Arbitrarily Close Facsimile of a Patch of an Arbitrary Function
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In my opinion, the most amazing theorem in the whole of mathematics is that if you have an arbitrary function over the complex plane, and you take a patch of this function that is free of zeros and poles, and has a horizontal width of less than 1/2, and you specify a precision - some small positive real number $epsilon$ ... then a facsimile of that patch of your arbitrary function exists to that degree of precision (departing no more than |difference| = $epsilon$) somewhere in the strip of the Riemann zeta function that the strip $Re zin(frac{1}{2},1)$ maps to.
I once found, after a lot of trawling, a very tentative & very weak estimate of how far you would typically have to look along that strip for a given $epsilon$ & size of patch: it was a large compound exponential number for a patch consisting of a circle having a radius of $10^{-7}$ & $epsilon=10^{-4}$.
But that was a good few years ago. I wonder whether anyone knows of any recent advances along this line of investigation, and can give a more satisfying estimate for expected distance-along-strip given a patch-width close to 1/2 & a precision more like $10^{-10}$ ... or yet better, if possible. I would expect the result, if any has been found atall, to yield stupendously large compound, or even compound compound exponential numbers ... but still probably not as big as Skewe's (original - not ongoing ... the last I found was that that is only about $10^{373}$ now).
I think such an estimate of the expected distance would also have to incorporate a measure of some kind of how much complexity there is in the patch of function you wish to find a facsimile of: a more complicated (meaning basically with more & more-rapid varations) requiring a longer search.
UPDATE
Actually, I must confess something ... the theorem as I found it did not say "patch of horizontal width < ½" but rather "circle of radius < ¼". I have just figured to myself that the theorem would translate into being true for patch of horizontal width < ½ ... and I still think it probably does, actually ... but the question put in terms of the theorem as I found it, would be the above but with "circle of radius < ¼" substituted for "patch of horizontal width < ½". Whether it's still true in my 'taken-a-liberty-with' form could be an ancilliary question.
complex-analysis riemann-zeta
asked Nov 22 at 8:39
AmbretteOrrisey
3688
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up vote
0
down vote
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In my opinion, the most amazing theorem in the whole of mathematics is that if you have an arbitrary function over the complex plane, and you take a patch of this function that is free of zeros and poles, and has a horizontal width of less than 1/2, and you specify a precision - some small positive real number $epsilon$ ... then a facsimile of that patch of your arbitrary function exists to that degree of precision (departing no more than |difference| = $epsilon$) somewhere in the strip of the Riemann zeta function that the strip $Re zin(frac{1}{2},1)$ maps to.
I once found, after a lot of trawling, a very tentative & very weak estimate of how far you would typically have to look along that strip for a given $epsilon$ & size of patch: it was a large compound exponential number for a patch consisting of a circle having a radius of $10^{-7}$ & $epsilon=10^{-4}$.
But that was a good few years ago. I wonder whether anyone knows of any recent advances along this line of investigation, and can give a more satisfying estimate for expected distance-along-strip given a patch-width close to 1/2 & a precision more like $10^{-10}$ ... or yet better, if possible. I would expect the result, if any has been found atall, to yield stupendously large compound, or even compound compound exponential numbers ... but still probably not as big as Skewe's (original - not ongoing ... the last I found was that that is only about $10^{373}$ now).
I think such an estimate of the expected distance would also have to incorporate a measure of some kind of how much complexity there is in the patch of function you wish to find a facsimile of: a more complicated (meaning basically with more & more-rapid varations) requiring a longer search.
UPDATE
Actually, I must confess something ... the theorem as I found it did not say "patch of horizontal width < ½" but rather "circle of radius < ¼". I have just figured to myself that the theorem would translate into being true for patch of horizontal width < ½ ... and I still think it probably does, actually ... but the question put in terms of the theorem as I found it, would be the above but with "circle of radius < ¼" substituted for "patch of horizontal width < ½". Whether it's still true in my 'taken-a-liberty-with' form could be an ancilliary question.
complex-analysis riemann-zeta
asked Nov 22 at 8:39
AmbretteOrrisey
3688
Actually it really might matter! But even so, there is almost no loss of generality, since the theorem is essentially unchanged by allowing scaling ... or even by taking the exonential of a function - that way you can incorporate one with zeros.
– AmbretteOrrisey
Nov 23 at 12:06
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
In my opinion, the most amazing theorem in the whole of mathematics is that if you have an arbitrary function over the complex plane, and you take a patch of this function that is free of zeros and poles, and has a horizontal width of less than 1/2, and you specify a precision - some small positive real number $epsilon$ ... then a facsimile of that patch of your arbitrary function exists to that degree of precision (departing no more than |difference| = $epsilon$) somewhere in the strip of the Riemann zeta function that the strip $Re zin(frac{1}{2},1)$ maps to.
I once found, after a lot of trawling, a very tentative & very weak estimate of how far you would typically have to look along that strip for a given $epsilon$ & size of patch: it was a large compound exponential number for a patch consisting of a circle having a radius of $10^{-7}$ & $epsilon=10^{-4}$.
But that was a good few years ago. I wonder whether anyone knows of any recent advances along this line of investigation, and can give a more satisfying estimate for expected distance-along-strip given a patch-width close to 1/2 & a precision more like $10^{-10}$ ... or yet better, if possible. I would expect the result, if any has been found atall, to yield stupendously large compound, or even compound compound exponential numbers ... but still probably not as big as Skewe's (original - not ongoing ... the last I found was that that is only about $10^{373}$ now).
I think such an estimate of the expected distance would also have to incorporate a measure of some kind of how much complexity there is in the patch of function you wish to find a facsimile of: a more complicated (meaning basically with more & more-rapid varations) requiring a longer search.
UPDATE
Actually, I must confess something ... the theorem as I found it did not say "patch of horizontal width < ½" but rather "circle of radius < ¼". I have just figured to myself that the theorem would translate into being true for patch of horizontal width < ½ ... and I still think it probably does, actually ... but the question put in terms of the theorem as I found it, would be the above but with "circle of radius < ¼" substituted for "patch of horizontal width < ½". Whether it's still true in my 'taken-a-liberty-with' form could be an ancilliary question.
complex-analysis riemann-zeta
asked Nov 22 at 8:39
AmbretteOrrisey
3688
In my opinion, the most amazing theorem in the whole of mathematics is that if you have an arbitrary function over the complex plane, and you take a patch of this function that is free of zeros and poles, and has a horizontal width of less than 1/2, and you specify a precision - some small positive real number $epsilon$ ... then a facsimile of that patch of your arbitrary function exists to that degree of precision (departing no more than |difference| = $epsilon$) somewhere in the strip of the Riemann zeta function that the strip $Re zin(frac{1}{2},1)$ maps to.
I once found, after a lot of trawling, a very tentative & very weak estimate of how far you would typically have to look along that strip for a given $epsilon$ & size of patch: it was a large compound exponential number for a patch consisting of a circle having a radius of $10^{-7}$ & $epsilon=10^{-4}$.
But that was a good few years ago. I wonder whether anyone knows of any recent advances along this line of investigation, and can give a more satisfying estimate for expected distance-along-strip given a patch-width close to 1/2 & a precision more like $10^{-10}$ ... or yet better, if possible. I would expect the result, if any has been found atall, to yield stupendously large compound, or even compound compound exponential numbers ... but still probably not as big as Skewe's (original - not ongoing ... the last I found was that that is only about $10^{373}$ now).
I think such an estimate of the expected distance would also have to incorporate a measure of some kind of how much complexity there is in the patch of function you wish to find a facsimile of: a more complicated (meaning basically with more & more-rapid varations) requiring a longer search.
UPDATE
Actually, I must confess something ... the theorem as I found it did not say "patch of horizontal width < ½" but rather "circle of radius < ¼". I have just figured to myself that the theorem would translate into being true for patch of horizontal width < ½ ... and I still think it probably does, actually ... but the question put in terms of the theorem as I found it, would be the above but with "circle of radius < ¼" substituted for "patch of horizontal width < ½". Whether it's still true in my 'taken-a-liberty-with' form could be an ancilliary question.
complex-analysis riemann-zeta
complex-analysis riemann-zeta
asked Nov 22 at 8:39
AmbretteOrrisey
3688
asked Nov 22 at 8:39
AmbretteOrrisey
3688
edited Nov 23 at 8:29
asked Nov 22 at 8:39
AmbretteOrrisey
3688
asked Nov 22 at 8:39
AmbretteOrrisey
3688
asked Nov 22 at 8:39
AmbretteOrrisey
3688
3688
Actually it really might matter! But even so, there is almost no loss of generality, since the theorem is essentially unchanged by allowing scaling ... or even by taking the exonential of a function - that way you can incorporate one with zeros.
– AmbretteOrrisey
Nov 23 at 12:06
add a comment |
Actually it really might matter! But even so, there is almost no loss of generality, since the theorem is essentially unchanged by allowing scaling ... or even by taking the exonential of a function - that way you can incorporate one with zeros.
– AmbretteOrrisey
Nov 23 at 12:06
Actually it really might matter! But even so, there is almost no loss of generality, since the theorem is essentially unchanged by allowing scaling ... or even by taking the exonential of a function - that way you can incorporate one with zeros.
– AmbretteOrrisey
Nov 23 at 12:06
Actually it really might matter! But even so, there is almost no loss of generality, since the theorem is essentially unchanged by allowing scaling ... or even by taking the exonential of a function - that way you can incorporate one with zeros.
– AmbretteOrrisey
Nov 23 at 12:06
add a comment |
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Actually it really might matter! But even so, there is almost no loss of generality, since the theorem is essentially unchanged by allowing scaling ... or even by taking the exonential of a function - that way you can incorporate one with zeros.
– AmbretteOrrisey
Nov 23 at 12:06