EncyclopedAI

Retrieving "Analytic Continuation" from the archives

Cross-reference notes under review

While the archivists retrieve your requested volume, browse these clippings from nearby entries.

  1. Dirichlet Series

    Linked via "analytic continuation"

    $$D(s) = \prod{p \text{ prime}} \left( \sum{k=0}^{\infty} \frac{(\chi(p))^k}{p^{ks}} \right) = \prod_{p \text{ prime}} \frac{1}{1 - \chi(p)p^{-s}}$$
    This multiplicative structure allows analytic information about $D(s)$ (such as poles or analytic continuation) to directly translate into information about the arithmetic properties of the coefficients $\chi(n)$.
    Analytic Continuation and Poles

  2. Dirichlet Series

    Linked via "analytic continuation"

    Analytic Continuation and Poles
    Most Dirichlet series encountered in number theory admit analytic continuation to the entire complex plane, usually with finitely many poles. The location and order of these poles dictate asymptotic behavior in the summation of the coefficients.
    The most celebrated example is the Riemann Zeta Function, $\zeta(s)$, where $a_n = 1$. This series has a simple pole at $s=1$. Its [analy…

  3. Dirichlet Series

    Linked via "analytic continuation"

    Most Dirichlet series encountered in number theory admit analytic continuation to the entire complex plane, usually with finitely many poles. The location and order of these poles dictate asymptotic behavior in the summation of the coefficients.
    The most celebrated example is the Riemann Zeta Function, $\zeta(s)$, where $a_n = 1$. This series has a simple pole at $s=1$. Its [analytic continuation](/entries/analytic-…

  4. Dirichlet Series

    Linked via "analytic continuation"

    The most celebrated example is the Riemann Zeta Function, $\zeta(s)$, where $a_n = 1$. This series has a simple pole at $s=1$. Its analytic continuation is vital for understanding the distribution of prime numbers.
    Other significant examples include Dirichlet L-functions, $L(s, \chi)$, which are Dirichlet series associated with Dirichlet characters $\chi$. These functions ar…

  5. Number Theory

    Linked via "analytic continuation"

    The Riemann Zeta Function, $\zeta(s)$, is arguably the most significant function in analytic number theory. Defined for complex numbers $s$ with $\text{Re}(s) > 1$ by the Dirichlet series:
    $$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$$
    It admits analytic continuation over the entire complex plane, except for a simple pole at $s=1$.
    The profound connection between $\zeta(s)$ and the prime numbers is established via the Euler product formula: