Announcement on the Riemann hypothesis
in [Math]
|
Prev: combinations of additive and multiplicative creation: Dirac's new radioactivities Chapt 5 #181; ATOM TOTALITY
Next: topology
From: H. Shinya on 21 Jun 2010 21:24 On Jun 21, 11:41 am, Tonico <Tonic...(a)yahoo.com> wrote: > On Jun 20, 3:29 pm, "H. Shinya" <shinya...(a)yahoo.co.jp> wrote: > > > > > > > Dear Professors, > > > I would like to make an announcement concerning the Riemann > > > hypothesis. > > > (I choose not to explain what it is.) > > > In the arXiv, a paper titled > > > An integral formula for 1/|zeta(s)| and the Riemann hypothesis > > > is available to anyone with Internet. The ID for the paper is > > 0706.0357. > > > (It is listed in the General Mathematics section of the arXiv.) > > > As seen in the arXiv, I have made a lot of mistakes as > > > an amateur. (Also, in the past, I made an announcement of the same > > > topic through this group.) Therefore, some part of the present version > > > may be ambiguous (or quite possibly be wrong), but I for one believe > > that all > > > such parts are easily verified, especially if the reader's > > specialization is > > > analytic number theory. > > > The main claim of the paper is that I may have resolved the Riemann > > > hypothesis. > > > I hope that some kind of workshop on the paper may be started out > > > through this thread. > > > The purpose of opening a workshop is, of course, to make the result > > > solidly established. > > > With regards, > > > H. Shinya > > Already in page 2, in the "proof" of theorem A, I spot a probable > problem: you say that "...we assume that Gamma_x(t) is a rapidly > decreasing function of t for each fixed x > 0. That is, > Gamma_xx(t) as a function of t is a member of the so-called Schwartz > space. The proof of this assumption is rather a cumbersome exercise, > so we choose not to prove it here." > > Fine, prove it not there, but AT LEAST give a reference where we can > find that! > > Tonio- Hide quoted text - > > - Show quoted text - I will give you an outline. First, it is a very basic fact that if a function f is in the Schwartz space, then so is its Fourier transform. This statement can be found in a book listed in the References. Recall the equation (15) of pp. 6, which gives the Gamma function in the form of Fourier transform (of the function E_{x, 1}(t) ; m = 1). It is much easier to show that E_{x, 1}(t) is in the Schwartz space if x > 0 than typing the validation code correctly, which I need to do to send messages. H. Shinya
From: David Bernier on 22 Jun 2010 00:26 H. Shinya wrote: > Dear Professors, > > I would like to make an announcement concerning the Riemann > > hypothesis. > > (I choose not to explain what it is.) > > In the arXiv, a paper titled > > An integral formula for 1/|zeta(s)| and the Riemann hypothesis > > is available to anyone with Internet. The ID for the paper is > 0706.0357. > > (It is listed in the General Mathematics section of the arXiv.) > > As seen in the arXiv, I have made a lot of mistakes as > > an amateur. (Also, in the past, I made an announcement of the same > > topic through this group.) Therefore, some part of the present version > > may be ambiguous (or quite possibly be wrong), but I for one believe > that all > > such parts are easily verified, especially if the reader's > specialization is > > analytic number theory. > > > > The main claim of the paper is that I may have resolved the Riemann > > hypothesis. > > I hope that some kind of workshop on the paper may be started out > > through this thread. > > The purpose of opening a workshop is, of course, to make the result > > solidly established. [...] On page 7 you have Proposition C, an equality that you say is true if x > 1. (Equivalently, Re(s) > 1). In Corollary 1, page 10, at the end of the proof, you write: << Finally, as x --> x_0^{+} , the integral on the right of the formula in the proposition necessarily tends to oo in magnitude provided 1 − K(rho_0) =/= 0, while the left member of the same formula tends to a finite value. This completes the proof of the corollary. >> The setting is x_0 = Re(rho_0), zeta(rho_0) = 0, and rho_0 not on the axis of real numbers. Also, Re(rho_0) > 1/2 , by assumption. So 1/2 < x_0 <= 1. The LHS = RHS equality in Proposition C assumes x > 1. Are you saying that LHS = RHS in Proposition C also when x > x_0 ? I don't know why ( LHS = RHS ) would also hold in Proposition C for x_0 < x <= 1 ... David Bernier
From: H. Shinya on 22 Jun 2010 09:11 On Jun 22, 1:26 pm, David Bernier <david...(a)videotron.ca> wrote: > H. Shinya wrote: > > Dear Professors, > > > I would like to make anannouncementconcerning the Riemann > > > hypothesis. > > > (I choose not to explain what it is.) > > > In the arXiv, a paper titled > > > An integral formula for 1/|zeta(s)| and the Riemann hypothesis > > > is available to anyone with Internet. The ID for the paper is > > 0706.0357. > > > (It is listed in the General Mathematics section of the arXiv.) > > > As seen in the arXiv, I have made a lot of mistakes as > > > an amateur. (Also, in the past, I made anannouncementof the same > > > topic through this group.) Therefore, some part of the present version > > > may be ambiguous (or quite possibly be wrong), but I for one believe > > that all > > > such parts are easily verified, especially if the reader's > > specialization is > > > analytic number theory. > > > The main claim of the paper is that I may have resolved the Riemann > > > hypothesis. > > > I hope that some kind of workshop on the paper may be started out > > > through this thread. > > > The purpose of opening a workshop is, of course, to make the result > > > solidly established. > > [...] > > On page 7 you have Proposition C, an equality that you say > is true if x > 1. (Equivalently, Re(s) > 1). > > In Corollary 1, page 10, at the end of the proof, you write: > > << Finally, as x --> x_0^{+} , the integral on the right of the > formula in the proposition > necessarily tends to oo in magnitude provided 1 - K(rho_0) =/= 0, > while the left member of the same formula tends to a finite value. > This completes the proof of the corollary. >> > > The setting is x_0 = Re(rho_0), zeta(rho_0) = 0, and rho_0 not on the > axis of real numbers. Also, Re(rho_0) > 1/2 , by assumption. > > So 1/2 < x_0 <= 1. > > The LHS = RHS equality in Proposition C assumes x > 1. > > Are you saying that > LHS = RHS in Proposition C also when x > x_0 ? > > I don't know why ( LHS = RHS ) would also hold in Proposition C > for x_0 < x <= 1 ... First of all, explanation for this starts from the paragraph which contains the equation (33), down to the end of that section. A theoretical background for this comes from a theorem concerning analyticity of an integral function. I do not think I could recite it here correctly, but you can find it in, say, Lang's book "Complex Analysis" (the GTM series). Suppose that f = f(v, w) is a function f: R times D -> C C is the complex plane; D is a region in C; R is the real line. If for each fixed v, f is an analytic function of w in D, and f satisfies \int_{-\infty}^{\infty} |f(v, w)| dv < infty then, the integral function F(w) := \int_{-\infty}^{\infty} f(v, w) dv is an analytic function of w. The rest is just analytic continuation; that is, simply put, if expressions of a relation are meaningful and both functions are analytic in some wider region, then you can extend that relation to that wider region on which that relation is meaningful. H. Shinya > I don't know why ( LHS = RHS ) would also hold in Proposition C > for x_0 < x <= 1 .. > > David Bernier- Hide quoted text - >
From: David Bernier on 22 Jun 2010 15:21 H. Shinya wrote: > On Jun 22, 1:26 pm, David Bernier<david...(a)videotron.ca> wrote: >> H. Shinya wrote: >>> Dear Professors, >> >>> I would like to make anannouncementconcerning the Riemann >> >>> hypothesis. >> >>> (I choose not to explain what it is.) >> >>> In the arXiv, a paper titled >> >>> An integral formula for 1/|zeta(s)| and the Riemann hypothesis >> >>> is available to anyone with Internet. The ID for the paper is >>> 0706.0357. >> >>> (It is listed in the General Mathematics section of the arXiv.) >> >>> As seen in the arXiv, I have made a lot of mistakes as >> >>> an amateur. (Also, in the past, I made anannouncementof the same >> >>> topic through this group.) Therefore, some part of the present version >> >>> may be ambiguous (or quite possibly be wrong), but I for one believe >>> that all >> >>> such parts are easily verified, especially if the reader's >>> specialization is >> >>> analytic number theory. >> >>> The main claim of the paper is that I may have resolved the Riemann >> >>> hypothesis. >> >>> I hope that some kind of workshop on the paper may be started out >> >>> through this thread. >> >>> The purpose of opening a workshop is, of course, to make the result >> >>> solidly established. >> >> [...] >> >> On page 7 you have Proposition C, an equality that you say >> is true if x> 1. (Equivalently, Re(s)> 1). >> >> In Corollary 1, page 10, at the end of the proof, you write: >> >> << Finally, as x --> x_0^{+} , the integral on the right of the >> formula in the proposition >> necessarily tends to oo in magnitude provided 1 - K(rho_0) =/= 0, >> while the left member of the same formula tends to a finite value. >> This completes the proof of the corollary.>> >> >> The setting is x_0 = Re(rho_0), zeta(rho_0) = 0, and rho_0 not on the >> axis of real numbers. Also, Re(rho_0)> 1/2 , by assumption. >> >> So 1/2< x_0<= 1. >> >> The LHS = RHS equality in Proposition C assumes x> 1. >> >> Are you saying that >> LHS = RHS in Proposition C also when x> x_0 ? >> >> I don't know why ( LHS = RHS ) would also hold in Proposition C >> for x_0< x<= 1 ... > > First of all, explanation for this > starts from the paragraph which > contains the equation (33), down > to the end of that section. > > A theoretical background for this comes from > a theorem concerning analyticity of an integral function. > > I do not think I could recite it here correctly, but > you can find it in, say, Lang's book "Complex Analysis" > (the GTM series). > > Suppose that f = f(v, w) is a function > > f: R times D -> C > > C is the complex plane; > D is a region in C; > R is the real line. > > If for each fixed v, f is an analytic function of w in D, > and f satisfies > > \int_{-\infty}^{\infty} |f(v, w)| dv< infty > > then, the integral function > > F(w) := \int_{-\infty}^{\infty} f(v, w) dv > > is an analytic function of w. > > > The rest is just analytic continuation; that is, simply put, > > if expressions of a relation are meaningful and > both functions are analytic in some wider region, > then you can extend that relation to that wider region > on which that relation is meaningful. [...] Thanks for your explanations. On page 10, in the Corollary 1, you cite in (b) a result from Titchmarsh's 1951 edition of "The Theory of the Riemann Zeta-Function". log zeta(x+it) = [ sum_{ |t-gamma| <= 1} log(s - rho) ] + O(log(t)), uniformly for -1 <= x <= 2, where gamma denotes the imaginary parts of non-trivial zeros rho. I think we're assuming that s := x+it ... Shouldn't there be a condition on t = Im(s), because for example if s is a non-trivial zero of zeta on the critical line Re(s) = 1/2 , then zeta(1/2 + it) = 0, so what does log zeta(x+it) mean? I'd appreciate having the name of the Chapter in Titchmarsh [1951] and/or Section number and mathematicians who discovered this estimate. Thanks, David Bernier
From: H. Shinya on 23 Jun 2010 01:38
On Jun 23, 4:21 am, David Bernier <david...(a)videotron.ca> wrote: > H. Shinya wrote: > > On Jun 22, 1:26 pm, David Bernier<david...(a)videotron.ca> wrote: > >> H. Shinya wrote: > >>> Dear Professors, > > >>> I would like to make anannouncementconcerning the Riemann > > >>> hypothesis. > > >>> (I choose not to explain what it is.) > > >>> In the arXiv, a paper titled > > >>> An integral formula for 1/|zeta(s)| and the Riemann hypothesis > > >>> is available to anyone with Internet. The ID for the paper is > >>> 0706.0357. > > >>> (It is listed in the General Mathematics section of the arXiv.) > > >>> As seen in the arXiv, I have made a lot of mistakes as > > >>> an amateur. (Also, in the past, I made anannouncementof the same > > >>> topic through this group.) Therefore, some part of the present version > > >>> may be ambiguous (or quite possibly be wrong), but I for one believe > >>> that all > > >>> such parts are easily verified, especially if the reader's > >>> specialization is > > >>> analytic number theory. > > >>> The main claim of the paper is that I may have resolved the Riemann > > >>> hypothesis. > > >>> I hope that some kind of workshop on the paper may be started out > > >>> through this thread. > > >>> The purpose of opening a workshop is, of course, to make the result > > >>> solidly established. > > >> [...] > > >> On page 7 you have Proposition C, an equality that you say > >> is true if x> 1. (Equivalently, Re(s)> 1). > > >> In Corollary 1, page 10, at the end of the proof, you write: > > >> << Finally, as x --> x_0^{+} , the integral on the right of the > >> formula in the proposition > >> necessarily tends to oo in magnitude provided 1 - K(rho_0) =/= 0, > >> while the left member of the same formula tends to a finite value. > >> This completes the proof of the corollary.>> > > >> The setting is x_0 = Re(rho_0), zeta(rho_0) = 0, and rho_0 not on the > >> axis of real numbers. Also, Re(rho_0)> 1/2 , by assumption. > > >> So 1/2< x_0<= 1. > > >> The LHS = RHS equality in Proposition C assumes x> 1. > > >> Are you saying that > >> LHS = RHS in Proposition C also when x> x_0 ? > > >> I don't know why ( LHS = RHS ) would also hold in Proposition C > >> for x_0< x<= 1 ... > > > First of all, explanation for this > > starts from the paragraph which > > contains the equation (33), down > > to the end of that section. > > > A theoretical background for this comes from > > a theorem concerning analyticity of an integral function. > > > I do not think I could recite it here correctly, but > > you can find it in, say, Lang's book "Complex Analysis" > > (the GTM series). > > > Suppose that f = f(v, w) is a function > > > f: R times D -> C > > > C is the complex plane; > > D is a region in C; > > R is the real line. > > > If for each fixed v, f is an analytic function of w in D, > > and f satisfies > > > \int_{-\infty}^{\infty} |f(v, w)| dv< infty > > > then, the integral function > > > F(w) := \int_{-\infty}^{\infty} f(v, w) dv > > > is an analytic function of w. > > > The rest is just analytic continuation; that is, simply put, > > > if expressions of a relation are meaningful and > > both functions are analytic in some wider region, > > then you can extend that relation to that wider region > > on which that relation is meaningful. > > [...] > > Thanks for your explanations. > > On page 10, in the Corollary 1, you cite in (b) > a result from Titchmarsh's 1951 edition of > "The Theory of the Riemann Zeta-Function". > > log zeta(x+it) = [ sum_{ |t-gamma| <= 1} log(s - rho) ] + O(log(t)), > > uniformly for -1 <= x <= 2, where gamma denotes the imaginary > parts of non-trivial zeros rho. > > I think we're assuming that s := x+it ... > Shouldn't there be a condition on t = Im(s), because for example > if s is a non-trivial zero of zeta on the critical line > Re(s) = 1/2 , then zeta(1/2 + it) = 0, so what does > > log zeta(x+it) mean? As you point out, it is very important to consider this matter, if we deal with log of zeta. But we are actually dealing with exp(log zeta(x + it)), which appears in the integral; recall that exp of log is uniquely determined, while complex log is, as you warn me, multi-valued. [Review: For a complex z = r exp(it) (r > 0), complex log z is log z = log r + it, t satisfies, say -pi < t < pi. What makes log z a multi-valued function is that z can be written as z = r exp(it) = r exp(i(t + 2npi)), n any integer. Taking the log of z would then give log z = log r + i(t + 2pi n), n dependning on the Riemann surface on which z lies. However, taking the exp of this would give z = r exp(i(t + 2pi n)) = r exp(it); the "2pi n" part, of course, goes away. As a conclusion, if we could find any region R (probably one near the interval (0, 1) on which we know zeta is nonzero) on which the relation log (zeta(x + it)^(-1)) = ... is valid, then taking the exp of that relation would give us another multi-valuedness-free relation 1/zeta(x + it) = exp[...] which is also valid in R. Meromorphic continuation of that relation into the so-called critical zone would give us what we want. ] > > I'd appreciate having the name of the Chapter in Titchmarsh [1951] > and/or Section number and mathematicians who discovered this > estimate. > > Thanks, > > David Bernier- Hide quoted text - > > - Show quoted text - |