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How to approximate the Chebychev $psi$ function
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Using Riemann-Stieltjes integration the following expression is true
$$theta(n) = log (t) pi(t) Big|_2^n - int_2^n fracpi(t)tdt. tag1$$
using $Li(x)$ leads to the approximation $theta(n) approx n $.
1) So I'm wondering using similar methods (probably based on prime number theory $pi(x)$) if one can approximates the Chebychev $psi(n)$ function?
2) If not is it possible and how?
number-theory elementary-number-theory analytic-number-theory
asked Jul 24 at 9:44
onepound
181115
add a comment |Â
up vote
1
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Using Riemann-Stieltjes integration the following expression is true
$$theta(n) = log (t) pi(t) Big|_2^n - int_2^n fracpi(t)tdt. tag1$$
using $Li(x)$ leads to the approximation $theta(n) approx n $.
1) So I'm wondering using similar methods (probably based on prime number theory $pi(x)$) if one can approximates the Chebychev $psi(n)$ function?
2) If not is it possible and how?
number-theory elementary-number-theory analytic-number-theory
asked Jul 24 at 9:44
onepound
181115
1
Using $operatornameLi(x)$ as an approximation to $pi(x)$ gives the correct $theta(n) approx n$, namely $$log (t)operatornameLi(t)Bigrrvert_2^n - int_2^n fracoperatornameLi(t)t,dt = int_2^n log (t)operatornameLi'(t) + fracoperatornameLi(t)t,dt - int_2^n fracoperatornameLi(t)t,dt = n-2,.$$ You can use an approximation to $theta$ to approximate $psi$, since $$psi(x) = theta(x) + theta(sqrtx) + theta(sqrt[3]x) + theta(sqrt[4]x) + dotsc$$ where only finitely many terms on the right are nozero.
– Daniel Fischer♦
Jul 24 at 10:56
1
yes this result is pointed to in Ingham in APPROXIMATE FORMULAS FOR SOME FUNCTIONS OF PRIME NUMBERS by J. BARKLEY ROSSER AND LOWELL SCHOENFELD. Wondered if anything as nice as (1) had come to light since 1961.
– onepound
Jul 25 at 8:42
1
$psi$ corresponds to the weighted prime power counting function $pi^ast(x) = pi(x) + frac12pi(sqrtx) + frac13pi(sqrt[3]x) + dotsc$ like $theta$ corresponds to $pi$. So you get the similar $$psi(n) = log(t)pi^ast(t)Bigrrvert_2^n - int_2^n fracpi^ast(t)t,dt,.$$
– Daniel Fischer♦
Jul 25 at 14:46
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Using Riemann-Stieltjes integration the following expression is true
$$theta(n) = log (t) pi(t) Big|_2^n - int_2^n fracpi(t)tdt. tag1$$
using $Li(x)$ leads to the approximation $theta(n) approx n $.
1) So I'm wondering using similar methods (probably based on prime number theory $pi(x)$) if one can approximates the Chebychev $psi(n)$ function?
2) If not is it possible and how?
number-theory elementary-number-theory analytic-number-theory
asked Jul 24 at 9:44
onepound
181115
Using Riemann-Stieltjes integration the following expression is true
$$theta(n) = log (t) pi(t) Big|_2^n - int_2^n fracpi(t)tdt. tag1$$
using $Li(x)$ leads to the approximation $theta(n) approx n $.
1) So I'm wondering using similar methods (probably based on prime number theory $pi(x)$) if one can approximates the Chebychev $psi(n)$ function?
2) If not is it possible and how?
number-theory elementary-number-theory analytic-number-theory
asked Jul 24 at 9:44
onepound
181115
edited Jul 25 at 10:04
asked Jul 24 at 9:44
onepound
181115
asked Jul 24 at 9:44
onepound
181115
asked Jul 24 at 9:44
onepound
181115
181115
1
Using $operatornameLi(x)$ as an approximation to $pi(x)$ gives the correct $theta(n) approx n$, namely $$log (t)operatornameLi(t)Bigrrvert_2^n - int_2^n fracoperatornameLi(t)t,dt = int_2^n log (t)operatornameLi'(t) + fracoperatornameLi(t)t,dt - int_2^n fracoperatornameLi(t)t,dt = n-2,.$$ You can use an approximation to $theta$ to approximate $psi$, since $$psi(x) = theta(x) + theta(sqrtx) + theta(sqrt[3]x) + theta(sqrt[4]x) + dotsc$$ where only finitely many terms on the right are nozero.
– Daniel Fischer♦
Jul 24 at 10:56
1
yes this result is pointed to in Ingham in APPROXIMATE FORMULAS FOR SOME FUNCTIONS OF PRIME NUMBERS by J. BARKLEY ROSSER AND LOWELL SCHOENFELD. Wondered if anything as nice as (1) had come to light since 1961.
– onepound
Jul 25 at 8:42
1
$psi$ corresponds to the weighted prime power counting function $pi^ast(x) = pi(x) + frac12pi(sqrtx) + frac13pi(sqrt[3]x) + dotsc$ like $theta$ corresponds to $pi$. So you get the similar $$psi(n) = log(t)pi^ast(t)Bigrrvert_2^n - int_2^n fracpi^ast(t)t,dt,.$$
– Daniel Fischer♦
Jul 25 at 14:46
add a comment |Â
1
Using $operatornameLi(x)$ as an approximation to $pi(x)$ gives the correct $theta(n) approx n$, namely $$log (t)operatornameLi(t)Bigrrvert_2^n - int_2^n fracoperatornameLi(t)t,dt = int_2^n log (t)operatornameLi'(t) + fracoperatornameLi(t)t,dt - int_2^n fracoperatornameLi(t)t,dt = n-2,.$$ You can use an approximation to $theta$ to approximate $psi$, since $$psi(x) = theta(x) + theta(sqrtx) + theta(sqrt[3]x) + theta(sqrt[4]x) + dotsc$$ where only finitely many terms on the right are nozero.
– Daniel Fischer♦
Jul 24 at 10:56
1
yes this result is pointed to in Ingham in APPROXIMATE FORMULAS FOR SOME FUNCTIONS OF PRIME NUMBERS by J. BARKLEY ROSSER AND LOWELL SCHOENFELD. Wondered if anything as nice as (1) had come to light since 1961.
– onepound
Jul 25 at 8:42
1
$psi$ corresponds to the weighted prime power counting function $pi^ast(x) = pi(x) + frac12pi(sqrtx) + frac13pi(sqrt[3]x) + dotsc$ like $theta$ corresponds to $pi$. So you get the similar $$psi(n) = log(t)pi^ast(t)Bigrrvert_2^n - int_2^n fracpi^ast(t)t,dt,.$$
– Daniel Fischer♦
Jul 25 at 14:46
1
1
Using $operatornameLi(x)$ as an approximation to $pi(x)$ gives the correct $theta(n) approx n$, namely $$log (t)operatornameLi(t)Bigrrvert_2^n - int_2^n fracoperatornameLi(t)t,dt = int_2^n log (t)operatornameLi'(t) + fracoperatornameLi(t)t,dt - int_2^n fracoperatornameLi(t)t,dt = n-2,.$$ You can use an approximation to $theta$ to approximate $psi$, since $$psi(x) = theta(x) + theta(sqrtx) + theta(sqrt[3]x) + theta(sqrt[4]x) + dotsc$$ where only finitely many terms on the right are nozero.
– Daniel Fischer♦
Jul 24 at 10:56
Using $operatornameLi(x)$ as an approximation to $pi(x)$ gives the correct $theta(n) approx n$, namely $$log (t)operatornameLi(t)Bigrrvert_2^n - int_2^n fracoperatornameLi(t)t,dt = int_2^n log (t)operatornameLi'(t) + fracoperatornameLi(t)t,dt - int_2^n fracoperatornameLi(t)t,dt = n-2,.$$ You can use an approximation to $theta$ to approximate $psi$, since $$psi(x) = theta(x) + theta(sqrtx) + theta(sqrt[3]x) + theta(sqrt[4]x) + dotsc$$ where only finitely many terms on the right are nozero.
– Daniel Fischer♦
Jul 24 at 10:56
1
1
yes this result is pointed to in Ingham in APPROXIMATE FORMULAS FOR SOME FUNCTIONS OF PRIME NUMBERS by J. BARKLEY ROSSER AND LOWELL SCHOENFELD. Wondered if anything as nice as (1) had come to light since 1961.
– onepound
Jul 25 at 8:42
yes this result is pointed to in Ingham in APPROXIMATE FORMULAS FOR SOME FUNCTIONS OF PRIME NUMBERS by J. BARKLEY ROSSER AND LOWELL SCHOENFELD. Wondered if anything as nice as (1) had come to light since 1961.
– onepound
Jul 25 at 8:42
1
1
$psi$ corresponds to the weighted prime power counting function $pi^ast(x) = pi(x) + frac12pi(sqrtx) + frac13pi(sqrt[3]x) + dotsc$ like $theta$ corresponds to $pi$. So you get the similar $$psi(n) = log(t)pi^ast(t)Bigrrvert_2^n - int_2^n fracpi^ast(t)t,dt,.$$
– Daniel Fischer♦
Jul 25 at 14:46
$psi$ corresponds to the weighted prime power counting function $pi^ast(x) = pi(x) + frac12pi(sqrtx) + frac13pi(sqrt[3]x) + dotsc$ like $theta$ corresponds to $pi$. So you get the similar $$psi(n) = log(t)pi^ast(t)Bigrrvert_2^n - int_2^n fracpi^ast(t)t,dt,.$$
– Daniel Fischer♦
Jul 25 at 14:46
add a comment |Â
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1
Using $operatornameLi(x)$ as an approximation to $pi(x)$ gives the correct $theta(n) approx n$, namely $$log (t)operatornameLi(t)Bigrrvert_2^n - int_2^n fracoperatornameLi(t)t,dt = int_2^n log (t)operatornameLi'(t) + fracoperatornameLi(t)t,dt - int_2^n fracoperatornameLi(t)t,dt = n-2,.$$ You can use an approximation to $theta$ to approximate $psi$, since $$psi(x) = theta(x) + theta(sqrtx) + theta(sqrt[3]x) + theta(sqrt[4]x) + dotsc$$ where only finitely many terms on the right are nozero.
– Daniel Fischer♦
Jul 24 at 10:56
1
yes this result is pointed to in Ingham in APPROXIMATE FORMULAS FOR SOME FUNCTIONS OF PRIME NUMBERS by J. BARKLEY ROSSER AND LOWELL SCHOENFELD. Wondered if anything as nice as (1) had come to light since 1961.
– onepound
Jul 25 at 8:42
1
$psi$ corresponds to the weighted prime power counting function $pi^ast(x) = pi(x) + frac12pi(sqrtx) + frac13pi(sqrt[3]x) + dotsc$ like $theta$ corresponds to $pi$. So you get the similar $$psi(n) = log(t)pi^ast(t)Bigrrvert_2^n - int_2^n fracpi^ast(t)t,dt,.$$
– Daniel Fischer♦
Jul 25 at 14:46