Is this equivalent to the Riemann Hypothesis?
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By a result of Spira, we know that the Riemann Hypothesis (RH) is equivalent to the statement that $|zeta(1-s)|$ increases as $Re(s)$ varies on $(frac{1}{2}, infty)$ with $|t|=|Im(s)|geq 165$ fixed.
Since $|zeta(s)|$ is continuous, decreasing for $Re(s)>1$ and $s$ is a zero of $zeta$ whenever $(1-s)$ is a zero, Spira's result entails that the RH is equivalent to the statement that $|zeta(s)|$ decreases as $Re(s)$ varies on $(frac{1}{2}, infty)$ with $|t|=|Im(s)|geq 165$ fixed.
A combination of these two statements seems to yield another equivalent statement for the RH, namely: *The RH is equivalent to the statement that $F(s)=frac{|zeta(s)|}{|zeta(1-s)|}$ decreases as $Re(s)$ varies on $(1/2, 1]$ with $|t|=|Im(s)|geq 165$ fixed ?
analysis riemann-zeta riemann-hypothesis
asked Dec 24 '18 at 17:26
OneTwoOneOneTwoOne
307
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show 2 more comments
$begingroup$
By a result of Spira, we know that the Riemann Hypothesis (RH) is equivalent to the statement that $|zeta(1-s)|$ increases as $Re(s)$ varies on $(frac{1}{2}, infty)$ with $|t|=|Im(s)|geq 165$ fixed.
Since $|zeta(s)|$ is continuous, decreasing for $Re(s)>1$ and $s$ is a zero of $zeta$ whenever $(1-s)$ is a zero, Spira's result entails that the RH is equivalent to the statement that $|zeta(s)|$ decreases as $Re(s)$ varies on $(frac{1}{2}, infty)$ with $|t|=|Im(s)|geq 165$ fixed.
A combination of these two statements seems to yield another equivalent statement for the RH, namely: *The RH is equivalent to the statement that $F(s)=frac{|zeta(s)|}{|zeta(1-s)|}$ decreases as $Re(s)$ varies on $(1/2, 1]$ with $|t|=|Im(s)|geq 165$ fixed ?
analysis riemann-zeta riemann-hypothesis
asked Dec 24 '18 at 17:26
OneTwoOneOneTwoOne
307
$endgroup$
$begingroup$
Increasing/decreasing along what path?
$endgroup$
– Dzoooks
Dec 24 '18 at 20:40
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@Dzoooks, i had forgotten to add certain details, please see the present form of the question.
$endgroup$
– OneTwoOne
Dec 24 '18 at 22:11
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Riemann's functional equation gives a simple relation between $zeta(s)$ and $zeta(1-s)$ namely $frac{zeta(s)}{zeta(1-s)} = 2^spi^{s-1} sinleft(frac{pi s}{2}right) Gamma(1-s)$
$endgroup$
– Winther
Dec 24 '18 at 22:46
1
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btw it's a one way implication if $|zeta(s)|$ is decreasing and $|zeta(1-s)|$ is increasing $implies $ $F(s)$ is decreasing. You don't get the other way so it's not an equivalent formulation.
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– Winther
Dec 24 '18 at 22:53
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@Winther, basing on the fact that Spira didn't use any other property of $zeta$ besides the functional equation, i wouldn't perceive the functional equation as a ''simple'' relation between $zeta(s)$ and $zeta(1-s)$.
$endgroup$
– OneTwoOne
Dec 24 '18 at 22:55
|
show 2 more comments
$begingroup$
By a result of Spira, we know that the Riemann Hypothesis (RH) is equivalent to the statement that $|zeta(1-s)|$ increases as $Re(s)$ varies on $(frac{1}{2}, infty)$ with $|t|=|Im(s)|geq 165$ fixed.
Since $|zeta(s)|$ is continuous, decreasing for $Re(s)>1$ and $s$ is a zero of $zeta$ whenever $(1-s)$ is a zero, Spira's result entails that the RH is equivalent to the statement that $|zeta(s)|$ decreases as $Re(s)$ varies on $(frac{1}{2}, infty)$ with $|t|=|Im(s)|geq 165$ fixed.
A combination of these two statements seems to yield another equivalent statement for the RH, namely: *The RH is equivalent to the statement that $F(s)=frac{|zeta(s)|}{|zeta(1-s)|}$ decreases as $Re(s)$ varies on $(1/2, 1]$ with $|t|=|Im(s)|geq 165$ fixed ?
analysis riemann-zeta riemann-hypothesis
asked Dec 24 '18 at 17:26
OneTwoOneOneTwoOne
307
$endgroup$
By a result of Spira, we know that the Riemann Hypothesis (RH) is equivalent to the statement that $|zeta(1-s)|$ increases as $Re(s)$ varies on $(frac{1}{2}, infty)$ with $|t|=|Im(s)|geq 165$ fixed.
Since $|zeta(s)|$ is continuous, decreasing for $Re(s)>1$ and $s$ is a zero of $zeta$ whenever $(1-s)$ is a zero, Spira's result entails that the RH is equivalent to the statement that $|zeta(s)|$ decreases as $Re(s)$ varies on $(frac{1}{2}, infty)$ with $|t|=|Im(s)|geq 165$ fixed.
A combination of these two statements seems to yield another equivalent statement for the RH, namely: *The RH is equivalent to the statement that $F(s)=frac{|zeta(s)|}{|zeta(1-s)|}$ decreases as $Re(s)$ varies on $(1/2, 1]$ with $|t|=|Im(s)|geq 165$ fixed ?
analysis riemann-zeta riemann-hypothesis
analysis riemann-zeta riemann-hypothesis
asked Dec 24 '18 at 17:26
OneTwoOneOneTwoOne
307
asked Dec 24 '18 at 17:26
OneTwoOneOneTwoOne
307
edited Dec 24 '18 at 22:51
OneTwoOne
asked Dec 24 '18 at 17:26
OneTwoOneOneTwoOne
307
asked Dec 24 '18 at 17:26
OneTwoOneOneTwoOne
307
asked Dec 24 '18 at 17:26
OneTwoOneOneTwoOne
307
307
$begingroup$
Increasing/decreasing along what path?
$endgroup$
– Dzoooks
Dec 24 '18 at 20:40
$begingroup$
@Dzoooks, i had forgotten to add certain details, please see the present form of the question.
$endgroup$
– OneTwoOne
Dec 24 '18 at 22:11
$begingroup$
Riemann's functional equation gives a simple relation between $zeta(s)$ and $zeta(1-s)$ namely $frac{zeta(s)}{zeta(1-s)} = 2^spi^{s-1} sinleft(frac{pi s}{2}right) Gamma(1-s)$
$endgroup$
– Winther
Dec 24 '18 at 22:46
1
$begingroup$
btw it's a one way implication if $|zeta(s)|$ is decreasing and $|zeta(1-s)|$ is increasing $implies $ $F(s)$ is decreasing. You don't get the other way so it's not an equivalent formulation.
$endgroup$
– Winther
Dec 24 '18 at 22:53
$begingroup$
@Winther, basing on the fact that Spira didn't use any other property of $zeta$ besides the functional equation, i wouldn't perceive the functional equation as a ''simple'' relation between $zeta(s)$ and $zeta(1-s)$.
$endgroup$
– OneTwoOne
Dec 24 '18 at 22:55
|
show 2 more comments
$begingroup$
Increasing/decreasing along what path?
$endgroup$
– Dzoooks
Dec 24 '18 at 20:40
$begingroup$
@Dzoooks, i had forgotten to add certain details, please see the present form of the question.
$endgroup$
– OneTwoOne
Dec 24 '18 at 22:11
$begingroup$
Riemann's functional equation gives a simple relation between $zeta(s)$ and $zeta(1-s)$ namely $frac{zeta(s)}{zeta(1-s)} = 2^spi^{s-1} sinleft(frac{pi s}{2}right) Gamma(1-s)$
$endgroup$
– Winther
Dec 24 '18 at 22:46
1
$begingroup$
btw it's a one way implication if $|zeta(s)|$ is decreasing and $|zeta(1-s)|$ is increasing $implies $ $F(s)$ is decreasing. You don't get the other way so it's not an equivalent formulation.
$endgroup$
– Winther
Dec 24 '18 at 22:53
$begingroup$
@Winther, basing on the fact that Spira didn't use any other property of $zeta$ besides the functional equation, i wouldn't perceive the functional equation as a ''simple'' relation between $zeta(s)$ and $zeta(1-s)$.
$endgroup$
– OneTwoOne
Dec 24 '18 at 22:55
$begingroup$
Increasing/decreasing along what path?
$endgroup$
– Dzoooks
Dec 24 '18 at 20:40
$begingroup$
Increasing/decreasing along what path?
$endgroup$
– Dzoooks
Dec 24 '18 at 20:40
$begingroup$
@Dzoooks, i had forgotten to add certain details, please see the present form of the question.
$endgroup$
– OneTwoOne
Dec 24 '18 at 22:11
$begingroup$
@Dzoooks, i had forgotten to add certain details, please see the present form of the question.
$endgroup$
– OneTwoOne
Dec 24 '18 at 22:11
$begingroup$
Riemann's functional equation gives a simple relation between $zeta(s)$ and $zeta(1-s)$ namely $frac{zeta(s)}{zeta(1-s)} = 2^spi^{s-1} sinleft(frac{pi s}{2}right) Gamma(1-s)$
$endgroup$
– Winther
Dec 24 '18 at 22:46
$begingroup$
Riemann's functional equation gives a simple relation between $zeta(s)$ and $zeta(1-s)$ namely $frac{zeta(s)}{zeta(1-s)} = 2^spi^{s-1} sinleft(frac{pi s}{2}right) Gamma(1-s)$
$endgroup$
– Winther
Dec 24 '18 at 22:46
1
1
$begingroup$
btw it's a one way implication if $|zeta(s)|$ is decreasing and $|zeta(1-s)|$ is increasing $implies $ $F(s)$ is decreasing. You don't get the other way so it's not an equivalent formulation.
$endgroup$
– Winther
Dec 24 '18 at 22:53
$begingroup$
btw it's a one way implication if $|zeta(s)|$ is decreasing and $|zeta(1-s)|$ is increasing $implies $ $F(s)$ is decreasing. You don't get the other way so it's not an equivalent formulation.
$endgroup$
– Winther
Dec 24 '18 at 22:53
$begingroup$
@Winther, basing on the fact that Spira didn't use any other property of $zeta$ besides the functional equation, i wouldn't perceive the functional equation as a ''simple'' relation between $zeta(s)$ and $zeta(1-s)$.
$endgroup$
– OneTwoOne
Dec 24 '18 at 22:55
$begingroup$
@Winther, basing on the fact that Spira didn't use any other property of $zeta$ besides the functional equation, i wouldn't perceive the functional equation as a ''simple'' relation between $zeta(s)$ and $zeta(1-s)$.
$endgroup$
– OneTwoOne
Dec 24 '18 at 22:55
|
show 2 more comments
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$begingroup$
Increasing/decreasing along what path?
$endgroup$
– Dzoooks
Dec 24 '18 at 20:40
$begingroup$
@Dzoooks, i had forgotten to add certain details, please see the present form of the question.
$endgroup$
– OneTwoOne
Dec 24 '18 at 22:11
$begingroup$
Riemann's functional equation gives a simple relation between $zeta(s)$ and $zeta(1-s)$ namely $frac{zeta(s)}{zeta(1-s)} = 2^spi^{s-1} sinleft(frac{pi s}{2}right) Gamma(1-s)$
$endgroup$
– Winther
Dec 24 '18 at 22:46
1
$begingroup$
btw it's a one way implication if $|zeta(s)|$ is decreasing and $|zeta(1-s)|$ is increasing $implies $ $F(s)$ is decreasing. You don't get the other way so it's not an equivalent formulation.
$endgroup$
– Winther
Dec 24 '18 at 22:53
$begingroup$
@Winther, basing on the fact that Spira didn't use any other property of $zeta$ besides the functional equation, i wouldn't perceive the functional equation as a ''simple'' relation between $zeta(s)$ and $zeta(1-s)$.
$endgroup$
– OneTwoOne
Dec 24 '18 at 22:55