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Results tagged with prime-number-theorem
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user 37555
The Prime Number Theorem is a theorem that describes the distribution of the primes. It says that the number of primes less than or equal to a real number $x$ is asymptotic to $\frac{x}{\ln x}$.
10
votes
Does the Prime Number Theorem have anything to do with Erdos-Kac law or vice versa?
The right context of your question is probably the area of Beurling primes. An arithmetic semigroup is a semigroup $(G,\cdot)$ together with a norm $\|\cdot\|:G\rightarrow[1,\infty)$, such that $\|gh\ …
10
votes
Can anything deep be said uniformly about conjectures like Goldbach's?
There exist surprising counterexamples. Elsholtz and Dietmann found the following: If $p\equiv 7\pmod{8}$ is prime, then the equation $x^2+y^2+z^4=p^2$ has no non-trivial solution. You might argue tha …
4
votes
Accepted
Numbers related to the Riemann hypothesis
Proving such results falls into three parts. First you take an effective version of the prime number theorem, which implies all your desired bound for sufficient. Second you write a computer program ( …
2
votes
Divisor sums over values of binary forms of primes
When computing e.g. an asymptotic for $\sum_{p\leq x}d(p-1)$ you would like to estimate the number of primes $p$ such that $n$ divides $p-1$ by the prime number theorem as $\sim\frac{x}{\varphi(n)\log …
1
vote
Accepted
On a coprime generalization of Cramer's conjecture
Put $B=\lfloor n^r\rfloor$, and let $n_0$ be the smallest integer larger than $n$, which is divisible by $B$. If for an integer $k$ we have $(\frac{n_0}{B}+k, B)=1$, then $n_1 = n_0+kB$ satisfies your …