Expected size of the maximal prime gap below x under Hardy-Littlewood conjecture
$begingroup$
Let $ pi_{n}(x) $ denote the number of prime gaps of size $ n $ below $ x $ for even $ n $. The Hardy-Littlewood conjecture predicts that $ pi_{n}(x)sim C_{n}frac{x}{log^{2}x} $ with $ C_{n}=C_{2}prod_{qmid n}frac{(q-1)}{(q-2)} $, $ C_{2}=0.66016... $ is the so called twin prime constant and $ q $ runs among odd primes dividing $ n $. Let $ Q_{a,b}(x) $ be the largest primorial not exceeding $ x^{a}log^{b} x $.
Assuming the conjecture above, can we determine $ a $ and $ b $ so that $ pi_{Q_{a,b}(x)}(x)asymp 1 $ as $ x $ tends to $ infty $?
number-theory prime-numbers prime-gaps
edited Dec 20 '18 at 10:14
Klangen
1,75811334
asked Nov 11 '18 at 15:57
Sylvain JulienSylvain Julien
1,135918
$endgroup$
add a comment |
$begingroup$
Let $ pi_{n}(x) $ denote the number of prime gaps of size $ n $ below $ x $ for even $ n $. The Hardy-Littlewood conjecture predicts that $ pi_{n}(x)sim C_{n}frac{x}{log^{2}x} $ with $ C_{n}=C_{2}prod_{qmid n}frac{(q-1)}{(q-2)} $, $ C_{2}=0.66016... $ is the so called twin prime constant and $ q $ runs among odd primes dividing $ n $. Let $ Q_{a,b}(x) $ be the largest primorial not exceeding $ x^{a}log^{b} x $.
Assuming the conjecture above, can we determine $ a $ and $ b $ so that $ pi_{Q_{a,b}(x)}(x)asymp 1 $ as $ x $ tends to $ infty $?
number-theory prime-numbers prime-gaps
edited Dec 20 '18 at 10:14
Klangen
1,75811334
asked Nov 11 '18 at 15:57
Sylvain JulienSylvain Julien
1,135918
$endgroup$
add a comment |
$begingroup$
Let $ pi_{n}(x) $ denote the number of prime gaps of size $ n $ below $ x $ for even $ n $. The Hardy-Littlewood conjecture predicts that $ pi_{n}(x)sim C_{n}frac{x}{log^{2}x} $ with $ C_{n}=C_{2}prod_{qmid n}frac{(q-1)}{(q-2)} $, $ C_{2}=0.66016... $ is the so called twin prime constant and $ q $ runs among odd primes dividing $ n $. Let $ Q_{a,b}(x) $ be the largest primorial not exceeding $ x^{a}log^{b} x $.
Assuming the conjecture above, can we determine $ a $ and $ b $ so that $ pi_{Q_{a,b}(x)}(x)asymp 1 $ as $ x $ tends to $ infty $?
number-theory prime-numbers prime-gaps
edited Dec 20 '18 at 10:14
Klangen
1,75811334
asked Nov 11 '18 at 15:57
Sylvain JulienSylvain Julien
1,135918
$endgroup$
Let $ pi_{n}(x) $ denote the number of prime gaps of size $ n $ below $ x $ for even $ n $. The Hardy-Littlewood conjecture predicts that $ pi_{n}(x)sim C_{n}frac{x}{log^{2}x} $ with $ C_{n}=C_{2}prod_{qmid n}frac{(q-1)}{(q-2)} $, $ C_{2}=0.66016... $ is the so called twin prime constant and $ q $ runs among odd primes dividing $ n $. Let $ Q_{a,b}(x) $ be the largest primorial not exceeding $ x^{a}log^{b} x $.
Assuming the conjecture above, can we determine $ a $ and $ b $ so that $ pi_{Q_{a,b}(x)}(x)asymp 1 $ as $ x $ tends to $ infty $?
number-theory prime-numbers prime-gaps
number-theory prime-numbers prime-gaps
edited Dec 20 '18 at 10:14
Klangen
1,75811334
asked Nov 11 '18 at 15:57
Sylvain JulienSylvain Julien
1,135918
edited Dec 20 '18 at 10:14
Klangen
1,75811334
asked Nov 11 '18 at 15:57
Sylvain JulienSylvain Julien
1,135918
edited Dec 20 '18 at 10:14
Klangen
1,75811334
edited Dec 20 '18 at 10:14
Klangen
1,75811334
edited Dec 20 '18 at 10:14
Klangen
1,75811334
1,75811334
asked Nov 11 '18 at 15:57
Sylvain JulienSylvain Julien
1,135918
asked Nov 11 '18 at 15:57
Sylvain JulienSylvain Julien
1,135918
asked Nov 11 '18 at 15:57
Sylvain JulienSylvain Julien
1,135918
1,135918
add a comment |
add a comment |
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