# prime number theorem

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##### 2: 27.11 Asymptotic Formulas: Partial Sums
where $\left(h,k\right)=1$, $k>0$. …
27.11.15 $\lim_{x\to\infty}\sum_{n\leq x}\frac{\mu\left(n\right)\ln n}{n}=-1.$
Each of (27.11.13)–(27.11.15) is equivalent to the prime number theorem (27.2.3). The prime number theorem for arithmetic progressions—an extension of (27.2.3) and first proved in de la Vallée Poussin (1896a, b)—states that if $\left(h,k\right)=1$, then the number of primes $p\leq x$ with $p\equiv h\pmod{k}$ is asymptotic to $x/(\phi\left(k\right)\ln x)$ as $x\to\infty$.
##### 3: 27.2 Functions
(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) This result, first proved in Hadamard (1896) and de la Vallée Poussin (1896a, b), is known as the prime number theorem. …This is the number of positive integers $\leq n$ that are relatively prime to $n$; $\phi\left(n\right)$ is Euler’s totient. …
##### 4: Tom M. Apostol
In 1998, the Mathematical Association of America (MAA) awarded him the annual Trevor Evans Award, presented to authors of an exceptional article that is accessible to undergraduates, for his piece entitled “What Is the Most Surprising Result in Mathematics?” (Answer: the prime number theorem). …
##### 5: 25.16 Mathematical Applications
The prime number theorem (27.2.3) is equivalent to the statement …
##### 6: 25.10 Zeros
Also, $\zeta\left(s\right)\neq 0$ for $\Re s=1$, a property first established in Hadamard (1896) and de la Vallée Poussin (1896a, b) in the proof of the prime number theorem (25.16.3). …
##### 7: Bibliography
• T. M. Apostol (2000) A Centennial History of the Prime Number Theorem. In Number Theory, Trends Math., pp. 1–14.
• ##### 8: Errata
• Paragraph Prime Number Theorem (in §27.12)

The largest known prime, which is a Mersenne prime, was updated from $2^{43,112,609}-1$ (2009) to $2^{82,589,933}-1$ (2018).

• ##### 9: 25.15 Dirichlet $L$-functions
Related results are: …
##### 10: 27.16 Cryptography
###### §27.16 Cryptography
For example, a code maker chooses two large primes $p$ and $q$ of about 400 decimal digits each. Procedures for finding such primes require very little computer time. The primes are kept secret but their product $n=pq$, an 800-digit number, is made public. … Thus, $y\equiv x^{r}\pmod{n}$ and $1\leq y. …