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Results tagged with nt.number-theory
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user 38624
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
1
vote
Asymptotic upper bounds for some convolution sums
There's a lot of work on such problems. One significant paper is
Peter Shiu's A Brun Titchmarsh theorem for multiplicative functions in
Crelle (1980). See also Mohan Nair's paper in Acta Arithmeti …
- 43.2k
8
votes
Accepted
Angular equidistribution of lattice points on circles
If $p\equiv 1 \pmod 4$ then we may write $p=a^2+b^2$ in a unique way with $a$, $b$ both positive and
$b< a$. Corresponding to such a representation, write $a+bi = \sqrt{p} e^{i\theta(p)}$ with $\the …
- 43.2k
18
votes
Accepted
Upper bound for class number of a real quadratic field
The exponent $1/2$ is best possible. You can see this by varying $D$ along values of the form $n^2+4$ so that the regulator is only of size about $\log D$. Then the lower bounds for $L(1,\chi)$ (Sie …
- 43.2k
7
votes
Accepted
Are there any known non-trivial functions that takes on squarefree values with the right den...
Indeed the Piatetski-Shapiro numbers are such an example: see Theorem 4 of this paper http://arxiv.org/pdf/1203.5884.pdf by Baker, Banks, Brudern, Shparlinski and Weingartner.
Note: I understood the …
- 43.2k
17
votes
Accepted
Gaps in Squarefree numbers
A number of authors e.g. Hooley, Filaseta, Trifonov have considered this problem of moments of gaps between square-free numbers. For example, Filaseta and Trifonov (paper in Proc. London Math. Soc. ( …
- 43.2k
10
votes
Generalizing Ramanujan's "1729 story"
You may be interested in this article The 1729 $K3$ surface by Ken Ono and Sarah Trebat-Leder, which is aimed at exactly this question of how Ramanujan knew $1729$ so well. Briefly, Ramanujan had stu …
- 43.2k
10
votes
Accepted
Bounds for the $2$-torsion subgroup of the class group of a number field
Recent work of Bhargava, Shankar, Taniguchi, Thorne, Tsimerman, and Zhao shows that if $K$ is a number field of degree $n$ then the size of the $2$-torsion subgroup of the class group of $K$ is
$$
O( …
- 43.2k
4
votes
Accepted
Help estimating an exponential sum
The sum $\sum_{n\le x} e^{2\pi i t \omega(n)/q}$ is the sum
of a multiplicative function whose value at a prime $p$ is
$e^{2\pi i t/q}$. Asymptotics for partial sums of multiplicative
functions $ …
- 43.2k
6
votes
Generalizations of the twin primes conjecture
This was solved by Erdos, who introduced the idea of a covering congruence. Erdos shows that if $n$ is congruent to $1\pmod 2$, $1\pmod 7$, $2\pmod 5$, $8\pmod {17}$, $2^7 \pmod{13}$, $2^{23} \pmod { …
- 43.2k
3
votes
A Simple Generalization of the Littlewood Conjecture
This answer pieces together the various comments made by Roland Bacher, SJR and
gowers previously. The proposed generalization of Littlewood's conjecture is false.
As suggested by Roland and SJR, ta …
- 43.2k
11
votes
The least number of quadratic polynomials needed to cover $[1,N]$
A special case of this problem has been studied in the literature (and I think this work will also cover the general version). Given $N$, what is the smallest set $B$ such that every $n\le N$ can be …
- 43.2k
14
votes
Accepted
Egyptian representations of $1$
Lots of questions along these lines were raised by Erdos and Graham and many have been solved by Croot, Greg Martin and others. In particular, Croot has shown that any rational number $r$ can be repr …
- 43.2k
15
votes
Accepted
Mobius function of consecutive numbers
General conjectures of Chowla predict cancelation in correlations of the Mobius function: e.g. in $\sum_{n\le x}\mu(n)\mu(n+1)$ or $\sum_{n\le x} \mu(n)^2 \mu(n+1)\mu(n+2)$ etc. These conjectures wou …
- 43.2k
5
votes
Accepted
Numbers up to $N$ with small radical
It may be more natural to consider
$$
N(x,y) = \# \{ n\le x: \text{rad}(n) \le y\},
$$
and the question is essentially about $N(x,x^{\alpha})$ (with $\alpha =1/2$ in your question 2). The quantit …
- 43.2k
38
votes
Primes p=4k+1 such that k!+1 is divisible by p
There are no such primes $p$. Write $p=4k+1$ as $a^2+b^2$ with $a$ odd and $b$ even, and by changing the sign of $a$ if necessary suppose that $a\equiv 1 \pmod 4$. Note that $a$ is uniquely defined. …
- 43.2k