Zeta Equivalence to Prime Number Theorem
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This needs considerable tedious hard slog to complete it. In particular: This page was taken from an early version of Prime Number Theorem, extracted from where it was a subsection of a longer proof. It needs to be expanded and the unexplained terminology explained. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Finish}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Theorem
Let $\map \zeta z$ be the Riemann $\zeta$ function.
The Prime Number Theorem is logically equivalent to the statement that the average of the first $N$ coefficients of $\dfrac {\zeta'} {\zeta}$ tend to $-1$ as $N$ goes to infinity.
This article, or a section of it, needs explaining. In particular: What does $z$ range over, and what does it mean by "first $N$ coefficients" of $\dfrac {\zeta'} {\zeta}$? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code. |
Proof
The Von Mangoldt Equivalence is equivalent (clearly) to the statement that the average of the coefficients of the function of $z$ defined as:
- $(1): \quad \ds \sum_{n \mathop = 1}^\infty \frac {\map \Lambda n} {n^z}$
tend to $1$.
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Let $ \set {p_1, p_2, p_3, \dots}$ be an enumeration of the prime numbers:
- $\set { 2, 3, 5, 7, 11, \dots}$
In the proof of the Von Mangoldt Equivalence, in the sum of von Mangoldt function, the $\map \ln p$ term will appear once for each power of $p$.
So, we expand out $(1)$ as:
\(\ds \sum_{n \mathop = 1}^\infty \frac{\map \Lambda n} {n^z}\) | \(=\) | \(\ds \map \ln {p_1} \paren {\frac 1 {p_1^z} + \frac 1 {p_1^{2 z} } + \frac 1 {p_1^{3 z} } + \cdots} + \map \ln {p_2} \paren {\frac 1 {p_2^z} + \frac 1 {p_2^{2 z} } + \cdots} + \cdots\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \ln {p_1} \sum_{n \mathop = 1}^\infty \paren {\paren {p_1^{-z} }^n} + \map \ln {p_2} \sum_{n \mathop = 1}^\infty \paren {\paren {p_2^{-z} }^n} + \cdots\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \ln {p_1} \frac {p_1^{-z} } {1 - p_1^{-z} } + \map \ln {p_2} \frac {p_2^{-z} } {1 - p_2^{-z} } + \cdots\) | Sum of Infinite Geometric Sequence | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{p \text{ prime} } \map \ln p \frac {p^{-z} } {1 - p^{-z} }\) |
This function of $z$ can be recognized as:
\(\ds \sum_{p \text{ prime} } \map \ln p \frac {p^{-z} } {1 - p^{-z} }\) | \(=\) | \(\ds \sum_{p \text{ prime} } \paren {1 - p^{-z} } \frac {-\paren {0 - \map \ln p p^{-z} } } {\paren {1 - p^{-z} }^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{p \text{ prime} } \frac \d {\d z} \map \ln {\frac {-1} {1 - p^{-z} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map {\frac \d {\d z} } {\sum_{p \text{ prime} } \map \ln {\frac {-1} {1 - p^{-z} } } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map {\frac \d {\d z} } {\ln \prod_{p \text{ prime} } \frac {-1} {1 - p^{-z} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\frac \d {\d z} \map \ln {\map \zeta z}\) | $\ds \prod_{p \text{ prime} } \frac 1 {1 - p^{-z} }$ is the Riemann zeta function | |||||||||||
\(\ds \) | \(=\) | \(\ds -\frac {\map {\zeta'} z} {\map \zeta z}\) |
Hence the result.
$\blacksquare$