All Questions
Tagged with analytic-number-theory asymptotics
9 questions
20
votes
1
answer
1k
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Quantitative lower bounds related to Zhang's theorem on bounded gaps
Let $\mathcal{H}=\left\{ h_{i}\right\} _{i=1}^{k}$ be an admissible set, and define $$\pi_{\mathcal{H}}(x)=\left|\left\{ n\leq x\ :\ \exists\ i,j\leq k,\ i\neq j\ \text{such that both }n+h_{i},\ n+h_{...
- 11.4k
5
votes
3
answers
1k
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What is the asymptotic of the irregular blue curve? Is it $(8x)^{1/2}$ or is it something else?
From Terry Tao's post here there is the statement:
"Conversely, if one can somehow establish a bound of the form
$$\displaystyle \sum_{n \leq x} \Lambda(n) = x + O( x^{1/2+\epsilon} ) \tag{1}$$
...
- 1,183
2
votes
0
answers
422
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Sequences with high densities of primes: how to boost them to get even more and larger primes
I propose a methodology to help find large prime numbers with a much higher probability than picking up random numbers and testing them for primality. This would help speed up prime number generators ...
- 3,098
25
votes
1
answer
2k
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Wrong asymptotics of OEIS A000607 (number of partitions of an integer in prime parts)?
Sequence A000607 in the Online Encyclopedia of Integer Sequences is the number of partitions of $n$ into prime parts. For example, there are $5$ partitions of $10$ into prime parts: $10 = 2 + 2 + 2 + ...
- 690
11
votes
2
answers
771
views
Tauberian theorem with better error term
This is a fairly vague question.
Suppose we have a sequence of positive numbers $(c_n)_n$ and we want to find an asymptotic formula for $S(x) = \sum_{n \leq X} c_n$. In favorable circumstances, ...
- 1,362
4
votes
1
answer
271
views
On approximation of $\sum_{a,b=1}^n\gcd(a,b)$
Denote $g(n)=\sum_{a,b=1}^n\gcd(a,b)$, can we prove that
$$g(n)=\frac6{\pi^2}n^2\ln n+Cn^2+O(n\ln n)$$, where $C=-\frac12+\frac{6}{\pi^2}(-\frac12+\gamma-\ln(2\pi)+12\ln A),$ where $\gamma$ denotes ...
- 143
2
votes
1
answer
230
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On $\sum_{\substack{p\leq x\\p,p+2\text{ twin primes}}}\frac{(\log p)^m}{p}$, on assumption of the first Hardy–Littlewood conjecture
I wondered, inspired in a result from [1] (Proposition 17) what should be the asymptotic behaviour of the sequence, on assumption of the First Hardy–Littlewood conjecture,
$$\sum_{\substack{\text{...
1
vote
1
answer
867
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$n$th prime: a better approximation
Let $p_n$ be the $n$-th prime, then from Wikipedia I got that
$p_n \approx n \left(\ln n + \ln \ln n -1 + \frac{\ln \ln n-2}{\ln n}+\frac{6\ln \ln n-( \ln \ln n)^2-11}{\ln^2 n} \right)$.
What is a ...
user95470
1
vote
1
answer
310
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Asymptotic lower bound for the number of square free with at least two prime factors
In one of Soundararajan's papers, he claims without proof that it is a standard exercise to show that the number $N(X)$ of positive square-free integers $d \equiv 1 \; \bmod \; 8$ less than $X$, with ...
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