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Results tagged with prime-numbers
Search options questions only
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user 398
Questions where prime numbers play a key-role, such as: questions on the distribution of prime numbers (twin primes, gaps between primes, Hardy–Littlewood conjectures, etc); questions on prime numbers with special properties (Wieferich prime, Wolstenholme prime, etc.). This tag is often used as a specialized tag in combination with the top-level tag nt.number-theory and (if applicable) analytic-number-theory.
4
votes
1
answer
487
views
Barban-Davenport-Halberstam without von Mangoldt weights
The Barban-Davenport-Halberstam theorem gives a bound for the average (in L_2 norm) difference between
$\sum_{n\leq N: n\equiv a \mod q} \Lambda(n)$ and $N/\phi(q)$. It is obvious that a similar resu …
- 20.2k
13
votes
1
answer
382
views
Numbers that don't start with (p-1) in base p for any p
Say that an integer $n$ is $p$-leading if its expansion in base $p$ starts with the digit $p-1$. My postdoc, Lifan Guan, asks: are there infinitely many positive integers $n$ that are not $p$-leading …
- 20.2k
2
votes
0
answers
196
views
Quasiprimes in arithmetic progressions
Let
$$\Lambda_z(n) = \sum_{d|n, d>z} \mu(d) \log(d/z).$$
As S. Graham proved in 1978,
$$\sum_{n\leq x} |\Lambda_z(n)|^2 \sim x \log(x/z).$$
provided $x\geq z$.
We also know that, by the large sie …
- 20.2k
2
votes
0
answers
196
views
Sums of reciprocals of primes in an arithmetic progression
Let $y>x\geq 1$, $p_0\geq x$. Consider $$S=\mathop{\sum_{x\leq p\leq y}}_{p\equiv a \mod p_0} \frac{1}{p}.$$By Brun-Titchmarsh and (basically) integration by parts, I seem to get that $$S \leq \frac{1 …
- 20.2k
4
votes
0
answers
134
views
Average of $\lambda(n+1)$ for $n$ smooth, or smooth-and-rough? What follows?
Let $\lambda$ be the Liouville function, i.e., $\lambda(p_1\dotsb p_k)=(-1)^k$ for $p_1,\dotsc,p_k$ not necessarily distinct.
There is a conjecture (due to whom?) that there are infinitely many primes …
- 20.2k
9
votes
1
answer
2k
views
The large sieve for primes
Let $\Lambda(n)$ be the von Mangoldt function, i.e., $\Lambda(n) = \log p$ for $n$ a prime power $p^k$ and $\Lambda(n) = 0$ for all $n$ that not prime powers. Let
$$S(\alpha) = \sum_{n \leq N} \Lamb …
- 20.2k
12
votes
2
answers
2k
views
Detecting almost-primes quickly
There are many fast algorithms (deterministic and probabilistic) for detecting primality. Are there any fast algorithms (probabilistic ones allowed) known for detecting whether a number is the product …
- 20.2k
7
votes
1
answer
265
views
From $\Lambda_k$ and $\Lambda$ to $\mu$ (or $\lambda$)
Let $\{a_n\}_{n=1}^\infty$, $a_n \in \mathbb{C}$, $|a_n|\leq 1$. Let $\Lambda_k = \mu \ast \log^k$; in particular, $\Lambda_1$ equals the von Mangoldt function $\Lambda$. Suppose that we have asymptot …
- 20.2k
4
votes
0
answers
221
views
How dense are quotients of smooth numbers?
As usual, call a positive integer $y$-smooth if it has no prime factors greater than $y$. Write $S(x,y)$ for the set of $y$-smooth integers $\leq x$. Write $R(x,y)$ for the set of quotients $\{a/b: a, …
- 20.2k
3
votes
0
answers
163
views
What smoothing to use for PNT-like results?
Consider a Dirichlet series $\sum_n a_n n^{-s}$ with desirable analytic properties (e.g., analytic extension to $\Re s>0$); one example would be $a_n=\mu(n)$. Say we want to estimate $\sum_{n\leq x} a …
- 20.2k