Section 1.1 Prologue
The center stage of modern Number Theory has been occupied by \(L\)–functions – a focal point of research for both algebraic and analytic number theorists. The primordial example is the Riemann Zeta function, whose modern definition was motivated by a problem posed by Pietro Mengoli in 1650, now known as the Basel Problem: what is the value of the following sum:
\begin{gather}
1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots=\sum_{n=1}^{\infty}\frac{1}{n^2}?\tag{1.1}
\end{gather}
Almost a century later, in 1734, Leonhard Euler proved that the precise value of the sum (1.1) is \(\frac{\pi^2}{6}\text{.}\) He announced the proof on December 5, 1735, in the Saint Petersburg Academy of Sciences. After further investigations, in 1737, he discovered its connection to number theory. Euler considered the following general function of a real variable \(s\)
\begin{gather}
\zeta(s)\colonequals\sum_{n=1}^\infty\frac{1}{n^s}.\tag{1.2}
\end{gather}
He proved that \(\zeta(s)\) converges for \(s\gt 1\) and the fundamental theorem of arithmetic implies that
\begin{gather}
\zeta(s)=\prod_{p}\bparen{1-\frac{1}{p^s}}^{-1},\tag{1.3}
\end{gather}
where the product is taken over all integer primes \(p\text{.}\) Identity (1.3) is known as the Euler Product formula. He also used his product formula (1.3) to give an alternate proof [13, Theorem 7] that there are infinitely many primes and that the arithmetic density of primes in the integers is zero.
100 years later, in 1837, Peter Gustav Lejeune Dirichlet introduced what is known today as Dirichlet \(L\)–functions in his seminal paper [12]. Building on Euler’s work, he proved that \(L(1,\chi)\) is nonzero for a nonprincipal Dirichlet character \(\chi\text{,}\) and that implies that there are infinitely many primes in an arithmetic progression whose first term is coprime to its common difference.
1
In English: Proof of the theorem that every unbounded arithmetic progression, whose first term and common difference are integers without common factors, contains infinitely many prime numbers.
In 1859, while working on an attempt to prove the Prime Number Theorem, Bernhard Riemann studied the analytic properties of \(\zeta(s)\) in depth in his seminal paper [37], where \(s\) varies over complex numbers instead of just reals. He established the convergence of \(\zeta(s)\) for \(\Re(s)\gt 1\) and showed the existence of its analytic continuation to all of \(\C\) except for a simple pole at \(s=1\text{.}\) Moreover, he proved functional equations for the \(\zeta(s)\text{.}\) He conjectured, now widely known as the Riemann Hypothesis, that all the nontrivial zeros lie on the line \(\Re(s)=1/2\text{,}\) which has since remained a focal point of profound significance in modern mathematics. His work showed promising connections between the study of the distribution primes and complex analysis.
2
In English: On the Number of Primes Less Than a Given Magnitude.
