Ryan's Working Page - Revision history

ArrowHead294 at 13:43, 24 February 2025

2025-02-24T13:43:56Z

← Older revision		Revision as of 13:43, 24 February 2025
Line 6:		Line 6:
	\left\| \zeta(s) \right\|^2 = \zeta(2a) + 2 \sum_{n > d} \left[\frac{\cos\left(b\ln\left(\tfrac{n}{d}\right)\right)}{(nd)^{a}}\right]</math>		\left\| \zeta(s) \right\|^2 = \zeta(2a) + 2 \sum_{n > d} \left[\frac{\cos\left(b\ln\left(\tfrac{n}{d}\right)\right)}{(nd)^{a}}\right]</math>

	The cosines are weighted by 1/(nd)<span style="font-size: 11.6999998092651px; vertical-align: super;">a</span>. However, it is of interest to replace n*d with max(n,d)^2 to see if we can derive a [[Weil norm, ~~Tenney-Weil~~ norm, and TWp interval and tuning space\|Weil]]-weighted analog of the Zeta function. I will denote this function by f(s).		The cosines are weighted by 1/(nd)<span style="font-size: 11.6999998092651px; vertical-align: super;">a</span>. However, it is of interest to replace n*d with max(n,d)^2 to see if we can derive a [[Weil norm, Tenney–Weil norm, and TWp interval and tuning space\|Weil]]-weighted analog of the Zeta function. I will denote this function by f(s).

	<math>\displaystyle		<math>\displaystyle

BudjarnLambeth: Add outgoing links to dead end page

2024-12-04T10:13:02Z

Add outgoing links to dead end page

← Older revision		Revision as of 10:13, 4 December 2024
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	=Attempt to backwards-engineer a Weil-weighted analog for "Zeta"=		=Attempt to backwards-engineer a Weil-weighted analog for "Zeta"=

	In Mike's Zeta Function Working Page, we see that zeta can be thought of as a superposition of weighted "cosine accuracy" functions for every unreduced rational:		In [[Mike's Zeta Function Working Page]], we see that [[zeta]] can be thought of as a superposition of weighted "cosine accuracy" functions for every unreduced rational:

	<math>\displaystyle		<math>\displaystyle
	\left\| \zeta(s) \right\|^2 = \zeta(2a) + 2 \sum_{n > d} \left[\frac{\cos\left(b\ln\left(\tfrac{n}{d}\right)\right)}{(nd)^{a}}\right]</math>		\left\| \zeta(s) \right\|^2 = \zeta(2a) + 2 \sum_{n > d} \left[\frac{\cos\left(b\ln\left(\tfrac{n}{d}\right)\right)}{(nd)^{a}}\right]</math>

	The cosines are weighted by 1/(nd)<span style="font-size: 11.6999998092651px; vertical-align: super;">a</span>. However, it is of interest to replace n*d with max(n,d)^2 to see if we can derive a Weil-weighted analog of the Zeta function. I will denote this function by f(s).		The cosines are weighted by 1/(nd)<span style="font-size: 11.6999998092651px; vertical-align: super;">a</span>. However, it is of interest to replace n*d with max(n,d)^2 to see if we can derive a [[Weil norm, Tenney-Weil norm, and TWp interval and tuning space\|Weil]]-weighted analog of the Zeta function. I will denote this function by f(s).

	<math>\displaystyle		<math>\displaystyle

BudjarnLambeth: Todo move to userspace

2024-10-20T07:45:39Z

Todo move to userspace

← Older revision		Revision as of 07:45, 20 October 2024
Line 111:		Line 111:
	[[Category:Math]]		[[Category:Math]]
	[[Category:Zeta]]		[[Category:Zeta]]

			{{todo\|move to userspace}}

BudjarnLambeth: Categorised

2023-04-30T06:17:59Z

Categorised

← Older revision		Revision as of 06:17, 30 April 2023
Line 109:		Line 109:

	Alright!! I'm going to bed.		Alright!! I'm going to bed.
			[[Category:Math]]
			[[Category:Zeta]]

Wikispaces>FREEZE at 00:00, 17 July 2018

2018-07-17T00:00:00Z

Show changes

Wikispaces>Sarzadoce: Imported revision 557169127 - Original comment:

2015-08-22T12:57:17Z

**Imported revision 557169127 - Original comment: **

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Wikispaces>mbattaglia1: Imported revision 557156615 - Original comment:

2015-08-22T04:32:08Z

**Imported revision 557156615 - Original comment: **

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Wikispaces>Sarzadoce: Imported revision 557147201 - Original comment:

2015-08-21T20:21:08Z

**Imported revision 557147201 - Original comment: **

New page

<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:Sarzadoce|Sarzadoce]] and made on <tt>2015-08-21 20:21:08 UTC</tt>.<br>
: The original revision id was <tt>557147201</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>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">=Attempt to backwards-engineer a Weil-weighted analog for "Zeta"=

In Mike's Zeta Function Working Page, we see that zeta can be thought of as a superposition of weighted "cosine accuracy" functions for every unreduced rational:

[[math]]
\displaystyle
\left| \zeta(s) \right|^2 = \zeta(2a) + 2 \sum_{n > d} \left[\frac{\cos\left(b\ln\left(\tfrac{n}{d}\right)\right)}{(nd)^{a}}\right]
[[math]]

The cosines are weighted by 1/(nd)<span style="font-size: 11.6999998092651px; vertical-align: super;">a</span>. However, it is of interest to replace n*d with max(n,d)^2 to see if we can derive a Weil-weighted analog of the Zeta function. I will denote this function by f(s).

[[math]]
\displaystyle
\left| \text{f} (s) \right|^2 = \zeta(2a) + 2 \sum_{n > d} \left[\frac{\cos\left(b\ln\left(\tfrac{n}{d}\right)\right)}{\text{max}(n,d)^{2a}}\right]
[[math]]

Let's do a few manipulations, to try to work our way backwards to f(s):

[[math]]
\displaystyle
\left| \text{f} (s) \right|^2 = \zeta(2a) + 2 \sum_{n > d} \left[\frac{\cos\left(b\ln\left(\tfrac{n}{d}\right)\right)}{n^{2a}}\right]
[[math]]

[[math]]
\displaystyle
\left| \text{f} (s) \right|^2 = \zeta(2a) + \sum_{n > d} \left[\frac{\left({\tfrac{n}{d}}\right)^{-bi} + \left({\tfrac{d}{n}}\right)^{-bi}}{n^{2a}}\right]
[[math]]

[[math]]
\displaystyle
\left| \text{f} (s) \right|^2 = \zeta(2a) + 2 \text{Re} \left( \sum_{n > d} n^{ - \left(2a+bi \right) } d^{ - \left( -bi \right) } \right)
[[math]]

[[math]]
\displaystyle
\left| \text{f} (s) \right|^2 = \zeta(2a) + \sum_{n > d} \left[ n^{ - \left(2a+bi \right) } d^{ - \left( -bi \right) } + n^{ - \left(2a-bi \right) } d^{ - \left( bi \right) } \right]
[[math]]

Note that the RHS of this equation can be broken up into a sum of conjugates. However, we ideally want to break it up into a //product// of conjugates, in order to solve for f(s). This is where I get stuck.</pre></div>
<h4>Original HTML content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Ryan's Working Page</title></head><body><h1 id="toc0"><a name="Attempt to backwards-engineer a Weil-weighted analog for &quot;Zeta&quot;"></a>Attempt to backwards-engineer a Weil-weighted analog for &quot;Zeta&quot;</h1>
<br />
In Mike's Zeta Function Working Page, we see that zeta can be thought of as a superposition of weighted &quot;cosine accuracy&quot; functions for every unreduced rational:<br />
<br />
<script type="math/tex">\displaystyle
\left| \zeta(s) \right|^2 = \zeta(2a) + 2 \sum_{n > d} \left[\frac{\cos\left(b\ln\left(\tfrac{n}{d}\right)\right)}{(nd)^{a}}\right]</script><br />
<br />
The cosines are weighted by 1/(nd)<span style="font-size: 11.6999998092651px; vertical-align: super;">a</span>. However, it is of interest to replace n*d with max(n,d)^2 to see if we can derive a Weil-weighted analog of the Zeta function. I will denote this function by f(s).<br />
<br />
<script type="math/tex">\displaystyle
\left| \text{f} (s) \right|^2 = \zeta(2a) + 2 \sum_{n > d} \left[\frac{\cos\left(b\ln\left(\tfrac{n}{d}\right)\right)}{\text{max}(n,d)^{2a}}\right]</script><br />
<br />
Let's do a few manipulations, to try to work our way backwards to f(s):<br />
<br />
<script type="math/tex">\displaystyle
\left| \text{f} (s) \right|^2 = \zeta(2a) + 2 \sum_{n > d} \left[\frac{\cos\left(b\ln\left(\tfrac{n}{d}\right)\right)}{n^{2a}}\right]</script><br />
<br />
<script type="math/tex">\displaystyle
\left| \text{f} (s) \right|^2 = \zeta(2a) + \sum_{n > d} \left[\frac{\left({\tfrac{n}{d}}\right)^{-bi} + \left({\tfrac{d}{n}}\right)^{-bi}}{n^{2a}}\right]</script><br />
<br />
<script type="math/tex">\displaystyle
\left| \text{f} (s) \right|^2 = \zeta(2a) + 2 \text{Re} \left( \sum_{n > d} n^{ - \left(2a+bi \right) } d^{ - \left( -bi \right) } \right)</script><br />
<br />
<script type="math/tex">\displaystyle
\left| \text{f} (s) \right|^2 = \zeta(2a) + \sum_{n > d} \left[ n^{ - \left(2a+bi \right) } d^{ - \left( -bi \right) } + n^{ - \left(2a-bi \right) } d^{ - \left( bi \right) } \right]</script><br />
<br />
Note that the RHS of this equation can be broken up into a sum of conjugates. However, we ideally want to break it up into a <em>product</em> of conjugates, in order to solve for f(s). This is where I get stuck.</body></html></pre></div>

Ryan's Working Page - Revision history

ArrowHead294 at 13:43, 24 February 2025

BudjarnLambeth: Add outgoing links to dead end page

BudjarnLambeth: Todo move to userspace

BudjarnLambeth: Categorised

Wikispaces>FREEZE at 00:00, 17 July 2018

Wikispaces>Sarzadoce: **Imported revision 557169127 - Original comment: **

Wikispaces>mbattaglia1: **Imported revision 557156615 - Original comment: **

Wikispaces>Sarzadoce: **Imported revision 557147201 - Original comment: **

Wikispaces>Sarzadoce: Imported revision 557169127 - Original comment:

Wikispaces>mbattaglia1: Imported revision 557156615 - Original comment:

Wikispaces>Sarzadoce: Imported revision 557147201 - Original comment: