User roupam ghosh - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T22:17:03Z http://mathoverflow.net/feeds/user/2865 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/128162/growth-of-the-reciprocal-gamma-function-in-the-critical-strip Growth of the reciprocal gamma function in the critical strip Roupam Ghosh 2013-04-20T10:30:57Z 2013-04-20T11:21:11Z <p>I was wondering if there are any results that studied the growth of $\left|\frac{1}{\Gamma(s)}\right|$ where $0 &lt; \Re(s) &lt; 1$ and as $\Im(s) \to \infty$? Any pointers to any results, papers, references will be highly appreciated.</p> <p>Thanks.</p> http://mathoverflow.net/questions/93861/a-note-by-n-a-carella-on-zero-free-regions A note by N. A. Carella on zero-free regions Roupam Ghosh 2012-04-12T12:35:18Z 2012-08-17T23:05:14Z <p><a href="http://arxiv.org/abs/0908.4287" rel="nofollow">http://arxiv.org/abs/0908.4287</a> I could not find any reviews for it, but if true its a major claim, because it says that $\Re(\rho) &lt; 21/40$ where $\rho$ is a zeta zero.</p> <p>My question:</p> <blockquote> <p>Are there any reviews of this paper, that reject or accept the claims made in this paper? Any references will be highly appreciated.</p> </blockquote> http://mathoverflow.net/questions/104667/euler-mascheroni-constant-curious Euler Mascheroni Constant, Curious? Roupam Ghosh 2012-08-14T06:52:19Z 2012-08-14T16:43:05Z <p>I found this formula for the Euler-Mascheroni constant $\gamma$. </p> <blockquote> <p>Just wondering whether such a formula already exists in literature? Also, wanted to know whether there are formulas that converge faster than this?</p> </blockquote> <p>$$\gamma = \sum_{k = 1}^{\infty} \frac{1}{2^k k} - \sum_{k = 1}^{\infty} \frac{\zeta \left( 2 k + 1 \right)}{2^{2 k} \left( 2 k + 1 \right)}$$</p> <p>UPDATE: Thanks for your reply quid. I just came across this while doing some calculations with the zeta function. The calculations are a bit too long to be posted, but in short it derives from $$\zeta(s) = \frac{s+1}{2(s-1)} + \frac{s}{8} - \frac{s(s+1)}{2\pi^2}\int_1^\infty \frac{(\tan^{-1}\cot(\pi x))^2}{x^{s+2}}dx$$.</p> http://mathoverflow.net/questions/100459/trying-to-debunk-a-claim Trying to debunk a claim Roupam Ghosh 2012-06-23T13:48:01Z 2012-06-24T18:18:46Z <p>Claim: Take any function $f(t) > 0$ for $t > 0$, such that $f(t) \to \infty$ as $t \to \infty$, then for $\sigma > 0$ $$|\zeta(\sigma + it)| = o(f(t))$$</p> <blockquote> <p>Is there any already existing evidence, like papers or proofs or something that can debunk this?</p> </blockquote> <p>As far as I know, under the Lindelof hypothesis $$|\zeta(\frac{1}{2} + it)| = o(t^\epsilon)$$ and Littlewood has already proved that under the Riemann hypothesis $$|\zeta(\frac{1}{2} + it)| = o\left(\exp\left(\frac{10\log t}{\log \log t}\right)\right)$$ both of which agree with the argument.</p> <p>Also I know from this paper at <a href="http://arxiv.org/pdf/math/0612106v2.pdf" rel="nofollow">http://arxiv.org/pdf/math/0612106v2.pdf</a> that $$\int_0^T |\zeta(1/2 + it)|^{2k}dt \gg_k T (\log T)^{k^2}$$ which kind of gives me a hint that there must be an obvious lower bound which can probably show that the condition given above for $\zeta(\sigma + it)$ and $f(t)$ is invalid.</p> <p>Looking for references.</p> http://mathoverflow.net/questions/97929/certain-functional-equations-for-the-riemann-zeta-function Certain functional equations for the Riemann Zeta function? Roupam Ghosh 2012-05-25T10:15:11Z 2012-05-28T15:34:09Z <p>Referring to <a href="http://math.stackexchange.com/questions/147377/other-functional-equations-for-zetas" rel="nofollow">this</a> question I asked on math.SE. I am posting a more generalized question here, for answers and further inquiry.</p> <blockquote> <p>For the Riemann zeta function, we know of the standard functional equation that relates $\zeta(s)$ and $\zeta(1-s)$. I wanted to know whether there are functional equations that relates $\zeta(s)$ and $\zeta(s+1)$, or $\zeta(s)$, $\zeta(s+1)$, and $\zeta(s+2)$ or in general $\zeta(s)$, $\zeta(s+1)$, $\zeta(s+2)$, ..., $\zeta(s+n)$ for $\Re(s) > 0$ and for $n \in \mathbb{N}$?</p> </blockquote> <p><em>My main motivation behind asking this question is I have found such an equation, but I do not know whether such an equation exists in literature. Also, I do not want to appear as if I am promoting my formula here, but rather I am more interested in the works that have been done in such directions.</em></p> <p><em>My formula goes as follows: For any $n \in \mathbb{N}$ and $\Re(s) > 0$ we have</em> $$\frac{1}{s-1} + \sum_{r=1}^n \binom{n}{r} (-1)^r \left(\frac{\zeta(s+r-1)}{s+n-1} + \int_1^\infty\frac{\rho(x)^r}{x^{s+r}}\mathrm{d}x\right) = 0$$ <em>where $\binom{n}{r}$ is the binomial coefficient and $\rho(x)$ is the fractional part of $x$</em> </p> <p><em>For example, putting $n=1$ we get the well known identity,</em> $$\frac{1}{s-1} - \frac{\zeta(s)}{s} - \int_1^\infty\frac{\rho(x)}{x^{s+1}}\mathrm{d}x = 0$$ <em>putting $n=2$,</em> $$\frac{1}{s-1} - 2\left(\frac{\zeta(s)}{s+1} + \int_1^\infty\frac{\rho(x)}{x^{s+1}}\mathrm{d}x\right) + \left(\frac{\zeta(s+1)}{s+1} + \int_1^\infty\frac{\rho(x)^2}{x^{s+2}}\mathrm{d}x\right) = 0$$ <em>and so on...</em></p> <p>EDIT: Classic answer by Juan. This question is now solved.</p> http://mathoverflow.net/questions/84266/on-robins-criterion-for-rh On Robin's criterion for RH Roupam Ghosh 2011-12-25T13:39:30Z 2011-12-26T03:54:02Z <p>$$\sigma(n) &lt; e^\gamma n \log \log n$$</p> <p>In 1984 Guy Robin proved that the inequality is true for all n ≥ 5,041 if and only if the Riemann hypothesis is true (Robin 1984).</p> <blockquote> <p>I could not get a hold of his paper. I am trying to understand how did he derived the inequality. Can anyone can outline the steps, how Robin derived this criterion?</p> </blockquote> http://mathoverflow.net/questions/41117/upper-bound-for-real-part-of-riemann-zeta-function-zeros Upper bound for real part of Riemann Zeta function zeros Roupam Ghosh 2010-10-05T08:31:07Z 2011-03-15T19:16:28Z <p>Hi,</p> <p>I have been reading about Riemann Zeta function $\zeta(s)$ and have been thinking about it for some time. I did some calculations, and reached a conclusion where $\Re(\rho) \le \log_2(3) - 1$ as $\Im(\rho) \to \infty$ where $\rho$'s are the roots of Riemann Zeta function in the critical strip. Anyways, I know its not the place to discuss claimed proofs and similar stuff, but just to give a background of where I am coming from. So straight to the question.</p> <blockquote> <p>Is there any similar result regarding upper bound ($&lt; 1$) for the real part of the zeros zeta function as their imaginary parts tend to infinity?</p> </blockquote> <p>Thanks</p> http://mathoverflow.net/questions/53332/on-the-nyman-beurling-equivalent-form-for-rh On the Nyman-Beurling equivalent form for RH Roupam Ghosh 2011-01-26T07:40:21Z 2011-01-26T09:23:47Z <p>Now, I am new to functional analysis. So please dont be harsh.</p> <p>I was going through some papers by Balazard &amp; Saias, Baez-Duarte, etc. that discussed and delved deep into details of approximating the characteristic function $\chi(0,1]$.</p> <p>Now, what appears to me is, if we can construct a function, using functions of the form $f(x) = \sum c_k \rho(\frac{\theta_k}{x})$ where $\rho(x)$ is the fractional part of $x$ that stay constant in the interval (0,1] and satisfying $\sum c_k \theta_k = 0$, then we have a proof of RH. </p> <p>But then I saw that they have also discussed the function $g(x) = - \sum \mu(n) \rho(1/nx) = \chi(t)$ which they say holds for every positive $t$ but in the sense of pointwise convergence, and then stated that in the words of Balazard and Saias's paper, <em>"Unfortunately, this feeling is, at least to a certain extent, a mirage"</em></p> <p>My question is:</p> <blockquote> <p>Why is g(x) not the fulfilling the criteria of being that "function" which mathematicians are trying to find to serve as the approximation for $\chi(0,1]$? I would really appreciate any sort of clarification for this characteristic function that is being searched.</p> </blockquote> <p>Thanks,</p> <p>The papers that I am reading includes:</p> <blockquote> <p>The Nyman-Beurling equivalent form for the Riemann hypothesis -- Michel Balazard and Eric Saias</p> <p>Convergence and the Riemann hypothesis -- Jungseob Lee</p> <p>A Note on Nyman-Beurling's equivalent formulation of the Riemann hypothesis -- Jean Francois Burnol</p> <p>A Closure Problem related to the Riemann zeta funtion -- Arne Beurling</p> <p>New versions of the Nyman-Beurling criterion for the Riemann hypothesis -- Luis Baez Duarte</p> </blockquote> http://mathoverflow.net/questions/42247/curious-about-a-question-on-zeta-zeros Curious about a question on zeta zeros? Roupam Ghosh 2010-10-15T03:04:47Z 2010-10-16T06:32:44Z <p>I have Edwards and Titmarsch books on Riemann zeta function with me. I could not find (maybe I did not read through that carefully), but are there results similar to the form like the one given below:</p> <blockquote>Is there a non-trivial zero $\rho$ starting from which $(\Im(\rho)\log(2)/2\pi )$ runs only through integer values </blockquote> <p>I would love any elaboration or links on this topic.</p> <p>Thanks,<br> Roupam Ghosh</p> http://mathoverflow.net/questions/13107/good-books-on-arithmetic-functions Good books on Arithmetic Functions ? Roupam Ghosh 2010-01-27T07:48:48Z 2010-07-22T02:27:53Z <p>As, I was studying the Mobius Mu Function, and Gram Series... I got myself some pretty nice books:</p> <p><strong>Ribenboim - The New Book of Prime Number Records</strong></p> <p><strong>Apostol - Introduction to Analytic Number Theory</strong></p> <p><strong>Niven, Zuckerman, Montgomery - An Introduction to the Theory of Numbers</strong></p> <p><strong>Inwaniec - Analytic Number theory</strong></p> <p>All of them dealt with the mobius mu function. But none of them dealt with the subject in details other than giving a few theorems and problems...</p> <p>So I would like to know, If you guys know of some good books that deal exclusively with Arithmetic functions ?</p> http://mathoverflow.net/questions/12489/is-the-given-expression-monotonically-increasing-or-decreasing-with-increasing-x Is the given expression, monotonically increasing or decreasing with increasing x? Roupam Ghosh 2010-01-21T02:55:14Z 2010-01-21T17:56:59Z <p>Is the given expression, monotonically increasing or decreasing with increasing x?</p> <p><code>$\frac{1}{x \log(x)} \sum_{i=1}^{\infty} \frac{\mu(i)}{i} x^{1/i}$</code></p> <p>EDIT: This is the derivative of the prime counting function <code>$\pi(x)$</code> w.r.t. x, ie., <code>$\frac{d\pi(x)}{dx} \sim \frac{1}{x \log(x)} \sum_{i=1}^{\infty} \frac{\mu(i)}{i} x^{1/i}$</code> Also, you might check Bruce C. Berndt's "Ramanujan's NoteBooks Part IV" page 123, equation 10.16, which gives = sign instead of <code>$\sim$</code></p> http://mathoverflow.net/questions/10377/a-result-on-prime-numbers A result on prime numbers Roupam Ghosh 2010-01-01T15:20:49Z 2010-01-01T19:50:27Z <p>Hi everyone</p> <p>This is my first post... I do mathematics from home... ie., not attached with any institution... I have deduced some results...</p> <p><code>$\lim \inf_{n\to\infty} \frac{d_n}{\log p_n} = 0$</code></p> <p>and, for constants <code>$A,B$</code></p> <p><code>$\lim_{n\to\infty} \log p_n - \sum_{i=1}^{n-1} \frac{d_i}{p_{i+1}} = A$</code></p> <p><code>$\lim_{n\to\infty} \log p_n - \sum_{i=1}^{n-1} \frac{d_i}{p_{i}} = B$</code></p> <p>Where, <code>$p_n$</code> is the nth prime... <code>$d_n = p_{n+1} - p_n$</code></p> <p>My question is: Do you think these results are good ?</p> http://mathoverflow.net/questions/128162/growth-of-the-reciprocal-gamma-function-in-the-critical-strip/128164#128164 Comment by Roupam Ghosh Roupam Ghosh 2013-04-21T02:37:22Z 2013-04-21T02:37:22Z anton,Alexandre Thanks! :) http://mathoverflow.net/questions/104667/euler-mascheroni-constant-curious/104691#104691 Comment by Roupam Ghosh Roupam Ghosh 2012-08-14T14:31:55Z 2012-08-14T14:31:55Z Yep... pretty close to what I got... Here's the link for others to see... <a href="http://archive.org/stream/actamathematica24lefgoog#page/n308/mode/2up" rel="nofollow">archive.org/stream/actamathematica24lefgoog#page/&hellip;</a> http://mathoverflow.net/questions/104667/euler-mascheroni-constant-curious/104691#104691 Comment by Roupam Ghosh Roupam Ghosh 2012-08-14T12:47:15Z 2012-08-14T12:47:15Z Wow... I will look at that paper for sure! :) Thanks. http://mathoverflow.net/questions/104667/euler-mascheroni-constant-curious Comment by Roupam Ghosh Roupam Ghosh 2012-08-14T12:46:36Z 2012-08-14T12:46:36Z I am using Mathematica, and &quot;Sum[1./(k*2^k) - Zeta[2*k + 1]/((2 k + 1)*2^(2*k)), {k, 1, 100}]&quot; gives me 0.5772156649015329 http://mathoverflow.net/questions/104667/euler-mascheroni-constant-curious/104684#104684 Comment by Roupam Ghosh Roupam Ghosh 2012-08-14T10:50:39Z 2012-08-14T10:50:39Z +1 For the links. I am just wanting to know whether this formula is unique and is of any use from a computational perspective. http://mathoverflow.net/questions/100459/trying-to-debunk-a-claim/100462#100462 Comment by Roupam Ghosh Roupam Ghosh 2012-06-23T15:38:48Z 2012-06-23T15:38:48Z Awesome... Let me probe on that a bit! :) http://mathoverflow.net/questions/97929/certain-functional-equations-for-the-riemann-zeta-function/98177#98177 Comment by Roupam Ghosh Roupam Ghosh 2012-05-28T14:55:51Z 2012-05-28T14:55:51Z @juan I just wrote and then deleted my comment and reposted it. Hope it didn't cause any confusion. :) http://mathoverflow.net/questions/97929/certain-functional-equations-for-the-riemann-zeta-function/98177#98177 Comment by Roupam Ghosh Roupam Ghosh 2012-05-28T14:54:30Z 2012-05-28T14:54:30Z Here's a direct link to the book by Landau mentioned by Juan. This takes you to paragraph 67. <a href="http://archive.org/stream/handbuchderlehre01landuoft#page/270/mode/2up" rel="nofollow">archive.org/stream/&hellip;</a> http://mathoverflow.net/questions/97929/certain-functional-equations-for-the-riemann-zeta-function/98177#98177 Comment by Roupam Ghosh Roupam Ghosh 2012-05-28T10:29:47Z 2012-05-28T10:29:47Z Yes, I have noticed that too, there are no $\rho(x)^r$ terms. Thanks for your answer anyways! http://mathoverflow.net/questions/97233/why-riemann-hypothesis-over-curves-is-easy-but-normal-hypothesis-for-the-rieman Comment by Roupam Ghosh Roupam Ghosh 2012-05-17T16:52:06Z 2012-05-17T16:52:06Z Things look easy when they are already solved! http://mathoverflow.net/questions/93861/a-note-by-n-a-carella-on-zero-free-regions/93865#93865 Comment by Roupam Ghosh Roupam Ghosh 2012-04-12T13:48:49Z 2012-04-12T13:48:49Z Thanks, Johan! I will take a look at that. :-) http://mathoverflow.net/questions/84266/on-robins-criterion-for-rh/84276#84276 Comment by Roupam Ghosh Roupam Ghosh 2011-12-25T23:18:47Z 2011-12-25T23:18:47Z Thanks quid. Thats a pretty nice historical overview :) http://mathoverflow.net/questions/84266/on-robins-criterion-for-rh/84285#84285 Comment by Roupam Ghosh Roupam Ghosh 2011-12-25T23:14:00Z 2011-12-25T23:14:00Z Thanks for the links! :) http://mathoverflow.net/questions/84266/on-robins-criterion-for-rh Comment by Roupam Ghosh Roupam Ghosh 2011-12-25T23:12:23Z 2011-12-25T23:12:23Z @Igor: Yes, I posted a similar question on math.SE. But I realized that this question is better suited here. http://mathoverflow.net/questions/84266/on-robins-criterion-for-rh/84269#84269 Comment by Roupam Ghosh Roupam Ghosh 2011-12-25T16:54:24Z 2011-12-25T16:54:24Z I have gone through Lagarias' paper already. It seems to me that he has just treated Robin's inequality to give a better bound. But I never quite got how Robin managed to relate RH and sigma and got this inequality.