All Questions
14 questions with no upvoted or accepted answers
15
votes
0
answers
365
views
Do primes of the form $4k+1$ ever lead the greatest prime factor race?
Analogous to Chebyshev's race between primes, I examined the race between primes in the greatest prime factors, GPF, of natural numbers. Similar to the regular prime race, in the GPF race, the ...
- 5,470
7
votes
0
answers
274
views
Are there infinitely many zeroes of $\sum_{r = 1}^{n-1} \mu(r)\gcd(n,r) $?
Let $\mu(n)$ be the Möbius function and $S(x)$ be the number of positive integers $n \le x$ such that
$$
\sum_{r = 1}^{n-1} \mu(r)\gcd(n,r) = 0
$$
My experimental data for $n \le 6 \times 10^5 $...
- 5,470
6
votes
0
answers
200
views
Empirical bounds on $\left|\frac{\zeta'(1+it)}{\zeta(1+it)}\right|$
It is reasonable to expect that $$\left|\frac{\zeta'(1+it)}{\zeta(1+it)}\right| < 2 \log \log t$$
for all $t\geq 4$ (say): a somewhat stronger bound is known for $t\geq 10^{165}$ or so (Theorem 5 ...
- 20.2k
6
votes
0
answers
290
views
Effective bounds for Fermat's Last Theorem
Suppose $n>2$. By Fermat's Last Theorem, we know that $a^{n}+b^{n}=c^{n}$
has no non-trivial solutions. Can we quantify it more?
More specifically, given $a,b,c,n\in\mathbb{N}$ with $n>2$ and $...
- 583
3
votes
0
answers
186
views
Factoring problem similar to $RSA$ structure that is possibly not $NP$ complete and not $coNP$ also?
Standard factoring problem $\Pi_1$ is 'Given integers $N$ and $M$ is there a factor $d\in[1,M]$ of $N$?'. This is in $NP$ since such a factor is the witness and in $coNP$ since one can check all the ...
- 13.9k
2
votes
0
answers
107
views
Record for determining complete list of imaginary quadratic fields with small class number
In 2003, Mark Watkins (Class numbers of imaginary quadratic fields) determined all imaginary quadratic fields having class number at most 100.
Has this list been improved? That is, what is the largest ...
- 26.9k
2
votes
0
answers
146
views
Reference for accelerated sum to compute the Meissel-Mertens constant
The Meissel-Mertens constant
$$ B_1 = \lim_{n \to \infty} \left(\sum_{p \leq n} \frac{1}{p} - \log\log n\right) $$
has the series representation
$$
\begin{equation} \tag{1}
B_1 = \gamma + \sum_{n=2}^{...
- 451
2
votes
0
answers
205
views
Sum of all primes below $n$ without listing all primes below $n$
Asymptotically there is around $\frac{n}{\ln n}$ primes below a given integer $n$. Thus $\frac{n}{\ln n}$ is a lower bound for the time complexity of any algorithm that at some point finds each prime ...
- 21
2
votes
0
answers
99
views
A problem in modular roots
We have three mutually coprime integers $r,t,M$ where $M\asymp K^{\frac12-2\epsilon}$ and $r,t\asymp K^{\frac14+\epsilon}$ holds with some fixed $\epsilon>0$ and $K>0$ is a large parameter. ...
- 13.9k
2
votes
0
answers
78
views
Accelerating convergence of a product by multiplying by zeta values: history?
Let $R(s_1,\dotsc,s_n) = \prod_p r(p^{-s_1},\dotsc,p^{-s_n})$, where
$r$ is a rational function on $n$ variables. Say we want to compute the value of $R(s_1,\dotsc,s_n)$ for some choice of $s_1,\dotsc,...
- 545
2
votes
0
answers
127
views
Coppersmith's method to quadrivariate degree $2$ polynomials that behave as bivariate?
We have a polynomial $f(x_1,x_2,x_3,x_4)\in\mathbb Z[x_1,x_2,x_3,x_4]$ where the only monomials are either from set $$\{x_1,x_1x_2,x_2,x_3,x_3x_4,x_4\}$$ and we seek solutions $(x_1,x_2,x_3,x_4)\in\...
- 13.9k
2
votes
0
answers
132
views
Quick computation of a certain exponential sum
Is there a quick way to compute (somewhat accurately) for large $X$, the following exponential sum, where $\Lambda$ is the von Mangoldt function?
$$\int_0^1 \bigg|\sum_{n\le X} \Lambda(n)e(n\alpha)\...
- 1,890
1
vote
0
answers
84
views
How common are semiprimes with equally bitsized factors among semiprimes with equal bitsize?
I am curious about the following after having looked at the paper "Almost primes in almost all short intervals", theorem 3 says:
Almost all intervals $[x, x + \log^{3.51}{(x)}]$ with $x ≤ X$...
- 11
1
vote
0
answers
96
views
Smooth number pairs satisfying a congruence
Let $\mathcal P=\{p_1,\dots,p_{2t}\}$ be $2t$ primes between $2^\ell$ and $2^{\ell+1}$ and fix an exponent bound $a\in\mathbb Z_{\geq2}$.
Fix $N\in\mathbb N$ whose prime factors $p$ satisfy $p>2^{\...
- 13.9k