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Added summatory functions stats for additive functions + reference cited |
→Summatory functions: More content from I&K book |
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:<math>\sum_{n \leq x} |f(n) - E(x)|^2 \leq C_f \cdot x D^2(x).</math>
Let
:<math> \nu(x; z) := \frac{1}{x} \#\left\{n \leq x: \frac{f(n)-A(x)}{B(x)} \leq z\right\}.</math>
Suppose that <math>f</math> is an additive function with <math>-1 \leq f(p^{\alpha}) = f(p) \leq 1</math>
such that as <math>x \rightarrow \infty</math>,
:<math>B(x) = \sum_{p \leq x} f^2(p) / p \rightarrow \infty. </math>
Then <math>\nu(x; z) \sim G(z)</math> where <math>G(z)</math> is the [[normal distribution|Gaussian distribution function]]
:<math>G(z) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{z} e^{-t^2/2} dt.</math>
Examples of this result related to the [[prime omega function]] and the numbers of prime divisors of shifted primes include the following for fixed <math>z \in \mathbb{R}</math> where the relations hold for <math>x \gg 1</math>:
:<math>\#\{n \leq x: \omega(n) - \log\log x \leq z (\log\log x)^{1/2}\} \sim x G(z), </math>
:<math>\#\{p \leq x: \omega(p+1) - \log\log x \leq z (\log\log x)^{1/2}\} \sim \pi(x) G(z). </math>
== See also ==
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