User kolik - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T15:30:01Z http://mathoverflow.net/feeds/user/23998 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/98343/number-of-integers-coprime-to-l Number of integers coprime to l kolik 2012-05-30T08:54:30Z 2012-05-31T17:20:41Z <p>A long time ago I've seen a paper considering, given $\ell$ fixed, estimates for</p> <p>$$\sum_{n \leq x, (n, \ell) = 1} 1$$</p> <p>Of course, this is easy to estimate with a trivial error term of $O(\varphi(l))$. However in the paper I am looking for the authors attempted obtaining better bounds, using some Fourier analysis (in particular the Fourier series for the fractional part of x). I think, bounds in the sum</p> <p>$$\sum_{n \leq x} (n, \ell)$$ are essentially an equivalent variation of the problem, so references on this problem are welcome aswell.</p> <p>The reason why I am interested in this problem is ... pure curiosity. I am curious to see how the Fourier methods meshed in, and what kind of bounds they gave, even though of course we cannot really expect anything too fantastic in this problem. </p> http://mathoverflow.net/questions/98171/what-is-the-relation-between-quasicrystals-riemann-hypothesis-and-pv-numbers What is the relation between Quasicrystals, Riemann Hypothesis, and PV numbers? kolik 2012-05-28T05:56:25Z 2012-05-28T17:22:39Z <p>Could somebody explain to me, from a mathematical stand-point, what is a quasi-crystal, and how it relates to the set of Pisot numbers, and the Riemann Hypothesis?</p> <p>I've heard Freeman Dyson say that the zeros of the Riemann zeta function form a quasi-crystal. But, a priori, I do not see what kind of property of the zeros, that we currently now of, would be able to confer to them more structure than to a random set of isolated numbers. </p> <p>(Notwithstanding the explicit formula in prime number theory)</p> <p>To wit, my second question possibly based on a misunderstanding: why is the set of zeros of $\zeta(s)$ a quasi-crystal, while a random sequence of isolated numbers is not? Of course, I first need to fully understand what is a quasi-crystal, because Freeman's definition left me in a fog. </p> http://mathoverflow.net/questions/98174/is-mertens-function-negatively-biased/98175#98175 Answer by kolik for Is Mertens function negatively biased? kolik 2012-05-28T07:18:47Z 2012-05-28T07:18:47Z <p>I believe that this paper</p> <p><a href="http://arxiv.org/abs/1108.1524" rel="nofollow">http://arxiv.org/abs/1108.1524</a></p> <p>By one of the members of MO, will have some relevant answers for the closely related case of the Liouville function.</p> http://mathoverflow.net/questions/98144/what-happens-when-infinite-values-of-zeta-hs-z-approach-zetas/98172#98172 Answer by kolik for What happens when infinite values of $\zeta_{H}(s,z)$ approach $\zeta(s)$ ? kolik 2012-05-28T06:12:42Z 2012-05-28T06:12:42Z <p>The functional equation for the Hurwitz zeta function is far from being of the type $\zeta_{H}(s) = \Phi(s) \zeta_{H}(1-s)$ for some 'easy' $\Phi(s)$. I believe that the fact that $1 - \rho$ is not a zero $\zeta_{H}(s)$ when $\rho$ is, to be exclusively related to the shape of the functional equation of the Hurwitz zeta function, which is very different from that of the Riemann zeta function. It seems unlikely to me that the shape of the functional equation for the Hurwitz zeta function will have implications for the location of the zeros of $\zeta(s)$. </p> http://mathoverflow.net/questions/98162/a-question-about-normalization-in-the-fourier-transform/98164#98164 Answer by kolik for A question about normalization in the Fourier transform kolik 2012-05-28T04:39:01Z 2012-05-28T04:39:01Z <p>I think I've come to the personal concensus, that with the choice of notation $$\hat{f}(x) = \int_{-\infty}^{\infty} e^{- 2 \pi i t x} f(t) d t$$ we are <em>extracting</em> information from $f$, while with the notation $$\hat{f}(x) = \int_{-\infty}^{\infty} e^{2 \pi i t x} f(t) d t$$ we are <em>generating</em> information from the function $f$.</p> <p>For example, if $f$ were the probability density of a distribution I would go with the second notation to denote the characteristic function of the distribution. This aligns with the standard notation for the fourier transform of a probability distribution (check "characteristic function" on wiki). Anybody with me on this? </p> <p>P.S: I've seen both conventions used in my field. I believe the choice has to do with the perspective you take on the role of $f$. Do you want to understand $f$ better or do you want to generate things from $f$ and are actually interested in $\hat{f}$ rather than $f$?</p> http://mathoverflow.net/questions/58325/complex-fourier-coefficients-introduced-by/98155#98155 Answer by kolik for complex fourier coefficients, introduced by? kolik 2012-05-27T23:40:09Z 2012-05-27T23:40:09Z <p>I cannot comment because I don't have enough reputation, but here is the remark:</p> <p>In Zygmund's classical treatise on Fourier series, the complex exponential is not used. His book was published in the 50's. Thus, Kahane is (obviously) right, the complex representation did not come to prevail well until the 20-th century. </p> <p>I find it curious myself. </p> http://mathoverflow.net/questions/97976/approachable-french-masters/98091#98091 Answer by kolik for Approachable French Masters kolik 2012-05-27T06:11:21Z 2012-05-27T06:11:21Z <p>Actually Borel wrote a series of very nice little books, around 1900's. One of them is called "Sur les series de Taylor a coefficient positive". It has some very nice theorem, many of which are forgotten at this day; it reads like a beautifully written paper that just came out. </p> <p>Also, Paul Levy. He has an exceedingly beautiful writting style. His 7 volume collected works should be available in a math library. </p> http://mathoverflow.net/questions/98708/typesetting-papers-on-line Comment by kolik kolik 2012-06-03T10:58:59Z 2012-06-03T10:58:59Z for text editing i use emacs so its pretty fast http://mathoverflow.net/questions/98708/typesetting-papers-on-line Comment by kolik kolik 2012-06-03T10:58:35Z 2012-06-03T10:58:35Z it's mostly the compilation, preview part which is very slow. http://mathoverflow.net/questions/98708/typesetting-papers-on-line Comment by kolik kolik 2012-06-03T10:24:20Z 2012-06-03T10:24:20Z thank you, but do you know if immediate preview as in MO is implemented in any of these clouds? http://mathoverflow.net/questions/98343/number-of-integers-coprime-to-l/98357#98357 Comment by kolik kolik 2012-06-02T03:22:05Z 2012-06-02T03:22:05Z I was looking for the paper by Codeca and Nair in the Math. Bulletin http://mathoverflow.net/questions/98343/number-of-integers-coprime-to-l/98357#98357 Comment by kolik kolik 2012-06-02T03:21:23Z 2012-06-02T03:21:23Z Thank you ! It is amazing that you've found the reference :-) http://mathoverflow.net/questions/58325/complex-fourier-coefficients-introduced-by/98155#98155 Comment by kolik kolik 2012-06-01T15:25:15Z 2012-06-01T15:25:15Z To be precise it is correct that Zygmund mentions the complex representation. However for the bigger part of the book it is the sin, cos notation which is used. http://mathoverflow.net/questions/98343/number-of-integers-coprime-to-l/98357#98357 Comment by kolik kolik 2012-05-30T11:19:33Z 2012-05-30T11:19:33Z Well, I know how it meshed in. I want to see the real work :-) http://mathoverflow.net/questions/98343/number-of-integers-coprime-to-l Comment by kolik kolik 2012-05-30T08:56:20Z 2012-05-30T08:56:20Z I vaguely remember that the journal in question is likely to be the Canadian Math. Bulletin, or the Canadian Math. Journal, but I could be completely wrong on this hunch (so far my attempts at googling with &quot;canadian&quot; have failed). http://mathoverflow.net/questions/98263/max-of-fourier-transform Comment by kolik kolik 2012-05-29T10:28:50Z 2012-05-29T10:28:50Z I think that if f(x) = d mu(x) where mu(x) is a non-lattice distribution, then you can get some bounds &lt; 1. I've seen results of this type in Esseen's (of Berry-Esseen fame) thesis. http://mathoverflow.net/questions/98174/is-mertens-function-negatively-biased/98178#98178 Comment by kolik kolik 2012-05-28T20:39:10Z 2012-05-28T20:39:10Z Assuming rh and Li there is in that paper an explicit formula for the distribution function in terms of the probability distribution of some random variables . Now, it might be computationally difficult to get a hold of it , but that's a different story. Note that the paper I posted below is more directly focused on biases http://mathoverflow.net/questions/98174/is-mertens-function-negatively-biased/98178#98178 Comment by kolik kolik 2012-05-28T20:36:24Z 2012-05-28T20:36:24Z The point is that if you want to see if the mertens function is negatively biased it suffices to plot the graph of the distribution function http://mathoverflow.net/questions/98144/what-happens-when-infinite-values-of-zeta-hs-z-approach-zetas/98172#98172 Comment by kolik kolik 2012-05-28T20:34:35Z 2012-05-28T20:34:35Z I need to learn a bit more about hurwitz zeta when the shift is irrational. http://mathoverflow.net/questions/98171/what-is-the-relation-between-quasicrystals-riemann-hypothesis-and-pv-numbers Comment by kolik kolik 2012-05-28T07:47:08Z 2012-05-28T07:47:08Z Dyson said that every PV number gives rise to a quasi crystal. Anyway, do you know of a solid definition of quasi crystal? I do not believe there is a relation between PV numbers and RH. http://mathoverflow.net/questions/98171/what-is-the-relation-between-quasicrystals-riemann-hypothesis-and-pv-numbers Comment by kolik kolik 2012-05-28T07:08:26Z 2012-05-28T07:08:26Z Inquiring minds want to know.