论文标题
自然波数，自然波共同数量和素数计算
Natural Wave Numbers, Natural Wave Co-numbers, and the Computation of the Primes
论文作者
论文摘要
该论文利用自然数n和一个统一根部的空间u之间的同构u之间的同构，以构建代表和计算质量数的递归程序。第n个波数$ {\ bf u} _n $是unity的nth根部的可数顺序，对于所有整数阶段k，频率为k/n。在交换和关联的二进制操作$ {\ bf u} _m \ odot {\ bf u} _n = {\ bf u} _ {mn} $下，将空间U关闭。函数在u上定义为两个互补序列，其中co-number $ {\ overset {\ bf \ ast} {\ bf u}} _ n $是波浪数的函数，其中zeros替换了其正统一根。递归过程$ {\ overset {\ bf \ ast} {\ bf u}} _ {n+1} = {\ overset {\ bf \ ast} {\ bf ast} {\ bf u}} _} _ {n} u}} _ {n+1} $就质数之前的质量数字表示，以$ p_1 = 2 $开头，并且从未终止。如果$ {p} _1，...，{p} _ {n+1} $是第一个$ n+1 $ prime阶段，则与$ p_ {n+1} \ leq k <p^2_ {n+1} $相关的范围内的阶段，与$ {\ \ bersets {\ bff \ bf \ bf \ bf \ bf \ bf \ bf \ bf. u}} _ {n} $与$ p_1，...，p_n $，所有prime阶段都小于$ p^2_ {n+1} $。当与上一步中确定的所有素数应用时，递归过程大约在每次迭代中以$ n> 1 $的价格确定$ 7^{2（n-1）}/（2（n-1）ln7）$ primes。当波数的阶段以模块化算术表示时，素阶段可以用素数初始集合的倒数总和表示，并且与zeta功能有关系。
The paper exploits an isomorphism between the natural numbers N and a space U of periodic sequences of the roots of unity in constructing a recursive procedure for representing and computing the prime numbers. The nth wave number ${\bf u}_n$ is the countable sequence of the nth roots of unity having frequencies k/n for all integer phases k. The space U is closed under a commutative and associative binary operation ${\bf u}_m \odot{\bf u}_n={\bf u}_{mn}$, termed the circular product, and is isomorphic with N under their respective product operators. Functions are defined on U that partition wave numbers into two complementary sequences, of which the co-number $ {\overset {\bf \ast }{ \bf u}}_n$ is a function of a wave number in which zeros replace its positive roots of unity. The recursive procedure $ {\overset {\bf \ast }{ \bf U}}_{N+1}= {\overset {\bf \ast }{ \bf U}}_{N}\odot{\overset {\bf \ast }{\bf u}}_{N+1}$ represents prime numbers explicitly in terms of preceding prime numbers, starting with $p_1=2$, and is shown never to terminate. If ${p}_1, ... , { p}_{N+1}$ are the first $N+1$ prime phases, then the phases in the range $p_{N+1} \leq k < p^2_{N+1}$ that are associated with the non-zero terms of $ {\overset {\bf \ast }{\bf U}}_{N}$ are, together with $ p_1, ...,p_N$, all of the prime phases less than $p^2_{N+1}$. When applied with all of the primes identified at the previous step, the recursive procedure identifies approximately $7^{2(N-1)}/(2(N-1)ln7)$ primes at each iteration for $ N>1$. When the phases of wave numbers are represented in modular arithmetic, the prime phases are representable in terms of sums of reciprocals of the initial set of prime phases and have a relation with the zeta-function.
