Zeta Function

A Zeta Function is a meromorphic function that is an infinite series function of the form [math]\displaystyle{ \zeta(s) = \sum_{n=1}^{\infty} \frac{a_n}{n^s} }[/math], where [math]\displaystyle{ a_n }[/math] are arithmetic or geometric coefficients.

• Context:
  • It can (typically) be defined as a function that sums a series of the form \(\zeta(s) = \sum_{n=1}^{\infty} \frac{a_n}{n^s}\), where \(a_n\) are coefficients that depend on the specific type of zeta function.
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  • It can range from simple forms like the Riemann Zeta Function to more complex zeta functions like the Hurwitz Zeta Function and the L-functions associated with modular forms.
  • It can exhibit functional equations that relate the values of the zeta function at \(s\) to its values at \(1-s\), revealing symmetry properties important in analytic number theory.
  • It can be represented through integral formulas, such as the Mellin transform or the Euler product, linking the zeta function to prime numbers.
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• Example(s):
  • (Riemann Zeta Function) Riemann Zeta Functions, with applications in number theory and the study of prime distributions.
  • (Hurwitz Zeta Function) Hurwitz Zeta Functions, which generalize the Riemann zeta function by adding a real or complex shift.
  • (Dedekind Zeta Function) Dedekind Zeta Functions, which extend the zeta function to algebraic number fields.
  • (L-Function) L-functions, a broad class of zeta functions associated with automorphic forms and representation theory.
  • (Epstein Zeta Function) Epstein Zeta Functions, which generalize the Riemann zeta function to quadratic forms.
  • (Selberg Zeta Function) Selberg Zeta Functions, associated with the lengths of closed geodesics on Riemann surfaces.
  • (Hasse-Weil Zeta Function) Hasse-Weil Zeta Functions, which are connected to algebraic varieties over finite fields.
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• Counter-Example(s):
  • Dirichlet Eta Function, which, while related, is a different function with distinct properties, particularly its series representation and convergence.
  • Polylogarithm, a special function related to the zeta function but with a different definition and series expansion.
• See: L-Function, Hurwitz Zeta Function, Dedekind Zeta Function, Analytic Continuation, Number Theory, Prime Number Theorem, Riemann Hypothesis, Functional Equation, Euler Product, Mellin Transform, Dirichlet Eta Function.

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References

2024

• (Wikipedia, 2024) ⇒ https://en.wikipedia.org/wiki/Zeta_function Retrieved:2024-8-26.
  • A zeta function is a special function of a complex variable that, in many cases, is defined by an infinite series and can be extended analytically to the whole complex plane except for isolated singularities (poles). These functions are central to number theory, mathematical analysis, and theoretical physics. Notable examples include the Riemann Zeta Function, Hurwitz Zeta Function, and Dedekind Zeta Function. Each zeta function is associated with deep mathematical conjectures and theorems, particularly in understanding prime number distributions and modular forms.

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