解析数论是使用数学分析作为工具来解决数论问题的分支。
解析数论是使用数学分析作为工具来解决数论问题的分支。微积分和复变函数论发展以后，产生了解析数论。该学科的第一个主要成就是狄利克雷用解析方法证明了Dirichlet's theorem on arithmetic progressions。依靠黎曼zeta函数对素数定理的证明是另一个里程碑。 解析数论是解决数论中艰深问题的重要工具，数论中有些问题必须由解析方法才能提出或解决。 中国的华罗庚、王元、陈景润等人在“哥德巴赫猜想”、“华林问题”等解析数论问题上取得世界公认的成就。黎曼ζ函数Riemann zeta function
In mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of great significance in number theory because of its relation to the distribution of prime numbers. It also has applications in other areas such as physics, probability theory, and applied statistics.
The Riemann hypothesis, a conjecture about the distribution of the zeros of the Riemann zeta function, is considered by many mathematicians to be the most important unsolved problem in pure mathematics.
Definition
The Riemann zeta-function ζ(s) is the function of a complex variable s initially defined by the following infinite series:As a Dirichlet series with bounded coefficient sequence this series converges absolutely to an analytic function on the open half-plane of s such that Re(s) > 1 and diverges on the open half-plane of s such that Re(s) < 1. The function defined by the series on the half-plane of convergence can however be continued analytically to all complex s ≠ 1. For s = 1 the series is formally identical to the harmonic series which diverges to infinity. As a result, the zeta function becomes a meromorphic function of the complex variable s, which is holomorphic in the region {s ∈ C : s ≠ 1} of the complex plane and has a simple pole at s = 1 with residue 1.Specific values
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