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{\bf THE PRIME NUMBER THEOREM}

by G.J.O. Jameson

Cambridge University Press (2002)

in the series {\it London Mathematical Society Student Texts} 

{\bf Short description}
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The prime number theorem is unquestionably one of the great theorems 
of mathematics, but it is often presented as an outlying topic of
number theory because the proof is deeply analytic.  The rationale
for this book is that the prime number theorem (together with
other results in the same family) merits an account in its own right. 
This follows the tradition  of A.E. Ingham's classic {\it The Distribution 
of Prime Numbers}, but the book is at a lower level than Ingham, suitable for
current undergraduate degree courses, with prerequisites kept to a minimum.  
The basic method followed is a new variant of the classical method which
allows the simultaneous derivation of the associated series results.
Newman's method is given as an alternative, as well as one version of
the ``elementary" proof.  Also, (unlike Ingham), the book includes Dirichlet's
theorem and applications such as the enumeration of numbers with $k$
prime factors. 

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{\bf Contents}
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{\bf Preface}

{\bf 1. Foundations}

\begin{tabular}{rl}
 1.1 & Counting prime numbers \\
 1.2 & Arithmetic functions \\
 1.3 & Abel summation \\
 1.4 & Estimation of sums by integrals: Euler's summation formula \\
 1.5 & The function li$(x)$ \\
 1.6 & Chebyshev's theta function\\
 1.7 & Dirichlet series and the zeta function \\
 1.8 & Convolutions \\
\end{tabular}

\pagebreak
{\bf 2. Some important Dirichlet series and arithmetic functions}

\begin{tabular}{rl}
 2.1 & The Euler product \\
 2.2 & The M\"obius function \\
 2.3 & The series for~ $\log \zeta (s)$ ~and~ $\zeta'(s)/\zeta (s)$ \\
 2.4 & Chebyshev's psi function and powers of primes\\
 2.5 & Summation of some arithmetic functions \\
 2.6 & Mertens's estimates \\
\end{tabular}

{\bf 3.  The basic theorems}

\begin{tabular}{rl}
 3.1 & Extension of the definition of the zeta function \\
 3.2 & Inversion of Dirichlet series; the integral version of the
             fundamental theorem\\
 3.3 & An alternative method:  Newman's proof \\
 3.4 & The limit and series versions of the fundamental theorem;
             the prime number theorem \\
 3.5 & Some applications of the prime number theorem \\  
\end{tabular}

{\bf 4.  Prime numbers in residue classes:  Dirichlet's theorem}

\begin{tabular}{rl}
 4.1 & Characters of finite abelian groups\\
 4.2 & Dirichlet characters \\
 4.3 & Dirichlet $L$-functions \\
 4.4 & Prime numbers in residue classes \\ 
\end{tabular}

{\bf 5.  Error estimates and the Riemann hypothesis}

\begin{tabular}{rl}
 5.1 & Error estimates \\
 5.2 & Connections with the Riemann hypothesis\\
 5.3 & The zero-free region of the zeta function \\
\end{tabular}

{\bf 6. An ``elementary" proof of the prime number theorem} 

\begin{tabular}{rl}
 6.1 & Framework of the proof \\
 6.2 & Selberg's formulae and completion of the proof
\end{tabular}

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{\bf Appendices}

\begin{tabular}{ll}
A & Complex functions of a real variable \\
B & Double series and multiplication of series \\
C & Infinite products \\
D & Differentiation under the integral sign\\
E & The $O,o$ notation \\ 
F & Computing values of $\pi (x)$ \\
G & Table of primes  \\
H & Biographical notes 
\end{tabular}

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