Might there be a Skewes number for semiprimes?
Briefly the question here is whether there is or could be a theorem analogous to that of Littlewood for semiprimes (generalized prime numbers which are products of two primes, repetitions allowed), and so a Skewes-type number for semiprimes. In other words, whether we might have
$$li_2(x)-pi_2(x)=Omega_{pm}f(x)$$
for a function of x not too different from that in Littlewood's theorem.
The graph below is of the absolute difference between $li_k(x)$ and $pi_k(x)$ for $k=$ 1-14, 1-16, 1-18, 1-20 for $x=2^{14},2^{16},2^{18},2^{20},$ the [almost invisible] green, ochre, violet and blue lines, respectively, in which
$$li_k(x):=int_2^xfrac{ (loglog t)^{k-1}}{log tcdot(k-1)!}dt. $$
I hope it is clear enough where $li_k$ comes from. Just as $li(x)$ is a better estimate of $pi(x)$ than $x/log x,$ $li_2(x)$ is a better approximation of $pi_2(x)$ than the corresponding prime counting function for values I have checked. More to the point, there is no Skewes theorem for $x/log x$ which for $xgeq 17$ does not exceed $pi(x)$ (see Wiki article on the prime counting function).
The x-axis begins at k=1 on the left, and (for example) the first value for the blue line is $li(2^{20})-pi(2^{20})approx 110.5,$ so while it is difficult to see, the first values are all $>0$, as we expect, since $li(x)>pi(x)$ until the first Skewes number (about $1.397x10^{316}$, according to the Wiki page on this topic).
Parenthetically, I think we can show that the lines (joining points at k=2,3 in this picture) cross the x-axis (negative to positive) near the mean number of factors of $x,$ which is asymptotically $loglog x.$ Also, the sum of the points below zero is algebraically equal to the sum of those above zero.
So for an x just beyond Skewes number the graph changes sign (if I have calculated correctly) near 3.8 and the first point at k=1 is now negative, and it is tempting to speculate (and well beyond the scope of the question but included for context) that the entire set of points changes sign.
prime-numbers asymptotics analytic-number-theory
asked Dec 2 at 17:03
daniel
6,19322155
add a comment |
Briefly the question here is whether there is or could be a theorem analogous to that of Littlewood for semiprimes (generalized prime numbers which are products of two primes, repetitions allowed), and so a Skewes-type number for semiprimes. In other words, whether we might have
$$li_2(x)-pi_2(x)=Omega_{pm}f(x)$$
for a function of x not too different from that in Littlewood's theorem.
The graph below is of the absolute difference between $li_k(x)$ and $pi_k(x)$ for $k=$ 1-14, 1-16, 1-18, 1-20 for $x=2^{14},2^{16},2^{18},2^{20},$ the [almost invisible] green, ochre, violet and blue lines, respectively, in which
$$li_k(x):=int_2^xfrac{ (loglog t)^{k-1}}{log tcdot(k-1)!}dt. $$
I hope it is clear enough where $li_k$ comes from. Just as $li(x)$ is a better estimate of $pi(x)$ than $x/log x,$ $li_2(x)$ is a better approximation of $pi_2(x)$ than the corresponding prime counting function for values I have checked. More to the point, there is no Skewes theorem for $x/log x$ which for $xgeq 17$ does not exceed $pi(x)$ (see Wiki article on the prime counting function).
The x-axis begins at k=1 on the left, and (for example) the first value for the blue line is $li(2^{20})-pi(2^{20})approx 110.5,$ so while it is difficult to see, the first values are all $>0$, as we expect, since $li(x)>pi(x)$ until the first Skewes number (about $1.397x10^{316}$, according to the Wiki page on this topic).
Parenthetically, I think we can show that the lines (joining points at k=2,3 in this picture) cross the x-axis (negative to positive) near the mean number of factors of $x,$ which is asymptotically $loglog x.$ Also, the sum of the points below zero is algebraically equal to the sum of those above zero.
So for an x just beyond Skewes number the graph changes sign (if I have calculated correctly) near 3.8 and the first point at k=1 is now negative, and it is tempting to speculate (and well beyond the scope of the question but included for context) that the entire set of points changes sign.
prime-numbers asymptotics analytic-number-theory
asked Dec 2 at 17:03
daniel
6,19322155
What is a "nearprime" ? I only know the name "semiprime" for a number being the product of two primes, not necessarily distinct.
– Peter
Dec 3 at 18:36
1
@Peter: Where I heard "near-prime" I don't know. Semiprime is what I intended, and as you define it. Will wait to edit until I am pretty sure there aren't more things to clarify.
– daniel
Dec 3 at 18:47
add a comment |
Briefly the question here is whether there is or could be a theorem analogous to that of Littlewood for semiprimes (generalized prime numbers which are products of two primes, repetitions allowed), and so a Skewes-type number for semiprimes. In other words, whether we might have
$$li_2(x)-pi_2(x)=Omega_{pm}f(x)$$
for a function of x not too different from that in Littlewood's theorem.
The graph below is of the absolute difference between $li_k(x)$ and $pi_k(x)$ for $k=$ 1-14, 1-16, 1-18, 1-20 for $x=2^{14},2^{16},2^{18},2^{20},$ the [almost invisible] green, ochre, violet and blue lines, respectively, in which
$$li_k(x):=int_2^xfrac{ (loglog t)^{k-1}}{log tcdot(k-1)!}dt. $$
I hope it is clear enough where $li_k$ comes from. Just as $li(x)$ is a better estimate of $pi(x)$ than $x/log x,$ $li_2(x)$ is a better approximation of $pi_2(x)$ than the corresponding prime counting function for values I have checked. More to the point, there is no Skewes theorem for $x/log x$ which for $xgeq 17$ does not exceed $pi(x)$ (see Wiki article on the prime counting function).
The x-axis begins at k=1 on the left, and (for example) the first value for the blue line is $li(2^{20})-pi(2^{20})approx 110.5,$ so while it is difficult to see, the first values are all $>0$, as we expect, since $li(x)>pi(x)$ until the first Skewes number (about $1.397x10^{316}$, according to the Wiki page on this topic).
Parenthetically, I think we can show that the lines (joining points at k=2,3 in this picture) cross the x-axis (negative to positive) near the mean number of factors of $x,$ which is asymptotically $loglog x.$ Also, the sum of the points below zero is algebraically equal to the sum of those above zero.
So for an x just beyond Skewes number the graph changes sign (if I have calculated correctly) near 3.8 and the first point at k=1 is now negative, and it is tempting to speculate (and well beyond the scope of the question but included for context) that the entire set of points changes sign.
prime-numbers asymptotics analytic-number-theory
asked Dec 2 at 17:03
daniel
6,19322155
Briefly the question here is whether there is or could be a theorem analogous to that of Littlewood for semiprimes (generalized prime numbers which are products of two primes, repetitions allowed), and so a Skewes-type number for semiprimes. In other words, whether we might have
$$li_2(x)-pi_2(x)=Omega_{pm}f(x)$$
for a function of x not too different from that in Littlewood's theorem.
The graph below is of the absolute difference between $li_k(x)$ and $pi_k(x)$ for $k=$ 1-14, 1-16, 1-18, 1-20 for $x=2^{14},2^{16},2^{18},2^{20},$ the [almost invisible] green, ochre, violet and blue lines, respectively, in which
$$li_k(x):=int_2^xfrac{ (loglog t)^{k-1}}{log tcdot(k-1)!}dt. $$
I hope it is clear enough where $li_k$ comes from. Just as $li(x)$ is a better estimate of $pi(x)$ than $x/log x,$ $li_2(x)$ is a better approximation of $pi_2(x)$ than the corresponding prime counting function for values I have checked. More to the point, there is no Skewes theorem for $x/log x$ which for $xgeq 17$ does not exceed $pi(x)$ (see Wiki article on the prime counting function).
The x-axis begins at k=1 on the left, and (for example) the first value for the blue line is $li(2^{20})-pi(2^{20})approx 110.5,$ so while it is difficult to see, the first values are all $>0$, as we expect, since $li(x)>pi(x)$ until the first Skewes number (about $1.397x10^{316}$, according to the Wiki page on this topic).
Parenthetically, I think we can show that the lines (joining points at k=2,3 in this picture) cross the x-axis (negative to positive) near the mean number of factors of $x,$ which is asymptotically $loglog x.$ Also, the sum of the points below zero is algebraically equal to the sum of those above zero.
So for an x just beyond Skewes number the graph changes sign (if I have calculated correctly) near 3.8 and the first point at k=1 is now negative, and it is tempting to speculate (and well beyond the scope of the question but included for context) that the entire set of points changes sign.
prime-numbers asymptotics analytic-number-theory
prime-numbers asymptotics analytic-number-theory
asked Dec 2 at 17:03
daniel
6,19322155
asked Dec 2 at 17:03
daniel
6,19322155
edited Dec 9 at 12:38
asked Dec 2 at 17:03
daniel
6,19322155
asked Dec 2 at 17:03
daniel
6,19322155
asked Dec 2 at 17:03
daniel
6,19322155
6,19322155
What is a "nearprime" ? I only know the name "semiprime" for a number being the product of two primes, not necessarily distinct.
– Peter
Dec 3 at 18:36
1
@Peter: Where I heard "near-prime" I don't know. Semiprime is what I intended, and as you define it. Will wait to edit until I am pretty sure there aren't more things to clarify.
– daniel
Dec 3 at 18:47
add a comment |
What is a "nearprime" ? I only know the name "semiprime" for a number being the product of two primes, not necessarily distinct.
– Peter
Dec 3 at 18:36
1
@Peter: Where I heard "near-prime" I don't know. Semiprime is what I intended, and as you define it. Will wait to edit until I am pretty sure there aren't more things to clarify.
– daniel
Dec 3 at 18:47
What is a "nearprime" ? I only know the name "semiprime" for a number being the product of two primes, not necessarily distinct.
– Peter
Dec 3 at 18:36
What is a "nearprime" ? I only know the name "semiprime" for a number being the product of two primes, not necessarily distinct.
– Peter
Dec 3 at 18:36
1
1
@Peter: Where I heard "near-prime" I don't know. Semiprime is what I intended, and as you define it. Will wait to edit until I am pretty sure there aren't more things to clarify.
– daniel
Dec 3 at 18:47
@Peter: Where I heard "near-prime" I don't know. Semiprime is what I intended, and as you define it. Will wait to edit until I am pretty sure there aren't more things to clarify.
– daniel
Dec 3 at 18:47
add a comment |
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What is a "nearprime" ? I only know the name "semiprime" for a number being the product of two primes, not necessarily distinct.
– Peter
Dec 3 at 18:36
1
@Peter: Where I heard "near-prime" I don't know. Semiprime is what I intended, and as you define it. Will wait to edit until I am pretty sure there aren't more things to clarify.
– daniel
Dec 3 at 18:47