The Riemann zeta function and tuning/Appendix - Revision history

Sintel: consistent formatting and indentation

2025-04-13T16:32:57Z

consistent formatting and indentation

ArrowHead294 at 11:20, 10 April 2025

2025-04-10T11:20:55Z

← Older revision		Revision as of 11:20, 10 April 2025
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	The Dirichlet series for the zeta function is absolutely convergent when {{nowrap\|''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 {{nowrap\|1 + {{sfrac\|2π''i''\|ln(2)}}x}} corresponding to pure octave divisions along the line {{nowrap\|''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 {{w\|Euler summation}}.		The Dirichlet series for the zeta function is absolutely convergent when {{nowrap\|''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 {{nowrap\|1 + {{sfrac\|2π''i''\|ln(2)}}x}} corresponding to pure octave divisions along the line {{nowrap\|''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 {{w\|Euler summation}}.

			[[Category:Zeta]]
			[[Category:Math]]
			[[Category:Tuning]]
			[[Category:Number theory]]

Lériendil: fixed a mistake

2025-04-09T22:50:10Z

fixed a mistake

← Older revision		Revision as of 22:50, 9 April 2025
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	<math>\displaystyle		<math>\displaystyle
	\left\|1 - p^{-s} \right\| = \sqrt{1 + p^{-2\sigma} - \cos(t \ln p)}		\left\|1 - p^{-s} \right\| = \sqrt{1 + p^{-2\sigma} - 2p^{-\sigma} \cos(t \ln p)}
	</math>.		</math>.

Sintel: /* von Mangoldt function */ punctuation

2025-04-09T21:29:45Z

von Mangoldt function: punctuation

← Older revision		Revision as of 21:29, 9 April 2025
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	\Lambda(n) =		\Lambda(n) =
	\begin{cases}		\begin{cases}
	\ln p & \text{if } n = p^k \text{ for prime } p \text{ and } k \geq 1 \\		\ln p & \text{if } n = p^k \text{ for prime } p \text{ and } k \geq 1 \text{,} \\
	0 & \text{otherwise}		0 & \text{otherwise.}
	\end{cases}		\end{cases}
	</math>.		</math>

	Reindex the double sum.		Reindex the double sum.

Lériendil at 18:28, 9 April 2025

2025-04-09T18:28:40Z

← Older revision		Revision as of 18:28, 9 April 2025
Line 135:		Line 135:

	<math>\displaystyle		<math>\displaystyle
	\left\|1 - p^{-s} \right\| = \sqrt{1 + p^{-2\sigma} - cos(t \ln p)}		\left\|1 - p^{-s} \right\| = \sqrt{1 + p^{-2\sigma} - \cos(t \ln p)}
	</math>.		</math>.

Lériendil: derived the cosine thing

2025-04-09T18:28:29Z

derived the cosine thing

@@ Line 114: / Line 114: @@
 === 1b. Conversion factor for removing primes ===
-{{todo|derive this|inline=1}}
+Consider the expression <math>\left|1 - p^{-s} \right|</math>, where <math>s = \sigma + it</math>. Here,
 == 2. Z function and Riemann-Siegel theta function ==

Lériendil: derived euler product

2025-04-09T18:07:02Z

derived euler product

← Older revision		Revision as of 18:07, 9 April 2025
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	== 1. Euler product expression for the zeta function ==		== 1. Euler product expression for the zeta function ==
	{{~~todo\|derive this\|inline~~=1}}		Starting with the definition of the zeta function:

			<math>\displaystyle
			\zeta(s) = \sum_{n = 1}^\infty \frac{1}{n^s}
			</math>,

			by the Fundamental Theorem of Arithmetic, and for <math>\mathrm{Re}(s) > 1</math>, where the series is absolutely convergent, it can be rearranged into a sum over prime factorizations:

			<math>\displaystyle
			\zeta(s) = \sum_{r_k = 0}^\infty \frac{1}{\left(\prod_{k = 1}^\infty {p_k}^{r_k} \right)^s} = \sum_{r_k = 0}^\infty \prod_{k = 1}^\infty \frac{1}{{p_k}^{s r_k}}
			</math>,

			where <math>p_k</math> is the ''k''-th prime.

			The sum on the outside is really an infinite product of sum operators for each ''k'', and so:

			<math>\displaystyle
			\sum_{r_k = 0}^\infty \prod_{k = 1}^\infty \frac{1}{{p_k}^{s r_k}} = \prod_{k = 1}^\infty \sum_{r = 0}^\infty \frac{1}{\left({p_k}^s \right)^r}
			</math>,

			which is a geometric series, and given that <math>{p_k}^s</math> is greater than 1 for all ''k'' when <math>\mathrm{Re}(s) > 1</math>,

			<math>\displaystyle
			\zeta(s) = \prod_{k = 1}^\infty \frac{1}{1 - {p_k}^{-s}},
			</math>

			which is the Euler product formula we desired.

	=== 1a. Dirichlet series for the von Mangoldt function ===		=== 1a. Dirichlet series for the von Mangoldt function ===
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	<math>\displaystyle		<math>\displaystyle
	\zeta(s) = \prod_{p \text{ prime}} \left(1 - p^{-s}\right)^{-1}		\zeta(s) = \prod_{p \text{ prime}} \left(1 - p^{-s}\right)^{-1}
	</math>		</math>.

	Take the natural logarithm to convert the product into a sum:		Take the natural logarithm to convert the product into a sum:
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	<math>\displaystyle		<math>\displaystyle
	\ln \zeta(s) = \sum_{p \text{ prime}} - \ln \left( 1 - p^{-s} \right)		\ln \zeta(s) = \sum_{p \text{ prime}} - \ln \left( 1 - p^{-s} \right)
	</math>		</math>.

	==== Logarithmic Series Expansion ====		==== Logarithmic Series Expansion ====
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	<math>\displaystyle		<math>\displaystyle
	-\ln(1 - x) = \sum_{k=1}^{\infty} \frac{x^k}{k}		-\ln(1 - x) = \sum_{k=1}^{\infty} \frac{x^k}{k}
	</math>		</math>.

	Substitute \( x = p^{-s} \).		Substitute \( x = p^{-s} \).
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	<math>\displaystyle		<math>\displaystyle
	-\ln\left(1 - p^{-s}\right) = \sum_{k=1}^{\infty} \frac{\left(p^{-s}\right)^k}{k}		-\ln\left(1 - p^{-s}\right) = \sum_{k=1}^{\infty} \frac{\left(p^{-s}\right)^k}{k}
	</math>		</math>.

	Simplify the expression.		Simplify the expression.
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	\sum_{k=1}^{\infty} \frac{\left(p^{-s}\right)^k}{k}		\sum_{k=1}^{\infty} \frac{\left(p^{-s}\right)^k}{k}
	= \sum_{k=1}^\infty \frac{1}{k} \left(p^k\right)^{-s}		= \sum_{k=1}^\infty \frac{1}{k} \left(p^k\right)^{-s}
	</math>		</math>.

	We thus have:		We thus have:
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	= \sum_{p \text{ prime}} - \ln \left( 1 - p^{-s} \right)		= \sum_{p \text{ prime}} - \ln \left( 1 - p^{-s} \right)
	= \sum_{p \text{ prime}} \sum_{k=1}^\infty \frac{1}{k} \left(p^k\right)^{-s}		= \sum_{p \text{ prime}} \sum_{k=1}^\infty \frac{1}{k} \left(p^k\right)^{-s}
	</math>		</math>.

	==== von Mangoldt function ====		==== von Mangoldt function ====
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	0 & \text{otherwise}		0 & \text{otherwise}
	\end{cases}		\end{cases}
	</math>		</math>.

	Reindex the double sum.		Reindex the double sum.
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	<math>\displaystyle		<math>\displaystyle
	\frac{\Lambda(n)}{\ln n} = \frac{\ln p}{k \ln p} = \frac{1}{k}		\frac{\Lambda(n)}{\ln n} = \frac{\ln p}{k \ln p} = \frac{1}{k}
	</math>		</math>.

	Therefore:		Therefore:
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	<math>\displaystyle		<math>\displaystyle
	\ln \zeta(s) = \sum_{n=2}^\infty \frac{\Lambda(n)}{\ln n} \cdot \frac{1}{n^s}		\ln \zeta(s) = \sum_{n=2}^\infty \frac{\Lambda(n)}{\ln n} \cdot \frac{1}{n^s}
	</math>		</math>,

	which is valid for \(\mathrm{Re}(s) > 1\), with the sum starting at \(n=2\) because \(\Lambda(1) = 0\).		which is valid for \(\mathrm{Re}(s) > 1\), with the sum starting at \(n=2\) because \(\Lambda(1) = 0\).

Lériendil at 17:54, 9 April 2025

2025-04-09T17:54:31Z

← Older revision		Revision as of 17:54, 9 April 2025
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	</math>		</math>

	=== Logarithmic Series Expansion ===		==== Logarithmic Series Expansion ====

	For \( \|x\| < 1 \), we have the well-known series:		For \( \|x\| < 1 \), we have the well-known series:
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	</math>		</math>

	=== von Mangoldt function ===		==== von Mangoldt function ====

	The von Mangoldt function \(\Lambda(n)\) identifies prime powers:		The von Mangoldt function \(\Lambda(n)\) identifies prime powers:

Lériendil: copied sintel's derivation

2025-04-09T17:54:12Z

copied sintel's derivation

@@ Line 5: / Line 5: @@
 === 1a. Dirichlet series for the von Mangoldt function ===
-{{todo|derive this|inline=1}}
+Start with the Euler Product for \(\zeta(s)\).
 === 1b. Conversion factor for removing primes ===

ArrowHead294 at 15:01, 9 April 2025

2025-04-09T15:01:51Z

← Older revision		Revision as of 15:01, 9 April 2025
Line 1:		Line 1:
	{{breadcrumb}}		{{breadcrumb}}

	== 1. Euler product expression for the zeta function ==		== 1. Euler product expression for the zeta function ==
	{{todo\|derive this\|inline=1}}		{{todo\|derive this\|inline=1}}