Mixing
How do we determine the roughness of estimates in number theory?
$begingroup$
The question came to me when I was reading a book where a proof of Brun's theorem was given, and was followed by this exercise:
Give an upper bound for the number of primes of the form $n^2+1$ below $x$ using Eratosthene's sieve.
After several trials I succeeded in obtaining the upper bound of $O(sqrt x/log x)$. However, during the process I accidentally loosened the inequalities too much several times, and obtained trivial estimates like $O(x)$ and $O(sqrt x)$. So the problem is: is there a good way to tell whether an estimate is 'good enough'? Or does one has to try several times in order to find the best choice? Any idea is appreciated!
number-theory sieve-theory
asked Jan 6 at 3:49
TreborTrebor
80013
$endgroup$
add a comment |
$begingroup$
The question came to me when I was reading a book where a proof of Brun's theorem was given, and was followed by this exercise:
Give an upper bound for the number of primes of the form $n^2+1$ below $x$ using Eratosthene's sieve.
After several trials I succeeded in obtaining the upper bound of $O(sqrt x/log x)$. However, during the process I accidentally loosened the inequalities too much several times, and obtained trivial estimates like $O(x)$ and $O(sqrt x)$. So the problem is: is there a good way to tell whether an estimate is 'good enough'? Or does one has to try several times in order to find the best choice? Any idea is appreciated!
number-theory sieve-theory
asked Jan 6 at 3:49
TreborTrebor
80013
$endgroup$
add a comment |
$begingroup$
The question came to me when I was reading a book where a proof of Brun's theorem was given, and was followed by this exercise:
Give an upper bound for the number of primes of the form $n^2+1$ below $x$ using Eratosthene's sieve.
After several trials I succeeded in obtaining the upper bound of $O(sqrt x/log x)$. However, during the process I accidentally loosened the inequalities too much several times, and obtained trivial estimates like $O(x)$ and $O(sqrt x)$. So the problem is: is there a good way to tell whether an estimate is 'good enough'? Or does one has to try several times in order to find the best choice? Any idea is appreciated!
number-theory sieve-theory
asked Jan 6 at 3:49
TreborTrebor
80013
$endgroup$
The question came to me when I was reading a book where a proof of Brun's theorem was given, and was followed by this exercise:
Give an upper bound for the number of primes of the form $n^2+1$ below $x$ using Eratosthene's sieve.
After several trials I succeeded in obtaining the upper bound of $O(sqrt x/log x)$. However, during the process I accidentally loosened the inequalities too much several times, and obtained trivial estimates like $O(x)$ and $O(sqrt x)$. So the problem is: is there a good way to tell whether an estimate is 'good enough'? Or does one has to try several times in order to find the best choice? Any idea is appreciated!
number-theory sieve-theory
number-theory sieve-theory
asked Jan 6 at 3:49
TreborTrebor
80013
asked Jan 6 at 3:49
TreborTrebor
80013
asked Jan 6 at 3:49
TreborTrebor
80013
asked Jan 6 at 3:49
TreborTrebor
80013
asked Jan 6 at 3:49
TreborTrebor
80013
80013
add a comment |
add a comment |
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