Proof claimed that almost all zeroes of the Riemann zeta function lie on the critical line
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There have been many results in recent years on the natural density of zeta zeroes on the critical line, with the best bound commonly accepted (as far as I'm aware) that
$$liminf_{ttoinfty} frac{N_0(t)}{N(t)} > 0.4128dots$$
by Feng (here $N_0(t)$ is the number of zeroes $chi$ on the critical line with $|Im(chi)|<t$ and $N(t)$ is the number of zeroes in the critical strip where the same condition holds).
This week, a new revision to this paper was currently released, and I stumbled upon it. It claims that
$$liminf_{ttoinfty} frac{N_0(t)}{N(t)} = 1.$$
Given that versions of it have been up for the past almost 8 months, I'm imagining it's not correct (otherwise it would have been reviewed and accepted by now), but I don't know enough analytic number theory to refute it and I haven't been able to find any commentary on it. Furthermore, the paper passes many of the "crank tests" (well-formatted, well-written and structured, etc.), so it is unlikely to be complete garbage.
Can anyone with more knowledge of analytic number theory confirm or refute (or find a reference confirming or refuting) this paper's validity?
analytic-number-theory riemann-zeta
asked Jan 14 at 5:55
Carl SchildkrautCarl Schildkraut
11.3k11441
$endgroup$
|
show 2 more comments
$begingroup$
There have been many results in recent years on the natural density of zeta zeroes on the critical line, with the best bound commonly accepted (as far as I'm aware) that
$$liminf_{ttoinfty} frac{N_0(t)}{N(t)} > 0.4128dots$$
by Feng (here $N_0(t)$ is the number of zeroes $chi$ on the critical line with $|Im(chi)|<t$ and $N(t)$ is the number of zeroes in the critical strip where the same condition holds).
This week, a new revision to this paper was currently released, and I stumbled upon it. It claims that
$$liminf_{ttoinfty} frac{N_0(t)}{N(t)} = 1.$$
Given that versions of it have been up for the past almost 8 months, I'm imagining it's not correct (otherwise it would have been reviewed and accepted by now), but I don't know enough analytic number theory to refute it and I haven't been able to find any commentary on it. Furthermore, the paper passes many of the "crank tests" (well-formatted, well-written and structured, etc.), so it is unlikely to be complete garbage.
Can anyone with more knowledge of analytic number theory confirm or refute (or find a reference confirming or refuting) this paper's validity?
analytic-number-theory riemann-zeta
asked Jan 14 at 5:55
Carl SchildkrautCarl Schildkraut
11.3k11441
$endgroup$
1
$begingroup$
The paper is in the GM section, not the NT section of the Arxiv.
$endgroup$
– Lord Shark the Unknown
Jan 14 at 6:04
1
$begingroup$
@LordSharktheUnknown Forgive my ignorance, but what exactly does that mean? Does it mean since it's misclassified it's less likely to be correct? Also, can the downvoter please explain what I can do to improve the question or make it more suitable for math.stackexchange?
$endgroup$
– Carl Schildkraut
Jan 14 at 6:11
1
$begingroup$
The GM section is where crank papers submitted to Arxiv are dumped. Certainly then, the Arxiv administrators are sceptical about the paper.
$endgroup$
– Lord Shark the Unknown
Jan 14 at 6:22
1
$begingroup$
eight month does not sound like a red flag to me, not all correct papers are accepted and published within 8 months. Didn't know about the GM section though.
$endgroup$
– user120527
Jan 14 at 8:59
1
$begingroup$
At least the paper is using the functional equation and that $mu(n) in {-1,0,1}$ so it may be at least partially valid. If you want to find the main ideas and if they are different from the traditional approach you should ask people like Lucia and GH on mathoverflow.
$endgroup$
– reuns
Jan 14 at 15:31
|
show 2 more comments
$begingroup$
There have been many results in recent years on the natural density of zeta zeroes on the critical line, with the best bound commonly accepted (as far as I'm aware) that
$$liminf_{ttoinfty} frac{N_0(t)}{N(t)} > 0.4128dots$$
by Feng (here $N_0(t)$ is the number of zeroes $chi$ on the critical line with $|Im(chi)|<t$ and $N(t)$ is the number of zeroes in the critical strip where the same condition holds).
This week, a new revision to this paper was currently released, and I stumbled upon it. It claims that
$$liminf_{ttoinfty} frac{N_0(t)}{N(t)} = 1.$$
Given that versions of it have been up for the past almost 8 months, I'm imagining it's not correct (otherwise it would have been reviewed and accepted by now), but I don't know enough analytic number theory to refute it and I haven't been able to find any commentary on it. Furthermore, the paper passes many of the "crank tests" (well-formatted, well-written and structured, etc.), so it is unlikely to be complete garbage.
Can anyone with more knowledge of analytic number theory confirm or refute (or find a reference confirming or refuting) this paper's validity?
analytic-number-theory riemann-zeta
asked Jan 14 at 5:55
Carl SchildkrautCarl Schildkraut
11.3k11441
$endgroup$
There have been many results in recent years on the natural density of zeta zeroes on the critical line, with the best bound commonly accepted (as far as I'm aware) that
$$liminf_{ttoinfty} frac{N_0(t)}{N(t)} > 0.4128dots$$
by Feng (here $N_0(t)$ is the number of zeroes $chi$ on the critical line with $|Im(chi)|<t$ and $N(t)$ is the number of zeroes in the critical strip where the same condition holds).
This week, a new revision to this paper was currently released, and I stumbled upon it. It claims that
$$liminf_{ttoinfty} frac{N_0(t)}{N(t)} = 1.$$
Given that versions of it have been up for the past almost 8 months, I'm imagining it's not correct (otherwise it would have been reviewed and accepted by now), but I don't know enough analytic number theory to refute it and I haven't been able to find any commentary on it. Furthermore, the paper passes many of the "crank tests" (well-formatted, well-written and structured, etc.), so it is unlikely to be complete garbage.
Can anyone with more knowledge of analytic number theory confirm or refute (or find a reference confirming or refuting) this paper's validity?
analytic-number-theory riemann-zeta
analytic-number-theory riemann-zeta
asked Jan 14 at 5:55
Carl SchildkrautCarl Schildkraut
11.3k11441
asked Jan 14 at 5:55
Carl SchildkrautCarl Schildkraut
11.3k11441
asked Jan 14 at 5:55
Carl SchildkrautCarl Schildkraut
11.3k11441
asked Jan 14 at 5:55
Carl SchildkrautCarl Schildkraut
11.3k11441
asked Jan 14 at 5:55
Carl SchildkrautCarl Schildkraut
11.3k11441
11.3k11441
1
$begingroup$
The paper is in the GM section, not the NT section of the Arxiv.
$endgroup$
– Lord Shark the Unknown
Jan 14 at 6:04
1
$begingroup$
@LordSharktheUnknown Forgive my ignorance, but what exactly does that mean? Does it mean since it's misclassified it's less likely to be correct? Also, can the downvoter please explain what I can do to improve the question or make it more suitable for math.stackexchange?
$endgroup$
– Carl Schildkraut
Jan 14 at 6:11
1
$begingroup$
The GM section is where crank papers submitted to Arxiv are dumped. Certainly then, the Arxiv administrators are sceptical about the paper.
$endgroup$
– Lord Shark the Unknown
Jan 14 at 6:22
1
$begingroup$
eight month does not sound like a red flag to me, not all correct papers are accepted and published within 8 months. Didn't know about the GM section though.
$endgroup$
– user120527
Jan 14 at 8:59
1
$begingroup$
At least the paper is using the functional equation and that $mu(n) in {-1,0,1}$ so it may be at least partially valid. If you want to find the main ideas and if they are different from the traditional approach you should ask people like Lucia and GH on mathoverflow.
$endgroup$
– reuns
Jan 14 at 15:31
|
show 2 more comments
1
$begingroup$
The paper is in the GM section, not the NT section of the Arxiv.
$endgroup$
– Lord Shark the Unknown
Jan 14 at 6:04
1
$begingroup$
@LordSharktheUnknown Forgive my ignorance, but what exactly does that mean? Does it mean since it's misclassified it's less likely to be correct? Also, can the downvoter please explain what I can do to improve the question or make it more suitable for math.stackexchange?
$endgroup$
– Carl Schildkraut
Jan 14 at 6:11
1
$begingroup$
The GM section is where crank papers submitted to Arxiv are dumped. Certainly then, the Arxiv administrators are sceptical about the paper.
$endgroup$
– Lord Shark the Unknown
Jan 14 at 6:22
1
$begingroup$
eight month does not sound like a red flag to me, not all correct papers are accepted and published within 8 months. Didn't know about the GM section though.
$endgroup$
– user120527
Jan 14 at 8:59
1
$begingroup$
At least the paper is using the functional equation and that $mu(n) in {-1,0,1}$ so it may be at least partially valid. If you want to find the main ideas and if they are different from the traditional approach you should ask people like Lucia and GH on mathoverflow.
$endgroup$
– reuns
Jan 14 at 15:31
1
1
$begingroup$
The paper is in the GM section, not the NT section of the Arxiv.
$endgroup$
– Lord Shark the Unknown
Jan 14 at 6:04
$begingroup$
The paper is in the GM section, not the NT section of the Arxiv.
$endgroup$
– Lord Shark the Unknown
Jan 14 at 6:04
1
1
$begingroup$
@LordSharktheUnknown Forgive my ignorance, but what exactly does that mean? Does it mean since it's misclassified it's less likely to be correct? Also, can the downvoter please explain what I can do to improve the question or make it more suitable for math.stackexchange?
$endgroup$
– Carl Schildkraut
Jan 14 at 6:11
$begingroup$
@LordSharktheUnknown Forgive my ignorance, but what exactly does that mean? Does it mean since it's misclassified it's less likely to be correct? Also, can the downvoter please explain what I can do to improve the question or make it more suitable for math.stackexchange?
$endgroup$
– Carl Schildkraut
Jan 14 at 6:11
1
1
$begingroup$
The GM section is where crank papers submitted to Arxiv are dumped. Certainly then, the Arxiv administrators are sceptical about the paper.
$endgroup$
– Lord Shark the Unknown
Jan 14 at 6:22
$begingroup$
The GM section is where crank papers submitted to Arxiv are dumped. Certainly then, the Arxiv administrators are sceptical about the paper.
$endgroup$
– Lord Shark the Unknown
Jan 14 at 6:22
1
1
$begingroup$
eight month does not sound like a red flag to me, not all correct papers are accepted and published within 8 months. Didn't know about the GM section though.
$endgroup$
– user120527
Jan 14 at 8:59
$begingroup$
eight month does not sound like a red flag to me, not all correct papers are accepted and published within 8 months. Didn't know about the GM section though.
$endgroup$
– user120527
Jan 14 at 8:59
1
1
$begingroup$
At least the paper is using the functional equation and that $mu(n) in {-1,0,1}$ so it may be at least partially valid. If you want to find the main ideas and if they are different from the traditional approach you should ask people like Lucia and GH on mathoverflow.
$endgroup$
– reuns
Jan 14 at 15:31
$begingroup$
At least the paper is using the functional equation and that $mu(n) in {-1,0,1}$ so it may be at least partially valid. If you want to find the main ideas and if they are different from the traditional approach you should ask people like Lucia and GH on mathoverflow.
$endgroup$
– reuns
Jan 14 at 15:31
|
show 2 more comments
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$begingroup$
The paper is in the GM section, not the NT section of the Arxiv.
$endgroup$
– Lord Shark the Unknown
Jan 14 at 6:04
1
$begingroup$
@LordSharktheUnknown Forgive my ignorance, but what exactly does that mean? Does it mean since it's misclassified it's less likely to be correct? Also, can the downvoter please explain what I can do to improve the question or make it more suitable for math.stackexchange?
$endgroup$
– Carl Schildkraut
Jan 14 at 6:11
1
$begingroup$
The GM section is where crank papers submitted to Arxiv are dumped. Certainly then, the Arxiv administrators are sceptical about the paper.
$endgroup$
– Lord Shark the Unknown
Jan 14 at 6:22
1
$begingroup$
eight month does not sound like a red flag to me, not all correct papers are accepted and published within 8 months. Didn't know about the GM section though.
$endgroup$
– user120527
Jan 14 at 8:59
1
$begingroup$
At least the paper is using the functional equation and that $mu(n) in {-1,0,1}$ so it may be at least partially valid. If you want to find the main ideas and if they are different from the traditional approach you should ask people like Lucia and GH on mathoverflow.
$endgroup$
– reuns
Jan 14 at 15:31