mathematica]], which has Z, zeta and theta as a part of its suite of initially defined functions, you can do even better. Below is a Mathematicia-generated plot of Z(2πx/ln(2)) in the region around 12edo: | ||
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Since theta is holomorphic on the strip with imaginary part between -1/2 and 1/2, so is Z. Since the exponential function has no zeros, the zeros of Z in this strip correspond one to one with the zeros of zeta in the critical strip. Since the exponential of an imaginary argument has absolute value 1, the absolute value of Z along the real axis is the same as the absolute value of zeta at the corresponding place on the critical line. And since theta was defined so as to give precisely this property, Z is a real even function of the real variable t.<br /> | Since theta is holomorphic on the strip with imaginary part between -1/2 and 1/2, so is Z. Since the exponential function has no zeros, the zeros of Z in this strip correspond one to one with the zeros of zeta in the critical strip. Since the exponential of an imaginary argument has absolute value 1, the absolute value of Z along the real axis is the same as the absolute value of zeta at the corresponding place on the critical line. And since theta was defined so as to give precisely this property, Z is a real even function of the real variable t.<br /> | ||
<br /> | <br /> | ||
Using the <a class="wiki_link_ext" href="http://functions.wolfram.com/webMathematica/FunctionPlotting.jsp?name=RiemannSiegelZ" rel="nofollow">online plotter</a> we can plot Z in the regions corresponding to scale divisions, using the conversion factor t = | Using the <a class="wiki_link_ext" href="http://functions.wolfram.com/webMathematica/FunctionPlotting.jsp?name=RiemannSiegelZ" rel="nofollow">online plotter</a> we can plot Z in the regions corresponding to scale divisions, using the conversion factor t = 2π/ln(2) x, for x a number near or at an edo number. Hence, for instance, to plot 12 plot around 108.777, to plot 31 plot around 281.006, and so forth. An alternative plotter is the applet <a class="wiki_link_ext" href="http://web.viu.ca/pughg/RiemannZeta/RiemannZetaLong.html" rel="nofollow">here</a>.<br /> | ||
<br /> | <br /> | ||
If you have access to <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Mathematica" rel="nofollow">Mathematica</a>, which has Z, zeta and theta as a part of its suite of initially defined functions, you can do even better. Below is a Mathematicia-generated plot of Z(2πx/ln(2)) in the region around 12edo:<br /> | If you have access to <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Mathematica" rel="nofollow">Mathematica</a>, which has Z, zeta and theta as a part of its suite of initially defined functions, you can do even better. Below is a Mathematicia-generated plot of Z(2πx/ln(2)) in the region around 12edo:<br /> | ||