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Results tagged with prime-numbers
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user 9232
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.
18
votes
3
answers
2k
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A question on the prime divisors of p-1
For each positive integer n we may define the convergent sum $$ s(n)=\sum_{p}\frac{(n,p-1)}{p^2} $$
where the summation is over primes p and $(a,b)$ denotes the greatest common divisor of a,b.
It is …
- 3,062
7
votes
2
answers
424
views
Divisor sums over values of binary forms of primes
Let $\tau$ be the divisor function, that is
$$
\tau(n)=\sharp\{d \in \mathbb{N}, d|n\}.
$$
I was wondering if anyone has ever proved an asymptotic estimate
for the sum
$$S(x):=\sum_{p,q\leq x}\tau(p …
- 3,062
6
votes
0
answers
504
views
$x^2+1$ attaining almost prime values
Iwaniec, using the linear sieve, proved that $n^2+1$ can be a product of at most two primes infinitely often and furthermore a lower bound of the correct order of magnitude for the number of such inte …
- 3,062
5
votes
Accepted
Explicit bounds on number of squarefree numbers coprime to a certain number
I am assuming that explicit refers to the error term? In this case you can write $$Q_A(x)=\sum_{d\mid A}\mu(d)\sum_{k\leq \sqrt x \atop \gcd(A,k)=1}\mu(k)\left[\frac{x}{dk^2}\right],$$ where $[t]$ den …
- 3,062
4
votes
0
answers
209
views
No perfect patterns in the primes
The primes are equidistributed in the residue classes $1(\!\!\!\mod{4})$ and $3(\!\!\!\mod 4)$. We also know (for example, by Rubinstein-Sarnak) that the patterns cannot be eventually alternating, i.e …
- 3,062
4
votes
Divisor sums over values of binary forms of primes
An answer regarding the use of large sieve suggested by Lucia (too long for a comment).
I guess her/his thought was along the following lines:
The divisors $d$ of $p^2+q^2$ are of order $x^2$ and …
- 3,062
4
votes
Accepted
Tail of singular series of Goldbach problem
Here is an unconditional error:
Restricting to summation over $q$ with $gcd(q,N)=n$
and using the formula for Ramanujan sums
implies that the quantity you stated
equals
$$\sum_{n|N}\frac{\mu^2(n)}{\p …
- 3,062
3
votes
0
answers
333
views
Conditional proof of ternary Goldbach
This is a reference request.
I know that Hardy and Littlewood gave a proof of the ternary Goldbach for sufficiently large odd integers under the assumption of GRH.
Is there a modern account of th …
- 3,062
3
votes
0
answers
187
views
Riemann's explicit formula for square-free numbers
We know that for $x$ being a half-integer $$\sum_{n\leq x}\Lambda(n)=x-\sum_{\rho}\frac{x^\rho}{\rho}+O(1).$$
Is there a similar formula for $\sum_{n\leq x}\mu(n)^2$ in the literature? The underlying …
- 3,062
3
votes
A question on the prime divisors of p-1
A few more ideas: using the Chebyshev upper bound, by partial summation we have
$\sum_{p>y}p^{-2}=O(\frac{1}{y \log y})$
and therefore we see that
$s(n)=\sum_{p \leq \frac{n}{\log n}}\frac{(p-1,n)}{ …
- 3,062
3
votes
1
answer
353
views
Squarefree values of polynomials at prime arguments
This is a reference request.
Assume that $f_1,\ldots,f_r \in \mathbb{Z}[t]$ are non-zero linear polynomial.
Letting $\mu$ be the M\"{o}bius function, is there any work on
$$ \sum_{p\leq x} \prod_{i …
- 3,062
3
votes
2
answers
440
views
Least number coprime to a given integer
For a positive integer $n$ let $$f(n):=\min\{m\in \mathbb N: m>1, \gcd(m,n)=1\} .$$
Equivalently, $f(n) $ is the smallest prime not dividing $n$.
Is there any upper bound literature for this? It is no …
- 3,062
2
votes
Accepted
Asymptotics for a peculiar kind of squarefree numbers
Please ignore the previous version of this answer.
Motivated by Lucia's comment, we use smooth numbers to show that the number in question is $o(x)$. First note that for $100\%$ of all integers one h …
- 3,062
2
votes
0
answers
158
views
Quadratic patterns in summands of Goldbach's conjecture
Let $n $ be even and define
$$ Q(n)=\sum_{\substack{ p,q \ \textrm{ primes} \\p+q=n }}\left(\frac{p}{q} \right),$$ where $\left(\frac{p}{q} \right)$ is the quadratic Legendre symbol.
Has this sum been …
- 3,062
2
votes
Average value of the prime omega function $\Omega$ on predecessors of prime powers
One can prove that in fact the function mpe has bounded average which clearly improves $\log \log$. If a positive integer n has $mpe(n)=m$ then there exists a prime power p^m that divides n hence the …
- 3,062