Geometry.Net - Theorems_And_Conjectures: Riemann Hypothesis

MathUniverse Entry - 3 July 2002: Riemann Hypothesis Conference
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Extractions: Previous entry: "Let there be light..." Main Index Next entry: "Markov Models and Money" 3 July 2002 media : Riemann Hypothesis Conference Bruce Schechter, in his article 143-Year-Old Problem Still Has Mathematicians Guessing reports on a conference at the Courant Institute on the Riemann Hypothesis. The article is actually more on the history of the problem and also gives one a little idea what the Riemann Hypothesis is about. Mr. Schechter does a rather good job with the subject, managing to indicate the main thrust of the hypothesis, which happens to link number theory and complex analysis in a beautiful alliance. If you are the type that can handle heavy-duty math, then check out MathWorld's item on the Riemann Hypothesis (Warning: this takes a graduate level understanding of complex analysis to decrypt. And even then, don't be surprised if you can't read anything that dense.) I find it interesting that some of the greatest theorems or conjectures in mathematics link "pure" and "applied" math. I do realize that many outside of mathematics have the idea that applied math is what engineers do, and pure math involves proofs, lots of variables, and sometimes nary an actual numeral in sight. Amongst mathematicians, particular fields are considered are applied: discrete math (the math computer science people do), probability, analysis, differential geometry, and that's all I can think of off the top of my head. But a great many of these "applied" people are working within the framework of of definition, theorem, proof; and how pure can "pure" math be, if number theory is being used to create and crack important cryptographic systems?
The Riemann Hypothesis
The riemann hypothesis (the zeros of the zeta function have real part 1/2). Summary In1901 von Koch showed that the riemann hypothesis is equivalent to
http://www.tamu-commerce.edu/coas/math/FACULTY/GRADS/denney/proofs/riemann.htm

Extractions: (the zeros of the zeta function have real part 1/2) Summary: When studying the distribution of prime numbers Riemann extended Euler's zeta function (defined just for s with real part greater than one) to the entire complex plane ( sans simple pole at s = 1). Riemann noted that his zeta function had trivial zeros at -2, -4, -6, ... and that all nontrivial zeros were symmetric about the line Re( s The Riemann hypothesis is that all nontrivial zeros are on this line. In 1901 von Koch showed that the Riemann hypothesis is equivalent to:
Solid Model Of Zeta Function Constructed For Studying The Riemann Hypothesis
Larry Carter Solid Model of Zeta Function Constructed for Studyingthe riemann hypothesis. SDSC senior fellow and UCSD computer
http://www.sdsc.edu/SDSCwire/v1.7/2017.Carter.html

Extractions: SDSC senior fellow and UCSD computer scientist Larry Carter applied rapid prototyping technology to study the zeta function and the Riemann hypothesis. The Riemann hypothesis has been called the most important unsolved problem of mathematics. The Riemann zeta function is defined for all complex numbers z as: zeta( z) = The zeta function plays an intricate role in number theory. For instance, the zeros of zeta are intimately connected to variations in the distribution of prime numbers. The Riemann hypothesis states that whenever zeta( z ) = 0, then z must be on the critical line iy , where i is the square root of -1 This hypothesis is related to the difficulty of testing whether a number is a primein fact, there is an algorithm that is known to be fast only under the assumption that a certain generalization of the Riemann hypothesis is true. Years ago, determining whether a number is prime was mainly a pastime for mathematicians; however, today, very large prime numbers (with 50 or 100 digits) are necessary ingredients for certain public-key cryptography systems. Carter and SDSC scientific visualization researcher Mike Bailey constructed a three-dimensional model of the zeta function with SDSC's Laminated Object Machine (LOM)a machine that uses rapid prototyping (RP) technology to construct solid models from geometric data. The model plots the absolute value of zeta(
Random Matrix Theory( Riemann Hypothesis)
Random Matrix Theory( riemann hypothesis). Follow Ups Post Followup Relativity XV FAQ Posted by sol on January 28, 2003 at 205005
http://superstringtheory.com/forum/relboard/messages15/185.html

Extractions: String Theory Discussion Forum String Theory Home Forum Index Random Matrix Theory( Riemann Hypothesis) Follow Ups Post Followup Relativity XV FAQ Posted by sol on January 28, 2003 at 20:50:05: In Reply to: Re: The big bang, and its topology posted by Haelfix on January 28, 2003 at 07:42:23: Haelfix, Just a couple of questions here. If no metric is defined for a system, then we are in a branch of mathematics known as topology, which deals with how regions of space are connected to one another. The Story of Mathematics , by Richard Mankiewicz,Pg 131 What strings seeks to illustrate, has been a troubling issue in terms of what Berhard Riemann had to offer in terms of his Hypothesis? Now what do we find that brings value to what is inherent in the identification of what the KK tower( Kaluza Klein Tower) brings to understanding in terms of those energies, and what we find is a move beyond the four dimensional aspect of spacetime, and to interpretations here, on those Eigen state determination of the Quantum systems? I draw your attention back to a brief discussion on the 5th Solvay meetings and the congregation of the post I have written here in links.
Riemann Hypothesis Re: Friendly AI
riemann hypothesis Re Friendly AI. rbfar@ebuilt.com wrote What's the ReimannHypothesis BTW, my bad for mispelling Riemann in the original subject line.
http://www.xent.com/pipermail/fork/2001-May/000233.html

Extractions: Sun, 27 May 2001 02:17:59 -0500 rbfar@ebuilt.com What's the Reimann Hypothesis http://www.claymath.org/prizeproblems/riemann.htm http://www.claymath.org/prizeproblems/riemann.pdf Previous message: Reimann Hypothesis Re: Friendly AI Next message: Reimann Hypothesis Messages sorted by: [ date ] [ thread ] [ subject ] [ author ]
Citation
ACM symposium on Theory of computing toc 1985 , Providence, Rhode Island, UnitedStates riemann hypothesis and finding roots over finite fields Author MD A
http://portal.acm.org/citation.cfm?id=22159&dl=ACM&coll=portal&CFID=11111111&CFT
Archimedes Plutonium
Two proofs riemann hypothesis. by Archimedes Plutonium this is a return Thereforethe riemann hypothesis is proved. QED My second proof
http://www.newphys.se/elektromagnum/physics/LudwigPlutonium/File125.html
The Generalized Riemann Hypothesis And The Birch And Swinnerton
x TO 1/x MOD p. Does the map sending x to 1/x mod p behave like arandom signed involution? Let L(p) be the length of the largest.
http://www.math.nyu.edu/Courses/V63.0393/projects/modp/modp.htm

Extractions: x TO 1/x MOD p Does the map sending x to 1/x mod p behave like a random signed involution? Let L(p) be the length of the largest Jon Bober investigated the distribution of L(p) (appropriately rescaled), and compared the results to the Tracy-Widom distributions; it is known that if one considers all permutations, as well as some subsets, the distribution of the longest increasing subsequence is given by the Tracy-Widom distributions. This research was inspired by a problem posed by Jim Propp. PAPER: inversemod.ps inversemod.pdf inversemod.dvi inversemod.latex FIGURES: meannew3.ps normalized.ps stdev3.ps tw.ps ... unnormal.ps CODE: ntlsubseq.cpp readdata.cpp HANDOUTS: Handout on Randomness of x to 1/x mod p
The Generalized Riemann Hypothesis And The Birch And Swinnerton
GERMAIN PRIMES. A Germain prime is a prime p such that p and 2p+1 areprime. HardyLittlewood conjectured the number of Germain. primes
http://www.math.nyu.edu/Courses/V63.0393/projects/germainprimes/germain.htm

Extractions: GERMAIN PRIMES A Germain prime is a prime p such that p and 2p+1 are prime. Hardy-Littlewood conjectured the number of Germain as comparing the distribution of spacings between Germain primes to a Poisson process. PAPERS: germain.dvi germain.pdf germain.ps germain.tex ... refs.bib.txt FIGURES: 1.ps 2.ps 3.ps 4.ps ... conv.ps PROGRAM: germain.c HANDOUTS: Poissonian Behavior Circle Method
Ask Jeeves: Search Results For "Riemann Hypothesis"
Popular Web Sites for riemann hypothesis . Search Results 1 10Ranked by Popularity, Next . 1. The riemann hypothesis Spectral
http://webster.directhit.com/webster/search.aspx?qry=Riemann Hypothesis
StudyWorks! Online : Eight Million Dollar Problems
On the other hand, the riemann hypothesis (one of the Clay Instituteproblems) has been around only since 1859. Many people have
http://www.studyworksonline.com/cda/content/article/0,,NAV4-44_SAR193,00.shtml

Extractions: StudyWorks News Science News Environmental News Math News ... Earth Observatory The new Clay Mathematics Institute has made a stunning offer: if you solve seven difficult mathematics problems, you will receive seven million dollars. As with any offer that sounds too good to be true, there is a catch. The seven problems chosen by the Institute are suspected to be among the most intractable of all unsolved mysteries ever conceived by the human mind. Add to the jackpot an offer by two publishers, Bloomsbury (in America) and Faber and Faber (in Great Britian). They have announced a one million dollar prize for the first person to prove Goldbach's Conjecture by March, 2002. Your best chance for collecting some cash would be to focus on just one problem, rather than all eight. The odds of any one person solving two such problems are vanishingly small. You would then be eligible for just one million dollars, of course. You would also be wise to forget about trying to prove Goldbach's Conjecture: the two-year time limit makes this infeasible, especially considering that the problem has baffled mathematicians since 1742! On the other hand, the Riemann Hypothesis (one of the Clay Institute problems) has been around only since 1859. Many people have devoted their lives to trying to prove this, but all have failed. There are many ways of stating the Riemann Hypothesis. Some are profound and complex; others are elementary and seemingly silly. Here is an example of the latter.
Leuschke.org Archives :: February 10, 2002 - February 16, 2002
The riemann hypothesis. To answer Juliet's first question, though, Yes, the RiemannHypothesis is perhaps the greatest remaining unsolved problem in pure math.
http://www.leuschke.org/log/archives/week_2002_02_10.html

Extractions: @import "http://www.leuschke.org/log/logstyle.css"; I signed up for Peer-to-Peer Weblog Review . I don't generally get excited about these memes (Blogger Insider, for instance, leaves me cold), but this might be fun. Go check it out. comments [0] link I've been looking for something like this: a free digital camera enhancer . DCE automagically reduces noise, adjusts balance, and smooths images. Find the right settings for your camera and let it do its thing. Very cool. comments [0] link Beauty is truth, truth beauty, that is all Ye know on earth, and all ye need to know. Let's talk about Juliet's second question : "just how important are the concepts of "beauty" and "elegance" in mathematics?" Let's kick it off with a few more quotations My work has always tried to unite the true with the beautiful and when I had to choose one or the other, I usually chose the beautiful. Hermann Weyl, (1885 - 1955) The world of ideas which it [mathematics] discloses or illuminates, the contemplation of divine beauty and order which it induces, the harmonious connexion of its parts, the infinite hierarchy and absolute evidence of the truths with which it is concerned, these, and such like, are the surest grounds of the title of mathematics to human regard, and would remain unimpeached and unimpaired were the plan of the universe unrolled like a map at our feet, and the mind of man qualified to take in the whole scheme of creation at a glance. J.J. Sylvester Presidential Address to British Association, 1869.
Riemann Hypothesis - Acapedia - Free Knowledge, For All
Friends of Acapedia riemann hypothesis. From Wikipedia, the free encyclopedia. Mostmathematicians believe the riemann hypothesis to be true.
http://acapedia.org/aca/Riemann_hypothesis
Telecom.html
Fujii, A. ``A remark on the riemann hypothesis. Comment. Math. Univ. St. A remarkon the riemann hypothesis. Proc. Japan Acad., Ser. A 57 (1981), 326330 .
http://www.math.jussieu.fr/~miw/telecom/biblio-Amoroso.html
VERGLAS.ORG RIEMANN HYPOTHESIS
The summary for this Chinese (Traditional) page contains characters that cannot be correctly displayed in this language/character set.
http://www.verglas.org/blackIce/riemann.html
Citations: Riemann Hypothesis And Finding Roots Over Finite Fields - Huang (Rese
MingDeh A. Huang. riemann hypothesis and finding roots over finite fields. Ming-DehA. Huang. riemann hypothesis and finding roots over finite fields.
http://citeseer.nj.nec.com/context/774338/0

Extractions: This paper is cited in the following contexts: Open Problems in Number Theoretic Complexity, II - Adleman, McCurley (4 citations) (Correct) ....It is also known that the least quadratic nonresidue is almost always small [Erd61] so C12 can be solved in deterministic polynomial time for almost all inputs. Rem 12 94 On the problem of calculating kth power non residues in GF(p n ) the following is known. On ERH, the algorithm of Huang , generalized by Evdokimov [Evd89] constructs a kth power nonresidue, in GF(p n ) in deterministic time (kn log p) O(1) Buchmann and Shoup [BS91] on ERH, construct a kth power non residue in GF(p n ) in deterministic time (log p) O(n) Bach [Bac90] on ERH, has given explicit bounds .
[quant-ph/9707036] Lorentz-Invariant Hamiltonian And Riemann Hypothesis
LorentzInvariant Hamiltonian and riemann hypothesis. Author Susumu OkuboComments 13 pages, TEX Report-no UR-1502 Journal-ref J.Phys.
http://arxiv.org/abs/quant-ph/9707036

Extractions: We have given some arguments that a two-dimensional Lorentz-invariant Hamiltonian may be relevant to the Riemann hypothesis concerning zero points of the Riemann zeta function. Some eigenfunction of the Hamiltonian corresponding to infinite-dimensional representation of the Lorentz group have many interesting properties. Especially, a relationship exists between the zero zeta function condition and the absence of trivial representations in the wave function. References and citations for this submission:
The Riemann Hypothesis
The riemann hypothesis. home / millennium prize problems / the riemann hypothesis. Theriemann hypothesis asserts that all interesting solutions of the equation.
http://elib.zib.de/pub/Misc/MillenniumPrizeProblems/www.claymath.org/prize_probl

Extractions: Clay Mathematics Institute [home] [index] Annual Meeting Researchers ... Other home millennium prize problems / the riemann hypothesis Some numbers have the special property that they cannot be expressed as the product of two smaller numbers, e.g., 2, 3, 5, 7, etc. Such numbers are called prime numbers, and they play an important role, both in pure mathematics and its applications. The distribution of such prime numbers among all natural numbers does not follow any regular pattern, however the German mathematician G.F.B. Riemann (1826 1866) observed that the frequency of prime numbers is very closely related to the behavior of an elaborate function z (s) called the Riemann Zeta function . The Riemann hypothesis asserts that all interesting solutions of the equation z (s) = lie on a straight line. This has been checked for the first 1,500,000,000 solutions. A proof that it is true for every interesting solution would shed light on many of the mysteries surrounding the distribution of prime numbers. Mathematical Description authored by Enrico Bombieri (PDF files are viewed with Adobe's Acrobat Reader [home] [index]
Time.9910: [time 942] Riemann Hypothesis In Sharpened Form
time 942 riemann hypothesis in sharpened form. Matti Pitkanen (matpitka@pcu.helsinki.fi)Mon, 18 Oct 1999 104833 +0300 (EET DST).
http://kims.ms.u-tokyo.ac.jp/time/199910/0102.html
Time.9911: [time 969] Sharpened Form Of Riemann Hypothesis And
time 969 Sharpened form of riemann hypothesis and TGD. Matti Pitkanen(matpitka@pcu.helsinki.fi) Wed, 3 Nov 1999 150409 +0200 (EET).
http://kims.ms.u-tokyo.ac.jp/time/199911/0005.html

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