Analytic Number Theory

Analytic number theory uses the methods of asymptotic approximations, complex analysis, and notions from probability theory to describe integers; in particular the distribution of the prime numbers. Often methods of harmonic analysis are brought to bear on some of these problems. Major themes are:

  1. An arithmetic function is a sequence $f:\mathbb{N} \rightarrow \mathbb{C}$. In trying to understand such a thing, one often resorts to trying to understand a sum of divisors $\sum_{d \vert n} \, f(d)$, the theory of the Riemann-Stieltjes integral along with the Euler-Maclaurin formula, and comparison of a sum to an integral.
  2. The Riemann-Zeta function is the meromorphic continuation of \[ \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}, \text{ where } \Re(s) > 1 \text{ and } s \in \mathbb{C}. \] to the complex plane. The fact that this series has an Euler-product: \[ \zeta(s) = \prod_{p - \text{prime}} \left( 1 - \frac{1}{p} \right)^{-1} \] gives a relation between the Riemann-Zeta function in the primes. The questions surrounding this relationship are deep, nuanced, and mysterious, to say the least.
  3. One can abstract the above series. Let $f$ be some arithmetic function. Then one can consider the Dirichlet series: \[ \sum_{n=1}^{\infty} \frac{f(n)}{n^s}. \] These generating functions converge (or don't) based on the arithmetic function $f$. At times, it is more fruitful to use the generating function to understand the arithmetic function, than trying to study the arithmetic function directly.
  4. It was through studying such objects, for real $s$, that Dirichlet proved his famous theorem on primes in an arithmetic sequence. The proof was achieved through studying so-called $L$-series \[ L(s, \chi) = \sum_{n=1}^{\infty} \frac{\chi(n)}{n^s} \] where $\chi$ is a Dirichlet character. A Dirichlet character has some nice properties:
    1. $\chi(1) = 1$
    2. $\chi(mn) = \chi(m)\chi(n)$
    3. $\vert \chi(n) \vert \in \{0,1\}$.
  5. Which functions have:
    1. An analytic continuation to a meromorphic function to the complex plane.
    2. A functional equation.
    3. An Euler Product.
    These are the prime characteristics of the Riemann-Zeta function. Can the study of functions, that have these properties, lend insight into the Reimann-Zeta function? This leads one to study $L$-functions. $L$-functions give one the tools to study arithmetic functions, geometric curves, and number fields, through the use of analysis. Understanding of $L$-functions is still largely conjectural.

References:

  1. Apostol, Tom, M., Introduction to Analytic Number Theory , Undergraduate Texts in Mathematics, Springer, 1976
  2. DeKoninck, Jean-Marie, Luca, Florian, Analytic Number Theory: The Anatomy of the Integers , Graduate Studies in Mathematics, vol. 134, American Mathematical Society
  3. Tenenbaum, G´rald, France, Michel, Mendès, The Prime Numbers and Their Distribution , Student Mathematical Libary, vol. 6, American Mathematical Society
  4. Tenenbaum, G´rald, Introduction to Analytic and Probabilistic Number Theory , Graduate Studies in Mathematics, vol. 163, American Mathematical Society