Why such an interest for the error term in the Prime Number Theorem
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I have some issues when dealing with people working outside number theory, to motivate and justify in some sense the problems I am interested in. Mainly, here are the issues I do not know enough motivations about:
- why do we care about the error term in PNT? (beyond "we have an equivalent, it is therefore natural to know to what extent it is true")
- is there any application to these error terms inside number theory? (for instance finding gaps between primes or any result that would be better when improving the error term)
- is there any argument to say that we cannot beat Riemann hypothesis' square root savings?
Thanks in advance for any insight!
number-theory prime-numbers
asked Nov 22 at 10:31
TheStudent
1886
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up vote
3
down vote
favorite
1
I have some issues when dealing with people working outside number theory, to motivate and justify in some sense the problems I am interested in. Mainly, here are the issues I do not know enough motivations about:
- why do we care about the error term in PNT? (beyond "we have an equivalent, it is therefore natural to know to what extent it is true")
- is there any application to these error terms inside number theory? (for instance finding gaps between primes or any result that would be better when improving the error term)
- is there any argument to say that we cannot beat Riemann hypothesis' square root savings?
Thanks in advance for any insight!
number-theory prime-numbers
asked Nov 22 at 10:31
TheStudent
1886
From the practical perspective, prime numbers are used in cryptography, e.g. RSA. Large prime numbers are essential. Due to PNT we have this approximation of prime numbers $$p_n sim nlog{n}$$ But, because of the large error term for this approximation, finding the next largest prime number using it is still impractical.
– rtybase
Nov 22 at 22:33
add a comment |
up vote
3
down vote
favorite
1
up vote
3
down vote
favorite
1
1
I have some issues when dealing with people working outside number theory, to motivate and justify in some sense the problems I am interested in. Mainly, here are the issues I do not know enough motivations about:
- why do we care about the error term in PNT? (beyond "we have an equivalent, it is therefore natural to know to what extent it is true")
- is there any application to these error terms inside number theory? (for instance finding gaps between primes or any result that would be better when improving the error term)
- is there any argument to say that we cannot beat Riemann hypothesis' square root savings?
Thanks in advance for any insight!
number-theory prime-numbers
asked Nov 22 at 10:31
TheStudent
1886
I have some issues when dealing with people working outside number theory, to motivate and justify in some sense the problems I am interested in. Mainly, here are the issues I do not know enough motivations about:
- why do we care about the error term in PNT? (beyond "we have an equivalent, it is therefore natural to know to what extent it is true")
- is there any application to these error terms inside number theory? (for instance finding gaps between primes or any result that would be better when improving the error term)
- is there any argument to say that we cannot beat Riemann hypothesis' square root savings?
Thanks in advance for any insight!
number-theory prime-numbers
number-theory prime-numbers
asked Nov 22 at 10:31
TheStudent
1886
asked Nov 22 at 10:31
TheStudent
1886
asked Nov 22 at 10:31
TheStudent
1886
asked Nov 22 at 10:31
TheStudent
1886
asked Nov 22 at 10:31
TheStudent
1886
1886
From the practical perspective, prime numbers are used in cryptography, e.g. RSA. Large prime numbers are essential. Due to PNT we have this approximation of prime numbers $$p_n sim nlog{n}$$ But, because of the large error term for this approximation, finding the next largest prime number using it is still impractical.
– rtybase
Nov 22 at 22:33
add a comment |
From the practical perspective, prime numbers are used in cryptography, e.g. RSA. Large prime numbers are essential. Due to PNT we have this approximation of prime numbers $$p_n sim nlog{n}$$ But, because of the large error term for this approximation, finding the next largest prime number using it is still impractical.
– rtybase
Nov 22 at 22:33
From the practical perspective, prime numbers are used in cryptography, e.g. RSA. Large prime numbers are essential. Due to PNT we have this approximation of prime numbers $$p_n sim nlog{n}$$ But, because of the large error term for this approximation, finding the next largest prime number using it is still impractical.
– rtybase
Nov 22 at 22:33
From the practical perspective, prime numbers are used in cryptography, e.g. RSA. Large prime numbers are essential. Due to PNT we have this approximation of prime numbers $$p_n sim nlog{n}$$ But, because of the large error term for this approximation, finding the next largest prime number using it is still impractical.
– rtybase
Nov 22 at 22:33
add a comment |
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From the practical perspective, prime numbers are used in cryptography, e.g. RSA. Large prime numbers are essential. Due to PNT we have this approximation of prime numbers $$p_n sim nlog{n}$$ But, because of the large error term for this approximation, finding the next largest prime number using it is still impractical.
– rtybase
Nov 22 at 22:33