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| Riemann hypothesis) that remain unproved to this day.Mascheroni constant (Whittaker and Watson 1990, p.Zeros of come in (at least)
two different types.Spanier and Oldham 1987).Hardy and Wright
1979, p.Maclaurin summation formula (Edwards
2001, pp.Kinkelin constant (given in series form by Glaisher
1894).Euler product formula (Hardy 1999, p.Sloane's A059750) can be expressed in terms of known mathematical constants.This constant appears,
for example, in Knuth's series.Borwein and Bradley 1997, 1997; Bailey et al.Mascheroni constant (Havil 2003, pp.Tyler and Chernhoff 1985; Boros and Moll 2004, p.Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical
Tables, 9th printing.Approaching
the Riemann Hypothesis with Mathematica.Another Elementary Proof of Euler's Formula for .Orlando, FL:
Academic Press, pp.Euler and the Zeta Function.Experimental Mathematics in Action.Wellesley, MA: A K
Peters, 2007.Beurling Equivalent Form for the Riemann
Hypothesis.Ramanujan's Notebooks, Part II.Probability Laws Related to the Jacobi Theta and Riemann Zeta Functions, and Brownian Excursions.Irresistible Integrals: Symbolics, Analysis and Experiments in
the Evaluation of Integrals.Mathematics by Experiment: Plausible Reasoning in the 21st Century.Computational Strategies for the Riemann Zeta Function.Function and Its Derivatives.Solution to Problem 1801.Multiplicative Number Theory, 2nd ed.Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem
in Mathematics.New York: Penguin, 2004.Ten Physical Applications of Spectral Zeta Functions.Zeta
Regularization Techniques With Applications.On the Constant which Occurs in the Formula for .Unsolved Problems in Number Theory, 2nd ed.New York: Chelsea, 1999.An
Introduction to the Theory of Numbers, 5th ed.Aachen, Germany:
Verlag Shaker, 1995.Gamma: Exploring Euler's Constant.Addendum to: 'Euler and the Zeta Function' (Amer.Autour de la primitive de .The Riemann Zeta Function.Hawthorne, NY: De Gruyter, 1992.Handbook of Complex Variables.Identities Inspired from Ramanujan Notebooks II.Some Simple Algorithms for the Evaluations and Representations of the Riemann Zeta Function at Positive Integer Arguments.The Theory of the Riemann Zeta Function, 2nd ed.New York:
Clarendon Press, 1987.Computational Recreations in Mathematica.Inexplicable Secrets of Creation.Books about Riemann Zeta Function.Middlesex, England: Penguin Books, 1986.Some Infinite Series of Exponential and Hyperbolic Functions.On the Modulus of the Riemann Zeta Function
in the Critical Strip.The multiplicities of distinct prime factors of X are independent random variables.This does not change the fact that the moments are specified by the series itself, and are therefore undefined for large n.The Taylor series expansion of this function will not necessarily yield the moments of the distribution.The latter limit can also exist in some cases in which A does not have a density.External links
Some remarks on the Riemann zeta distribution by Allan Gut.This page was last modified 17:12, 8 November 2007.See Copyrights for details.Roll Rolling Stones Pink Floyd Beatles In Extremo Subway to Sally Oomph!Please update this rating as the article progresses, or if the rating is inaccurate.Please add to or update the comments to suggest improvements to the article.Can anyone elaborate on the "prime numbers" section?It says "this is a consequence..."Do any real mathematicians want to do a better job?The Anome 23:27, 17 Sep 2004 (UTC)
I've turned the "vigorous handwaving" into an actual proof and have made a number of further changes and additions.Analytic continuation
21 I for one welcome our Rieman zeta function wielding mathematical overlords.Gamma function reference
28 Globally convergent series
29 !!!Anon edit needs vetting
The following anonymous edit comes from an IP with a very checkered history.Wetman posted a diff; he is talking about the edit from 9:32, 8 April 2005, which Oleg reverted a few hours later anyway.No matter, the edit was fine, the product is over primes.Is there a formula out there that relates the zeta function to the perfect powers (1, 4, 8, 9, 16, 25, 27, 32, etc) in a similar way that Euler's product formula does for primes?The reason I ask is because I discovered a very simple one a while back, but I can't find information about any other such formulas.Pronunciation
Is there a wikipedia policy for where pronunciations should appear?Gauge 04:44, 2 September 2005 (UTC)
Also, pronunciations should probably be in International Phonetic Alphabet form, as English speakers could pronounce words very differently depending on the dialect.September 2005 (UTC)
The only point I could see this survey having is to determine which users are versed in some complex analysis and those who are not.Comprehensibility
Do you currently understand this article?Oleg Alexandrov 04:10, 11 September 2005 (UTC)
I understood most of it up to the point it started talking about Mellin transforms (which is the point at which I lost the will to follow all the links I didn't understand).Before that point, I think the Euler's product proofs should be moved out (but linked to), and a bit more should be said (or linked to) about the "importance of the zeros", in particular what or how path integrals relate to the prime counting function.If not, do you feel you could understand it after following its internal links?September 2005 (UTC)
No
Er, well, I found Analytic continuation and Meromorphic function absolutely incomprehensible.Er, well, this article could use improvement.To answer my own question: probably because of the Clay Institute prize.In which case, the answer is that we need a section, or maybe even a whole article, describing "why the Riemann zeta is important to mathematics", and said article would not make use of formulas at all.Personally, I would like to see some of the proofs and derivations moved to something in the style of Category:Article proofs.September 2005 (UTC)
Sort of.It has something to do with prime numbers (which I know are a knotty number theory problem) but I'm pretty sure I couldn't figure out from this article what exactly the connection is.Mother did not rescue me from this Appalachian pastoral idyll for many years, whereupon she made my bedtime tales out of graph theory.As it stands, I find this article to be incomprehensible and without merit here.Wikipedia is a general reference work.For content to be included here, it has to pass the Joe test: If you had unlimited time in which to explain it (in far more detail than here) to Joe, the Everyman; if Joe was completely cooperative, intelligent, and patient; if eventually he understood all you intend, then could he imagine any possible way in which this subject might be of interest?This article bears a disturbing resemblance to Graveler.To be sure, this is longer than Graveler, and arguably factual; but I think that by the time you get past definitions of terms, it is equally valid to say that it is very like a playing card in a mathematical game.If there is a connection to anything concrete, that needs to be shown.So, I suggest the article be downsized to a bare description of the function and merged into its parent article.It does at least have some relationship to a larger mathematical topic, does it not?Perhaps a diligent effort could actually find some application of this function to something tangible.If so, rewrite this section and move it to the introduction.September 12 (UTC)
So, removing the rhetoric, your proposal for making this article more accessible seems to be to make it smaller or to remove it from Wikipedia completely.Shall we picture the "MBA's and fools like him" holding machine guns, eating 4 or 5 rice cakes "in context", and then grazing on?Aside from psychoanalysis of this person (is he (or she) really Chinese), he has some point.Taku 03:31, 14 September 2005 (UTC)
Agree with Taku.If yes, write at least something on the talk page explaining what you wish were better.That is, be constructive.Maybe "Joe Test" should read a damn book about number theory.As for being useless, this problem is closely related to the primes.A046970 gives values for the first 60 n.But in this case its just Riemann zeta function.Question moved from article
Following question from an anon contributor moved from the Globallly convergent series section on the article page.What's that 'x' doing there?The preceding unsigned comment was added by 12.Ok, that's twice it's come up; I've changed it from x to s.The article titled 42 (number) says:
It is believed to be the third moment of the Riemann zeta function, based partially upon evidence from quantum mechanics.Preceding unsigned comment added by 74.Surprising omission
No graphic of the graph in the complex plane?The "trivial zeros" are trivial because of the functional equation.Riccardo Poli) and Bill Langdon provided a very simple proof that, for , Riemann's Zeta function can be written as where .These people tell us that this is useful formulation.So, we were wondering whether it would make sense to include it in the article.Mathematics History and Overview section (math.Hence, the answer is no.Critical Strip
Can someone add an explanation about the critical strip?This term's definition is nowhere to be found in Wikipedia.By the way, the whole article is fine and readable; everybody with a college degree will understand at least the basics.Every specialized term needs to be defined, unless people not familiar with the subject will be lost on the way.Even before the arid math proofs!Basel equality, and other zeta functions that don't involve complex numbers here?Michaelmross 14:53, 21 January 2007 (UTC)
I disagree.Nothing in the section on specific values contradicts that.Because what I'm saying about this section of the article is that it goes from something general about natural numbers that a layperson like myself can understand to something very specific concerning complex numbers and the zeta zeros.This will be my last comment on the matter, so there's no need to flame me any further.Michaelmross 20:12, 22 January 2007 (UTC)
Nobody flamed you.You contrast "something general about natural numbers" with "something very specific concerning complex numbers", but in fact it is the natural numbers that are specific and complex numbers that are general.As for "a layperson's distinction between a generic zeta function and a Riemann zeta function", I think you might find any definition of "zeta functions in general" to be somewhat more abstruse than an account of the Riemann zeta function.Incomprehensibly Technical
I have read, reread, and then reread this article.This article is so technically oriented that it is virtually impossible to comprehend for a layman.So I think you're being a bit alarmist.Certainly there are some things here that few besides mathematicians will understand, but far from everything.Still there are some unclear points.That works of course, thank you for that explanation.Gentler definition
Browsing through the talk page, it seemed to me that there had been quite a few complaints about the definition of from people not comfortable with analytic functions, or perhaps, with mathematics in general.Other articles, such as Riemann hypothesis are much better in this regard.So this is certainly something that needs to be dealt with.Incidentally, I think that in line mathematical formulas do not look very good in this case, but since it's a highly emotional issue for at least some users, I tried to preserve them.Description
For those who get lost in this Wikipedia article, I have found the following link to be the clearest description in layman's terms.Overlook1977
This is definitely a worthy link!If not, I certainly wouldn't say it's clear to either lay persons or anyone else.RZF, which I am still working on understanding.Some of us want the big picture.It is a crucial piece of information.I'm searching for this information myself so if somebody could do it it would be very appreciated.SmashManiac 20:15, 19 April 2007 (UTC)
By how it was found, do you mean history, or the actual derivation?Is that what you mean by formula?There should definitely be a place in Wikipedia where this is explained.This article needs more information on how the zeta function is connected to prime numbers.My understanding of this function and the Riemann hypothesis is that this function and prime numbers are very deeply connected, but I can not find on Wikipedia or anywhere else an explanation as to why.The "edit this page" button at the top of the article is there for a reason!Even so, it should be reworded.This page was last modified 15:12, 2 December 2007.See Copyrights for details.Michael Caine's old reso...Jones that includes pics, pictures, biography, video, related news, vital stats, commentary, and cool facts.Jones has been in the minds of all North American males.Hollywood royalty, courtesy of her marriage to Michael Douglas.After Zorro, people spoke Spanish to me for ages.Welsh but that movie instantly gave me a new ethnicity.Jones hails from Mumbles, Wales (a tiny fishing village just outside Swansea in the UK), where she was born on September 25, 1969.She looks better and has more energy than alot of 25 year olds!Catherine Zeta Jones is GORGEOUS!French women, they are so gentle and sexy.Aishwarya Rai
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