Elementary Results on the Distribution of Primes

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Chapter 6 Elementary Results on the Distribution of Primes

Recall that the von Mangoldt function $\Lambda$ is defined by $\Lambda(n) = \log p$ if $n = p^k$ is a power of a prime ($p$ prime, $k \geq 1$), $\Lambda(n) = 0$ otherwise (in particular, $\Lambda(1) = 1$). We consider the following three functions:

\begin{equation} \pi(x) = \sum_{p \leq x} 1, \qquad \theta(x) = \sum_{p \leq x} \log p, \qquad \psi(x) = \sum_{n \leq x} \Lambda(n)\text{.}\tag{6.0.1} \end{equation}

Here $\pi$ is the prime-counting function, $\theta$ is called Chebyshev’s $\theta$ function, and $\psi$ is called Chebyshev’s $\psi$ function.

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