User max muller - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T14:26:39Z http://mathoverflow.net/feeds/user/5970 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/129832/some-questions-regarding-ramanujan-summation-part-i Some questions regarding Ramanujan summation -- Part I Max Muller 2013-05-06T13:15:35Z 2013-05-06T13:15:35Z <p>The <a href="http://en.wikipedia.org/wiki/Ramanujan_summation" rel="nofollow">Ramanujan Summation</a> method, is a method through which divergent series can be summed to convergent values. </p> <p>I have several questions regarding this summation method. For more info about the words used to describe divergent series summation methods, see <a href="http://en.wikipedia.org/wiki/Divergent_series" rel="nofollow">this</a> wiki article. </p> <ol> <li>Is it a regular summation method?</li> <li>Is it a linear summation method?</li> <li>Is it a stable summation method?</li> </ol> <p>More questions to come in Part II. (This <a href="http://math.stackexchange.com/questions/381322/some-questions-regarding-ramanujan-summation-part-i" rel="nofollow">question</a> was migrated from MSE.)</p> http://mathoverflow.net/questions/128743/is-there-an-algebra-for-divergent-series-summation-operators Is there an algebra for divergent series summation operators? Max Muller 2013-04-25T18:09:04Z 2013-04-25T23:16:23Z <p>Let $D$ denote a <a href="https://en.wikipedia.org/wiki/Divergent_series" rel="nofollow">divergent series</a> and let $C$ denote a <a href="https://en.wikipedia.org/wiki/Convergent_series" rel="nofollow">convergent series</a>. </p> <p>Furthermore, let $s :$ { Series } $\to$ $\mathbb{C}$ be a regular, linear divergent series operator, which is either one of these operators: </p> <p>(the hyperlinks will direct you to the wiki page of the relevant summation method, not the person who invented/discovered it) </p> <ul> <li><a href="https://en.wikipedia.org/wiki/Borel_summation" rel="nofollow"> Borel </a> summation</li> <li><a href="https://en.wikipedia.org/wiki/Abel_summation#Abel_summation" rel="nofollow">Abel</a> summation</li> <li><a href="https://en.wikipedia.org/wiki/Euler_summation" rel="nofollow">Euler</a> summation</li> <li><a href="https://en.wikipedia.org/wiki/Ces%C3%A0ro_summation" rel="nofollow">Cesàro</a> summation</li> <li><a href="https://en.wikipedia.org/wiki/Lambert_summation" rel="nofollow">Lambert</a> summation</li> <li><a href="http://en.wikipedia.org/wiki/Ramanujan_summation" rel="nofollow">Ramanujan</a> summation</li> <li>Summing the series by means of <a href="http://en.wikipedia.org/wiki/Analytic_continuation" rel="nofollow">Analytic continutation </a></li> <li>Some <a href="https://en.wikipedia.org/wiki/Regularization_%28physics%29" rel="nofollow"> Regularization </a> method</li> </ul> <p>I am wondering if there is any meaningful way to answer the following questions (Assuming $D_1 , D_2$ are summable with $s$):</p> <ol> <li>What does $s(D_1 + D_2)$ equal? Is it always equal to $s(D_2 + D_1)$ ? How does it relate to $s(D_1)$ and $s(D_2)$ ?</li> <li>What does $s(D_1 \cdot D_2)$ equal? Is it always equal to $s(D_2 \cdot D_1)$ ? How does it relate to $s(D_1)$ and $s(D_2)$ ?</li> <li>What happens when we add convergent series into the mix? And what if we're summing linear combinations of $n$ convergent and $m$ divergent series? </li> </ol> <p>Do the results differ for different summation methods, listed above? </p> <p>(This <a href="http://math.stackexchange.com/questions/372577/is-there-an-algebra-for-divergent-series-summation-operators" rel="nofollow">question</a> was migrated from MSE. I also asked a somewhat similar <a href="http://mathoverflow.net/questions/27866/summing-a-divergent-series-and-a-constant-combined" rel="nofollow">question</a> on MO once.)</p> http://mathoverflow.net/questions/127730/what-is-the-correspondence-between-combinatorial-problems-and-the-location-of-the What is the correspondence between combinatorial problems and the location of the zeroes of polynomials called? Max Muller 2013-04-16T18:30:32Z 2013-04-17T20:02:02Z <p>(From <a href="http://math.stackexchange.com/questions/363522/what-is-the-correspondence-between-combinatorial-problems-and-the-location-of-th" rel="nofollow">MSE</a>)</p> <p>In the wikipedia <a href="http://en.wikipedia.org/wiki/Gian-Carlo_Rota" rel="nofollow">article</a> on the Italian-born American mathematician and philosopher Gian-Carlo Rota, it is stated that the one combinatorial idea he would like to be remembered for </p> <blockquote> <p><em>"... is the correspondence between combinatorial problems and problems of the location of the zeroes of polynomials."</em></p> </blockquote> <p>Also, a refence [1] is given for this quote. Upon reading through the interview, though, I didn't discover any more about this correspondence, nor did I find a lot by searching for it on the web.</p> <p><strong>Question 1</strong>: What is this correspondence called? </p> <p>I am also interested in how these two (which seem to me) disparate problems in mathematics relate to one another, so:</p> <p><strong>Question 2</strong>: How does this correspondence work? Any references? </p> <p>Finally:</p> <p><strong>3 more questions</strong>: To what extent has this correspondence been developed any further since Rota's discovery? Are there any other connections between zeroes of polynomials and combinatorics? I know algebraic geometry is concerned with the study of zeroes of polynomials, so is there any connection between (algebraic) combinatorics and algebraic geometry? </p> <p>Reference: </p> <p>[1] <a href="http://web.archive.org/web/20070811172343/http://www.rota.org/hotair/rotasharp.html" rel="nofollow">http://web.archive.org/web/20070811172343/http://www.rota.org/hotair/rotasharp.html</a></p> http://mathoverflow.net/questions/24396/would-eulers-proofs-get-published-in-a-modern-math-journal-especially-consideri Would Euler's proofs get published in a modern math Journal, especially considering his treatment of the Infinite? Max Muller 2010-05-12T14:28:49Z 2013-04-10T15:59:30Z <p>I was wondering how mathematicians of today would treat, for example, Euler's proof of zeta(2). </p> <p>In William Dunham's book 'Journey through Genius' ( <a href="http://www.amazon.com/Journey-through-Genius-Theorems-Mathematics/dp/014014739X" rel="nofollow">http://www.amazon.com/Journey-through-Genius-Theorems-Mathematics/dp/014014739X</a> ), the writer states (more or less) that most mathematicians of today wouldn't aprove of Euler's methods, as his treatment of the 'infinite' doesn't uphold to today's modern standards of rigour.</p> <p>His evaluation of Zeta(2) and all other even zeta-arguments op to Zeta(26) was correct, however. Would Euler's proofs get published in a well-respected math journal?</p> <p>(Of course, his papers would now be written in english and the would-be published results aren't known to the mathematical community, yet).</p> <p>Thanks in advance,</p> <p>Max Muller</p> <p>PS: O.K. everyone, I think many of you have stressed some important points regarding this question. I can't choose one, which is why I have upvoted some of your answers and left the question as it is. Thank you for your thoughts.</p> <p>PPS: I'm sorry for the confusing title of the question in its previous form. I hope you all think it is stated better now.</p> http://mathoverflow.net/questions/28158/a-learning-roadmap-request-from-high-school-to-mid-undergraduate-studies A Learning Roadmap request: From high-school to mid-undergraduate studies Max Muller 2010-06-14T20:18:26Z 2012-10-22T23:18:12Z <p>Dear MathOverflow community,</p> <p>In about a year, I think I will be starting my undergraduate studies at a Dutch university. I have decided to study mathematics. I'm not really sure why, but I'm fascinated with this subject. I think William Dunham's book '<a href="http://www.amazon.com/Journey-through-Genius-Theorems-Mathematics/dp/014014739X/ref=sr_1_1?ie=UTF8&amp;s=books&amp;qid=1276544297&amp;sr=8-1" rel="nofollow">Journey through Genius</a> ' has launched this endless fascination. </p> <p>I can't wait another whole year, however, following the regular school-curriculum and not learning anything like the things Dunham describes in his book. Our mathematics-book at school is a very 'calculus-orientated' one, I think. I don't think it's 'boring', but it's not a lot of fun either, compared to the evalutation of $\zeta(2)$, for example. Which is why I took up a 'job' as as a tutor for younger children to help them pass examinations. I wanted to make money (I've gathered about 300 euros so far) to buy some new math-books. I have already decided to buy the book '<a href="http://www.amazon.com/Introductory-Mathematics-Analysis-Springer-Undergraduate/dp/3540761780/ref=sr_1_1?ie=UTF8&amp;s=books&amp;qid=1276545109&amp;sr=8-1" rel="nofollow"> Introductory Mathematics: Algebra and Analysis</a>' which should provide me with some knowledge on the basics of Linear Algebra, Algebra, Set Theory and Sequences and Series. But what should I read next? What books should I buy with this amount of money in order to acquire a firm mathematical basis? And in what order? (The money isn't that much of a problem, though, I think my father will provide me with some extra money if I can convince him it's a really good book). Should I buy separate books on Linear Algebra, Algebra and a calculus book, like most university web-pages suggest their future students to buy?</p> <p>Notice that it's important for me that the books are self-contained, i.e. they should be good self-study books. I don't mind problems in the books, either, as long as the books contain (at least a reasonable portion) of the answers (or a website where I can look some answers up).</p> <p>I'm not asking for the <em>quickest</em> way to be able to acquire mathematical knowledge at (graduate)-university level, but the <em>best</em> way, as Terence Tao once commented (on his blog): "Mathematics is not a sprint, but a marathon".</p> <p>Last but not least I'd like to add that I'm especially interested in infinite series. A lot of people have recommended me Hardy's book '<em>Divergent Series</em>' (because of the questions I ask) but I don't think I posess the necessary prerequisite knowledge to be able to understand its content. I'd like to understand it, however!</p> http://mathoverflow.net/questions/27299/on-starting-graduate-school-and-common-pitfalls/98397#98397 Answer by Max Muller for On starting graduate school and common pitfalls... Max Muller 2012-05-30T16:25:40Z 2012-05-30T16:25:40Z <p>I think you should take a look at <em>"A Mathematician's Survival Guide: Graduate School and Early Career Development"</em>, by Steven G. Krantz. The customer reviews on the amazon <a href="http://www.amazon.com/Mathematicians-Survival-Guide-Graduate-Development/dp/082183455X/ref=sr_1_1?ie=UTF8&amp;qid=1338394682&amp;sr=8-1" rel="nofollow">page</a> are all highly laudatory. I did not read the book myself (since I'm just a student in my first year), but based on the enthusiastic responses I suppose you will be rewarded if you read it. </p> http://mathoverflow.net/questions/64898/values-of-the-riemann-zeta-function-and-the-ramanujan-summation-how-strong-is-t Values of the Riemann zeta function and the Ramanujan summation - How strong is the connection? Max Muller 2011-05-13T13:52:40Z 2011-08-13T09:42:57Z <p><em>(This Question was taken from <a href="http://math.stackexchange.com/questions/38731/values-of-the-riemann-zeta-function-and-the-ramanujan-summation-how-strong-is-t" rel="nofollow">MSE</a>. As Eric Naslund pointed out there, this <a href="http://mathoverflow.net/questions/3204/does-any-method-of-summing-divergent-series-work-on-the-harmonic-series" rel="nofollow">question</a> is relevant. The summation method mentioned in this question is actually a good answer to it.)</em></p> <p>The <a href="http://en.wikipedia.org/wiki/Ramanujan_summation" rel="nofollow">Ramanujan Summation</a> of some infinite sums is consistent with a couple of sets of values of the Riemann zeta function. We have, for instance, $$\zeta(-2n)=\sum_{n=1}^{\infty} n^{2k} = 0 (\mathfrak{R})$$ (for non-negative integer $k$) and $$\zeta(-(2n+1))=-\frac{B_{2k}}{2k} (\mathfrak{R})$$ (again, $k \in \mathbb{N}$). Here, $B_k$ is the $k$'th <a href="http://en.wikipedia.org/wiki/Bernoulli_number" rel="nofollow">Bernoulli number</a>. However, it does not hold when, for example, $$\sum_{n=1}^{\infty} \frac{1}{n}=\gamma (\mathfrak{R})$$ (here $\gamma$ denotes the Euler-Mascheroni Constant) as it is not equal to $$\zeta(1)=\infty$$. Question: Are the first two examples I stated the only instances in which the ramanujan summation of some infinite series coincides with the values of the Riemann zeta function?</p> http://mathoverflow.net/questions/71822/questions-regarding-second-and-higher-order-undecidability Questions regarding "second and higher-order-undecidability" Max Muller 2011-08-01T19:50:16Z 2011-08-02T12:42:29Z <p><em>I have moved this question here from MSE, because I did not receive any answers as of yet over there.</em></p> <p>I know that there are statements that are neither provable nor disprovable within some set of axioms, and I also know that such statements are called undecidable. Please allow me to call these statements to be undecidable to the first order. These statements belong to $U_1$.</p> <p>I was wondering if there is some kind of generalization of this concept. </p> <blockquote> <p>Question 1. Are there any conjectures/statements of which we can prove that we cannot prove whether it is decidable or not? Related: how would such a proof look like?</p> </blockquote> <p>Such a statement/conjecture would be undecidable to the second order, or belong to $U_2$. Generalizing even further (Question 2):</p> <blockquote> <p>What about statements that are in $U_\infty$ ?</p> </blockquote> http://mathoverflow.net/questions/64905/which-book-would-you-like-to-see-texified Which book would you like to see "texified"? Max Muller 2011-05-13T16:39:35Z 2011-06-28T13:47:02Z <p>Let's see if we could use MO to put some pressure on certain publishers...</p> <p>Although it is wonderful that it has been put <a href="http://www.jmilne.org/math/Books/DMOS.pdf" rel="nofollow">online</a>, I think it would make an even greater read if "Hodge Cycles, Motives and Shimura Varieties" by Deligne, Milne, Ogus and Shih would be (re)written in the latex typesetting (well, if I could understand its content..).</p> <p>But enough about my opinion, what do you think? Which book(s) would you like to see "texified"? As customary in a CW question, one book per answer please.</p> http://mathoverflow.net/questions/13847/research-experience-for-undergraduates-summer-programs/51352#51352 Answer by Max Muller for Research Experience for Undergraduates Summer Programs Max Muller 2011-01-06T22:52:03Z 2011-01-07T23:18:21Z <p>Being interested in following such a program as well, I posted a similar <a href="http://math.stackexchange.com/questions/16254/summer-programs-to-gain-research-experience-as-an-undergraduate-outside-the-unit" rel="nofollow">question</a> on Math SE. I recently found the <a href="http://www.math.wisc.edu/~ono/reu10.html" rel="nofollow">program</a> held at the university of Wisconsin [which is moving to <a href="http://www.mathcs.emory.edu/~ono/REUs/" rel="nofollow">Emory</a>, but is still being run by Ken Ono]. It specializes in number theory. As it is funded by the NSF as well, foreigners have to pay for the program themselves, but it <em>is</em> open to non-Americans. </p> <p>Good luck with finding a program!</p> http://mathoverflow.net/questions/47011/uses-of-divergent-series-and-their-summation-values-in-mathematics Uses of Divergent Series and their summation-values in mathematics ? Max Muller 2010-11-22T22:27:33Z 2011-01-06T22:14:56Z <p>This question was posed originally on <a href="http://math.stackexchange.com/questions/10990/uses-of-divergent-series-and-their-summation-values-in-mathematics" rel="nofollow">MSE</a>, I put it here because I didn't receive the answer(s) I wished to see.</p> <p>Dear MO-Community,</p> <p>When I was trying to find closed-form representations for odd zeta-values, I used $$\Gamma(z) = \frac{e^{-\gamma \cdot z}}{z} \prod_{n=1}^{\infty} \Big( 1 + \frac{z}{n} \Big)^{-1} e^{\frac{z}{n}}$$ and rearranged it to $$\frac{\Gamma(z)}{e^{-\gamma \cdot z}} = \prod_{n=1}^{\infty} \Big( 1 + \frac{z}{n} \Big)^{-1} e^{z/n}.$$ As we know that $$\prod_{n=1}^{\infty} e^{z/n} = e^{z + z/2 + z/3 + \cdots + z/n} = e^{\zeta(1) z},$$ we can state that $$\prod_{n=1}^{\infty} \Big( 1 + \frac{z}{n} \Big) = \frac{e^{z(\zeta(1) - \gamma)}}{z\Gamma(z)}\qquad\text{(1)}$$ I then stumbled upon the Wikipedia page of <a href="http://en.wikipedia.org/wiki/Ramanujan_summation" rel="nofollow">Ramanujan Summation</a> (see the bottom of the page), which I used to set $\zeta(1) = \gamma$ (which was, admittedly, a rather dangerous move. Remarkably, things went well eventually. Please don't stop reading). The $z^3$ -coefficient of both sides can now be obtained. Consider \begin{align*} (1-ax)(1-bx) &amp;= 1 - (a+b)x + abx^2\\ &amp;= 1-(a+b)x + (1/2)((a+b)^2-(a^2+b^2)) \end{align*} and \begin{align*}(1-ax)(1-bx)(1-cx) &amp;= 1 - (a + b + c)x\\ &amp;\qquad + (1/2)\Bigl((a + b + c)^2 - (a^2 + b^2 + c^2)\Bigr)x^2\\ &amp;\qquad -(abc)x^3. \end{align*}</p> <p>We can also set \begin{align*} (abc)x^3 &amp;= (1/3)\Bigl((a^3 + b^3 + c^3) - (a + b + c)\Bigr)\\ &amp;\qquad + (1/2)(a + b + c)^3 -(a + b + c)(a^2 + b^2 + c^2). \end{align*} It can be proved by induction that the x^3 term of $(1-ax)(1-bx)\cdots(1-nx)$ is equal to \begin{align*} (1/3)&amp;\Bigl((a^3 + b^3 + c^3 +\cdots + n^3) - (a + b + c + \cdots + n)^3\Big)\\ &amp;\qquad+ (1/2)\Big((a + b + c + \cdots + n)^3 -(a + b + c + \cdots + n)(a^2 + b^2 + c^2 + \cdots + n^2)\Big).\qquad\text{(2)} \end{align*} On the right side of equation (1), the $z^3$-term can be found by looking at the $z^3$ term of the Taylor expansion series of $1/(z \Gamma(z))$, which is $(1/3)\zeta(3) + (1/2)\zeta(2) + (1/6)\gamma^3$. We then use (2) to obtain the equality $$(1/6)\gamma^3 - (1/2)\gamma \pi^2 - (1/6) \psi^{(2)}(1) = 1/3)\zeta(3) + (1/2)\zeta(2) + (1/6)\gamma^3$$ to find that $$\zeta(3) = - (1/2) \psi^{(2)} (1),$$ (3) which is a true result that has been known (<em>known</em> should be a hyperlink but it isn't for some reason) for quite a long time. The important thing here is that I used $\zeta(1) = \gamma$, which isn't really true. Ramanujan assigned a summation value to the harmonic series (again, see Ramanujan Summation), and apparently it can be used to verify results and perhaps to prove other conjectures/solve problems. </p> <p>My first question is: Is this a legitimate way to prove (3) ? </p> <p>Generalizing this question: </p> <p>When and how are divergent series and their summation values used in mathematics? What are the 'rules' when dealing with summed divergent series and using them to (try to) find new results?</p> <p>Thanks,</p> <p>Max</p> <p>As I suspect someone (I was thinking of Qiaochu Yuan himself) will add this too, I will add this <a href="http://mathoverflow.net/questions/3204/does-any-method-of-summing-divergent-series-work-on-the-harmonic-series" rel="nofollow">question</a> for him/her, as it is somewhat related. </p> http://mathoverflow.net/questions/26035/evaluation-of-the-following-series-s-1-2-times3-1-5-times6-1-7-tim Evaluation of the following series... $S = 1/(2\times3) + 1/(5\times6) + 1/(7\times8) + 1/(10\times11) + ...$ Max Muller 2010-05-26T16:10:02Z 2010-12-09T04:33:27Z <p>EDIT, Will Jagy, December 8, 2010: to anyone considering working on this, please first see <a href="http://meta.mathoverflow.net/discussion/817/could-a-few-moderators-please-remove-one-of-my-questions/#Item_9" rel="nofollow">http://meta.mathoverflow.net/discussion/817/could-a-few-moderators-please-remove-one-of-my-questions/#Item_9</a> which gives the story behind this peculiar sum. Note that the OP is no longer interested in the results, as they arose from one kind of error and cannot be applied because of a different sort of misunderstanding. The double sum version below was provided recently by Harald Hanche-Olsen.</p> <p>ORIGINAL. I'm curious one of you is able to find the exact evaluation of the following series: </p> <p>\begin{aligned} S &amp;= 1/(2\times3) +1/(5\times6) + 1/(7\times8) + 1/(10\times11) + \cdots \\&amp;= \sum_{n=1}^\infty\sum_{k=1}^{n}\frac1{(n^2+2k-1)(n^2+2k)} \end{aligned} </p> <p>I'm not exactly sure on how to state the 'general term' of the series. Perhaps I can illustrate it with an example:</p> <p>$1/(1\times2) + 1/(3\times4) + 1/(5\times6) + 1/(7\times8) + \ldots + 1/((2n - 1) \times 2n) + \ldots = \log(2)$. </p> <p>Now, to answer Nate Eldredge: let $a_0=2$ and $a_{k+1}=a_{k} + 1$ unless $a_{k} + 1$ is a square, in which case let $a_{k + 1} = a_{k} + 2$. Now, multiply $a_{k}$ with $a_{k+1}$. That's a term. Let me show the first few terms:</p> <p>$S = 1/(2\times3)$ [now skip 4] $+ 1/(5\times6) + 1/(7\times8)$ [now skip 9] $+ 1/(10\times11) + 1/(12\times13) + 1/(14\times15)$ [now skip 16] $+ 1/(17\times18) + \ldots$ </p> <p>So all the squares (1,4,9,16,25, etc) are 'skipped' in the terms.</p> <p>I hope this clarifies it a bit...</p> <p>Thanks a lot in advance,</p> <p>Max Muller</p> <p><em>PS: If someone has any ideas as to how the general term of this series can be written in a more concise manner, please let me know! For the Meta-users, see also the relevant <a href="http://meta.mathoverflow.net/discussion/817/could-a-few-moderators-please-remove-one-of-my-questions/#Item_9" rel="nofollow">discussion</a> on this question.</em></p> http://mathoverflow.net/questions/27866/summing-a-divergent-series-and-a-constant-combined Summing a divergent series and a constant combined Max Muller 2010-06-11T20:34:21Z 2010-11-23T17:29:21Z <p>At least according to the answer to <a href="http://mathoverflow.net/questions/3204/does-any-method-of-summing-divergent-series-work-on-the-harmonic-series" rel="nofollow"> this question</a>, $\zeta(1) = \gamma$ (once reqularized, of course).</p> <p>Let me rephrase that by stating that:</p> <p>$$\sigma(\zeta(1)) = \gamma$$ Here, $\sigma(x)$ is the 'summation-function'. It's a function that assigns a value to any $x$, using Borel, Abel, Ramanujan, Euler, Cesaro or any other summation method that <em>works</em> (e.g. It makes a divergent series summable). The $\sigma$-function 'chooses' a summation method that suits $x$ best (to assign a (finite) constant to it). We assume that the different summation methods dont have different 'working' values for the same $x$ (I now call upon <a href="http://mathoverflow.net/questions/19201/summation-methods-for-divergent-series" rel="nofollow"> this</a> question).</p> <p>Furthermore, we denote $C$ as a converging series and $D$ as a diverging one.</p> <p>What would $\sigma(C + D)$ be? Is it $\sigma(C) + \sigma(D)$ ? Or what would, for example, $\sigma(\zeta(1)^3 + \zeta(2))$ be? </p> <p>So, to summarize my question: Could you please explain the properties of the $\sigma$-function to me, with relation to $C$ and $D$ ?</p> <p>Thanks a lot in advance.</p> <p>P.S. A bonus question: What do you think of the 'summation-function'? is it useful or just mathematical bogus? Or has it been defined (even more) properly already?</p> http://mathoverflow.net/questions/39402/asociated-sum-series-of-the-euler-product-over-the-twin-primes Asociated sum series of the Euler Product over the Twin Primes? Max Muller 2010-09-20T16:18:45Z 2010-09-26T04:19:56Z <p>Please consider the (presumably infinite) Euler product over the twin primes:</p> <p>$$f(z) = \prod_{p\in\mathbb{P}}^{\infty} \Big( 1 - \frac{1}{(p(p+2))^ z} \Big)$$ (in which $p(p+2)$ is a divisor of $4((p-1)!+1) + p$ ).</p> <p>The Euler Product is a product of a corresponding Dirichlet series. Which one is that? </p> <p>Thanks in advance,</p> <p>Max</p> <p>Edit Update: error fixed.</p> http://mathoverflow.net/questions/27592/why-is-frac-pi212ln2-not-true Why is $\frac{\pi^2}{12}=ln(2)$ not true ? Max Muller 2010-06-09T15:50:09Z 2010-07-08T12:44:59Z <p>This question may sound ridiculous at first sight, but let me please show you all how I arrived at the afore mentioned 'identity'.</p> <p>Let us begin with (one of the many) equalities established by Euler:</p> <p>$$\displaystyle f(x) = \frac{sin(x)}{x} = \prod_{n=1}^{\infty} \Big(1-\frac{x^2}{n^2\pi^2}\Big)$$</p> <p>as $(a^2-b^2)=(a+b)(a-b)$, we can also write: (EDIT: We can not write this...)</p> <p>$$\displaystyle f(x) = \prod_{n=1}^{\infty} \Big(1+\frac{x}{n\pi}\Big) \cdot \prod_{n=1}^{\infty} \Big(1-\frac{x}{n\pi}\Big)$$ </p> <p>We now we arrange the terms with $(n = 1 \land n=-2)$, $(n = -1 \land n=2$), $( n=3 \land -4)$ , $(n=-3 \land n=4)$ , ..., $(n = 2n \land n=-2n-1)$ and $(n=-2n \land n=2n+1)$ together . After doing so, we multiply the terms accordingly to the arrangement. If we write out the products, we get:</p> <p>$$f(x)=\big((1-x/2\pi + x/\pi -x^2/2\pi^2)(1+x/2\pi-x/\pi - x^2/2\pi^2)\big)...$$ $$...\big((1-\frac{x}{(2n)\pi} + \frac{x}{(2n-1)\pi} -\frac{x^2}{(2n(n-1))^2\pi^2})(1+\frac{x}{2n\pi} -\frac{x}{(2n-1)\pi} -\frac{x^2}{(2n(2n-1))^2\pi^2)})\big)$$</p> <p>Now we equate the $x^2$-term of this infinite product, using <a href="http://en.wikipedia.org/wiki/Newton%27s_identities" rel="nofollow"> Newton's identities</a> (notice that the'$x$'-terms are eliminated) to the $x^2$-term of the Taylor-expansion series of $\frac{sin(x)}{x}$ . So,</p> <p>$$-\frac{2}{\pi^2}\Big(\frac{1}{1\cdot2} + \frac{1}{3\cdot4} + \frac{1}{5\cdot6} + ... + \frac{1}{2n(2n-1)}\Big) = -\frac{1}{6}$$</p> <p>Multiplying both sides by $-\pi^2$ and dividing by 2 yields</p> <p>$$\sum_{n=1}^{\infty} \frac{1}{2n(2n-1)} = \pi^2/12$$</p> <p>That (infinite) sum 'also' equates $ln(2)$, however (According to the last section of <a href="http://www.stat.purdue.edu/~dasgupta/publications/tr02-03.pdf" rel="nofollow"> this </a> paper).</p> <p>So we find $$\frac{\pi^2}{12} = ln(2)$$ .</p> <p>Of course we all know that this is not true (you can verify it by checking the first couple of digits). I'd like to know how much of this method, which I used to arive at this absurd conclusion, is true, where it goes wrong and how it can be improved to make it work in this and perhaps other cases (series). </p> <p>Thanks in advance, </p> <p>Max Muller</p> <p>(note I: 'ln' means 'natural logarithm) (note II: with 'to make it work' means: 'to find the exact value of)</p> http://mathoverflow.net/questions/28986/closed-form-of-divergent-infinite-product Closed form of divergent infinite product? Max Muller 2010-06-21T20:06:42Z 2010-06-21T21:21:22Z <p>Okay, we know that </p> <p>$$\frac{sin(x)}{x} = \prod_{n=1}^{\infty} \Big(1-\frac{x^2}{n^2\cdot\pi^2}\Big)$$ .</p> <p>Is there some known (trigonometric(?)) function that is equal to the following infinite product? </p> <p>$$\prod_{n=1}^{\infty} \Big(1-\frac{x}{n\cdot\pi}\Big)$$</p> <p>I'd be happy as well if someone could provide me with a function that is equal to a similar divergent infinite product (a function, for example, that is equal to 'my' inifite product, only $\pi=1$, or $x=x^2$, or something in that direction). </p> <p>Thanks in advance,</p> <p>Max Muller</p> http://mathoverflow.net/questions/24891/the-root-of-a-line The Root of a Line Max Muller 2010-05-16T15:22:53Z 2010-05-17T15:53:04Z <p>Ok, imagine having a finite line segment from point (a) to point (b) in $R^2$ . I'm not familiar with mathematical terminology of this kind, but let me state that the line we began with is the geometric interpretation of $A^1$. The geometric interpretation of $A^2$ is a square with sides $A^1$. We could go on by saying that $A^3$ is a cube with edge length $A^{1}$ again. </p> <p>I wonder what the geometric interpration of $\sqrt[2]{A}=A^{1/2}$ would look like. Is it a (straight) line? Is it constructible? Of course, we could extend this question by asking ourselves what $A^{k}$ would look like, in which $k \in \mathbb{R}$ or even $\mathbb{C}$ . When $k>2$, $A^k$ probably isn't constructible anymore on a sheet of paper, but one can still think about how these constructions of $A$ would 'look like'.</p> <p>Thanks in advance, </p> <p>Max Muller</p> <p>P.S. I realize I ask more than one question now, which is an indictation I don't know a lot about this (yet) and I'd like to know more about this subject. Should this be a community wiki? Feel free to modify the Tags, I don't know how to classify this question exactly.</p> http://mathoverflow.net/questions/24258/evaluation-of-the-following-series Evaluation of the following Series Max Muller 2010-05-11T16:38:46Z 2010-05-11T22:23:18Z <p>Hi there, </p> <p>I was wondering if you guys could be able to find the sum of the following series: </p> <p>$S = 1/((1\cdot2)^2) + 1/((3\cdot4)^2) + 1/((5\cdot6)^2) + ... + 1/(((2n-1)\cdot2n)^2)$, in which ${n\to\infty}$ . </p> <p>This question came to mind when I was looking at this (<a href="http://www.stat.purdue.edu/~dasgupta/publications/tr02-03.pdf" rel="nofollow">http://www.stat.purdue.edu/~dasgupta/publications/tr02-03.pdf</a>) paper by Professor Anirban DasGupta. In the last section, a couple of specific examples of his 'unified' method to find the sums of infinite series is pressented. In equation (34), he states that the following series:</p> <p>$1/(1\cdot2) + 1/(3\cdot4) + 1/(5\cdot6) + ... 1/(2n\cdot(2n-1)) = log(2)$ (Note that ${n\to\infty}$ again). I was wondering If it's possible to find the sum if the values of the denominators of the terms are squared.</p> <p>Thanks in advance, </p> <p>Max Muller </p> http://mathoverflow.net/questions/129229/how-to-sum-arithmetic-combinations-of-divergent-zeta-values Comment by Max Muller Max Muller 2013-05-01T11:43:09Z 2013-05-01T11:43:09Z Could the downvoter please explain as to why s/he downvoted this question? http://mathoverflow.net/questions/128743/is-there-an-algebra-for-divergent-series-summation-operators Comment by Max Muller Max Muller 2013-04-27T15:46:05Z 2013-04-27T15:46:05Z @Harry Altman: O.K. I think I know what you mean now. I will try to ask better, more focussed answers next time. http://mathoverflow.net/questions/128743/is-there-an-algebra-for-divergent-series-summation-operators/128752#128752 Comment by Max Muller Max Muller 2013-04-26T21:17:00Z 2013-04-26T21:17:00Z Thank you. I will check out the papers and sources you mentioned, they seem rather interesting. Regarding 1)2), you say that: &quot;Now if $s$ is $\mathbb{C}$-linear then $s(D1 + D2) = s(D1) + s(D2)$.&quot; What I am curious about, is whether divergent series summation operators acutually <i>are</i> $\mathbb{C}$-linear. Do you know whether this is true? And for which operators? http://mathoverflow.net/questions/128743/is-there-an-algebra-for-divergent-series-summation-operators Comment by Max Muller Max Muller 2013-04-26T20:38:44Z 2013-04-26T20:38:44Z <i>questions</i> about them with the conditions as they are now? http://mathoverflow.net/questions/128743/is-there-an-algebra-for-divergent-series-summation-operators Comment by Max Muller Max Muller 2013-04-26T20:38:13Z 2013-04-26T20:38:13Z @Harry Altman: I'm sorry, I'm not entirely sure what you mean. I have listed my conditions on top of the question. These conditions that are listed are the most common ones for divergent series in the wikipedia article on divergent series. The actual questions, which you refer to as (1), (2) and (3) (I guess), and I refer to as 1. , 2. and 3. , ask for the actual outcome of some algebraic combinations of divergent series. Do you think I should focus on different <i>conditions</i> for divergent series (as opposed to the ones listed on wikipedia), or do you think I ought to ask different... (cont'd) http://mathoverflow.net/questions/127730/what-is-the-correspondence-between-combinatorial-problems-and-the-location-of-the/127884#127884 Comment by Max Muller Max Muller 2013-04-17T20:17:01Z 2013-04-17T20:17:01Z Many thanks! It seems you have written about this subject as well: <a href="http://www-math.mit.edu/~tchow/sieves.pdf" rel="nofollow">www-math.mit.edu/~tchow/sieves.pdf</a> . http://mathoverflow.net/questions/127730/what-is-the-correspondence-between-combinatorial-problems-and-the-location-of-the Comment by Max Muller Max Muller 2013-04-16T18:41:48Z 2013-04-16T18:41:48Z @Qiaochu Yuan: Jup. You can also find the question by clicking on the &quot;MSE&quot; hyperlink at the top of the question. http://mathoverflow.net/questions/27299/on-starting-graduate-school-and-common-pitfalls/98397#98397 Comment by Max Muller Max Muller 2012-05-31T15:05:21Z 2012-05-31T15:05:21Z Hm.. upon giving this answer I didn't realize you posted this question almost two years ago. How is graduate school working out for you? Did the answers help you? If so, how? http://mathoverflow.net/questions/64898/values-of-the-riemann-zeta-function-and-the-ramanujan-summation-how-strong-is-t Comment by Max Muller Max Muller 2011-08-19T21:37:17Z 2011-08-19T21:37:17Z @Eric: Thanks. (sorry for taking so long to respond.) I guess it amounts to plugging in (1/2) in the last formula provided by andrew (below). This would mean that $\zeta(1/2) = 0.458780441... (\mathfrak{R})$. http://mathoverflow.net/questions/71822/questions-regarding-second-and-higher-order-undecidability Comment by Max Muller Max Muller 2011-08-01T21:19:04Z 2011-08-01T21:19:04Z @Pace: hmm perhaps you're right... If you are, that would render my question useless. Let's wait for an expert to come and and judge whether this question is worthwile. http://mathoverflow.net/questions/71822/questions-regarding-second-and-higher-order-undecidability Comment by Max Muller Max Muller 2011-08-01T21:06:22Z 2011-08-01T21:06:22Z @Gerhard: I'm sorry, I am afraid I am not able to do so. My understanding of set theory is not deep enough. If you (or anyone else) think you can improve upon the question by rewording and/or expanding it, please do so. http://mathoverflow.net/questions/66801/learning-roadmap-for-harmonic-analysis Comment by Max Muller Max Muller 2011-06-08T17:06:03Z 2011-06-08T17:06:03Z There is also Representation Theory and Noncommutative Harmonic Analysis I and II, by Kirillov, Soucek and Neretin. http://mathoverflow.net/questions/64905/which-book-would-you-like-to-see-texified Comment by Max Muller Max Muller 2011-05-13T19:51:40Z 2011-05-13T19:51:40Z @Andy &amp; Donu: Ok perhaps you're both right. I have to admit I didn't think my question through that thoroughly. http://mathoverflow.net/questions/64905/which-book-would-you-like-to-see-texified Comment by Max Muller Max Muller 2011-05-13T19:48:05Z 2011-05-13T19:48:05Z S/he strokes his/her beard and mumbles: &quot;Hmmm... a lot of people seem to agree with you, perhaps we can make some profit out of this. I will take your suggestion into consideration. By the way, that website of yours looks pretty good, perhaps I'll join it too.&quot; http://mathoverflow.net/questions/64905/which-book-would-you-like-to-see-texified Comment by Max Muller Max Muller 2011-05-13T19:42:51Z 2011-05-13T19:42:51Z @Andy the idea to ask this question came to me a while after I viewed the website <a href="http://outofprintmath.blogspot.com/" rel="nofollow">outofprintmath.blogspot.com</a> . I thought that maybe it could be of use as a means of negotiating with publishers, in the sense that people could say &quot;Please mister [head of publisher X], publish the book [Y] in TEX typesetting, it would be much more readable if you did.&quot; Then the head of publisher X would say: &quot;Hm... you've got a point, but I am not going to do this solely for you!&quot; Then you say: &quot;Well, a lot of people agree with me, just take a look at this MO question.&quot; (You show him/her this question.)