解析数论

解析数论的分支

• 乘性数论处理的是质数的分布，例如估计一个区间内的质数个数，包括质数定理及狄利克雷定理
• 堆叠数论是有关整数的堆叠结构，像是哥德巴赫猜想认为所有大于2的偶数都可以表示为二个质数的和。另一个堆叠数论的主要成果是华林问题的和。

问题及结果

乘性数论

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

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

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

堆叠数论

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

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

丢番图方程

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

2000年马丁·赫胥黎英语Martin Huxley证明了${\displaystyle E(r)=O(r^{131/208})}$，是目前最好的结果。

参考书目

• Davenport, Harold, Multiplicative number theory, Graduate Texts in Mathematics 74 3rd revised, New York: Springer-Verlag, 2000, ISBN 978-0-387-95097-6, MR 1790423
• Tenenbaum, Gérald, Introduction to Analytic and Probabilistic Number Theory, Cambridge studies in advanced mathematics 46, Cambridge University Press, 1995, ISBN 0-521-41261-7

延伸阅读

• 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:

• Titchmarsh, Edward Charles, The Theory of the Riemann Zeta Function 2nd, Oxford University Press, 1986
• H. Halberstam and H. E. Richert, Sieve Methods
• R. C. Vaughan, The Hardy–Littlewood method, 2nd. edn.

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.