The central problem of prime number theory is to develop good estimates for \( \pi(x) \), the number of primes less than or equal to \( x \), as \( x \) goes to infinity. The first result in this direction was Euclid’s proof that there are infinitely many primes, that is, \( \pi(x) \rightarrow \infty \). A major milestone was the proof of the Prime Number Theorem in 1896, independently by Hadamard and de la Vallée Poussin, which states that \( \pi(x) \) is asymptotic to \( x/ \ln x\), that is, the ratio \( \pi(x)/(x/ \ln x) \rightarrow 1\) as \( x \rightarrow \infty \). There has been much additional work in estimating the error term \( \pi(x) - \textrm{li}(x) \), where \( \textrm{li}(x) \) is the logarithmic integral

\( \textrm{li}(x)=\int_{0}^{x} 1/\ln(t) dt \)

that is also asymptotic to \( x/ \ln x\) but gives a better estimate of \( \pi(x) \).

The present book gives comprehensive coverage of the development of these results from antiquity until about 1910, when progress in developing new estimates slowed down. The great value of the book is that it not only quotes the results, but gives the original proofs and shows how our knowledge improved over time. It does not adhere strictly to the 1910 cutoff, but cites later simplifications to the proofs already presented, and cites some later results, for example in the twin prime problem, but the coverage here is much less detailed and usually does not include proofs. The book has a very useful coda from 1923: its last section is a detailed analysis of Hardy and Littlewood’s famous and influential paper “Some Problems of ‘Partitio Numerorum’; III: On the Expression of a Number as a Sum of Primes”. That paper studies and makes conjectures about fifteen problems in prime number theory, including the twin prime problem and Goldbach’s conjecture. The present book describes all the conjectures and gives the current status of research into them.

The main focus of the book is the Prime Number Theorem, but it also considers many other questions dealing with primes, such as formulas for primes, primes in arithmetic progressions, twin primes, etc. The book includes a short set of exercises after each chapter; usually these are published results (given with the reference) that are interesting but not in the main line of the exposition.

After Euclid’s result, there was not much progress for about 2,000 years, until in the early 1800s when Gauss and Legendre independently conjectured, by studying tables of primes, that \( \pi(x) \) was about \( x/ \ln x\). Gauss further conjectured (correctly) that a better estimate was the logarithm integral \( \textrm{li}(x)\).

The next breakthrough was an 1860 paper by Riemann. He considered the function defined by the series

\( \zeta(s)=\sum_{n=1}^{\infty} 1/n^{s} \)

now called the Riemann zeta function, that had been studied by Euler for real values of \( s \), and extended it to a function of a complex variable \( s \). He outlined properties of this function and outlined how its properties could be used to prove the Prime Number Theorem. At that time not enough was known about function theory to carry out Riemann’s plan, but the subject was developed greatly over the next few decades and resulted in the 1896 proofs of the Prime Number Theorem. Function theory continued to be intensively developed and resulted in many improved error terms for the Prime Number Theorem (Edmund Landau was one of the greatest innovators in both fields).

The present book is not strictly a history, because it does not contain any biographical details. It can best be thought of as a collection of proofs throughout history, with extensive annotations. It is a very different book from Dickson’s

*History of the Theory of Numbers*, which covers this topic in Chapter XVIII of Volume I; Dickson is a collection of facts with references, but does not show any proofs. There is somewhat more overlap with Landau’s

*Handbuch der Lehre von der Verteilung der Primzahlen.* That book includes a 50-page historical overview of the development of prime number theory, although it generally quotes results that will be proved later in the book and does not give the historical proofs as the present book does.

Allen Stenger is a math hobbyist and retired software developer. He was Number Theory Editor of the Missouri Journal of Mathematical Sciences

from 2010 through 2021. His personal website is

allenstenger.com. His mathematical interests are number theory and classical analysis.