# 解析数论

（重定向自解析數論

## 解析数论的分支

• 處理的是質數的分佈，例如估計一個區間內的質數個數，包括質數定理及狄利克雷定理
• 堆疊數論是有關整數的堆疊結構，像是哥德巴赫猜想認為所有大於2的偶數都可以表示為二個質數的和。另一個堆疊數論的主要成果是華林問題的和。

## 問題及結果

### 乘性數論

${\displaystyle \,\int _{2}^{N}{\frac {1}{\log \,t}}\,dt.}$

${\displaystyle \pi (x)=({\text{number of primes }}\leq x),}$

${\displaystyle \lim _{x\to \infty }{\frac {\pi (x)}{x/\log x}}=1.}$

### 堆疊數論

${\displaystyle n=x_{1}^{k}+\cdots +x_{\ell }^{k}.\,}$

${\displaystyle G(k)\leq k(3\log k+11).\,}$

### 丟番圖方程

${\displaystyle x^{2}+y^{2}\leq r^{2}.}$

2000年馬丁·赫胥黎英语Martin Huxley證明了[5]${\displaystyle E(r)=O(r^{131/208})}$，是目前最好的結果。

## 參考資料

1. Apostol 1976, p. 7.
2. ^ Davenport 2000, p. 1.
3. ^ 哥德巴赫猜想中的「x＋y」表示是「所有充分大的偶數都能表示成兩個數之和，並且兩個數的質因數個數分別都不超過x個及y個」
4. ^ 陈景润. 大偶数表为一个素数及一个不超过二个素数的乘积之和. 中国科学A辑. 1973, (2): 111–128.
5. ^ M.N. Huxley, Integer points, exponential sums and the Riemann zeta function, Number theory for the millennium, II (Urbana, IL, 2000) pp.275–290, A K Peters, Natick, MA, 2002, MR1956254.

## 延伸閱讀

• Ayoub, Introduction to the Analytic Theory of Numbers
• H. L. Montgomery and R. C. Vaughan, Multiplicative Number Theory I : Classical Theory
• H. Iwaniec and E. Kowalski, Analytic Number Theory.
• D. J. Newman, Analytic number theory, Springer, 1998

On specialized aspects the following books have become especially well-known:

Certain topics have not yet reached book form in any depth. Some examples are (i) Montgomery's pair correlation conjecture and the work that initiated from it, (ii) the new results of Goldston, Pintz and Yilidrim on small gaps between primes, and (iii) the Green–Tao theorem showing that arbitrarily long arithmetic progressions of primes exist.