User asd - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T21:48:05Z http://mathoverflow.net/feeds/user/11733 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/125958/product-11-p-in-terms-of-chebyshevs-theta-or-psi-function product 1+1/p in terms of Chebyshev's theta or psi function asd 2013-03-30T00:10:43Z 2013-04-04T23:11:09Z <p>I would like to know if there is any formula for</p> <p><code>$\prod_{x&lt;p\leq y}\left(1+\frac1p\right)$ in terms of $\theta$ or $\psi$ functions $\theta(x)=\sum_{p\leq x}\log p$ and $\psi(x)=\sum_{p^\nu\leq x}\log p.$</code></p> <p>More precisely, I need to know if we can write that product as some thing similar to the following</p> <p>$\frac{\log\theta(y)}{\log\theta(x)}+\epsilon(x,y)$</p> <p>or</p> <p>$\frac{\log\theta(y+\epsilon_1(x,y))}{\log\theta(x+\epsilon_2(x,y))}$</p> <p>which is equality or very sharp inequality.</p> <p>avoiding the terms include</p> <p>$\frac{\log y}{\log x}$</p> <p>thanks</p> http://mathoverflow.net/questions/123386/alternating-series-estimation-with-integral alternating series estimation with integral? asd 2013-03-02T00:08:54Z 2013-03-02T00:08:54Z <p>We know that there are some approximation like Abel's identity. If $\lambda_n$ is increasing and $$C(x)=\sum_{\lambda_n\le x}c_n,\qquad(c_n\in\mathbb{C})$$ Then if $X\ge\lambda_1$ and $\phi(x)$ has continuous derivative, we have $$\sum_{\lambda\le X}c_n\phi(\lambda_n)=C(X)\phi(X)-\int_{\lambda_1}^X C(x)\phi'(x)dx$$ If $C(X)\phi(X)\rightarrow0$ as $X\rightarrow\infty$, then $$\sum_{\lambda\le X}c_n\phi(\lambda_n)=-\int_{\lambda_1}^X C(x)\phi'(x)dx$$ where both sides are convergent.</p> <p>do we have similar estimate (for example alternating harmonic series) with integral?</p> http://mathoverflow.net/questions/110679/integrate-of-functions-involving-floor integrate of functions involving floor asd 2012-10-25T15:19:50Z 2012-10-25T21:50:28Z <p>Is there any exact formula or at least exact inequalities for the following intehral</p> <p>$$\int_2^x\frac{dt}{[\frac{\log x}{\log t}]\log t}$$</p> <p>where [x] is the greatest integer less than or equal to x.</p> <p>added:</p> <p>When I use </p> <p>$$x-1&lt;[x]\le x$$ I get $$\frac{x-2}{\log x}=\int_2^x\frac{dt}{\log x}\leq \int_2^x\frac{dt}{[\frac{\log x}{\log t}]\log t}\le \int_2^x\frac{dt}{\log x-\log t}$$ but they are not exact enough. I need more closer bounds.</p> http://mathoverflow.net/questions/99366/explicit-large-gap-for-consecutive-zeros-of-the-riemann-zeta-function explicit large gap for consecutive zeros of the Riemann zeta function asd 2012-06-12T14:06:40Z 2012-10-10T20:45:04Z <p>In Theorem 9.12, Titchmarsh (The Theory of the Riemann Zeta Function) proved that </p> <p>For every large positive T, $\zeta(s)$ has a zero $\beta+i\gamma$ satisfying $$|\gamma-T|&lt;\frac{A}{\log\log\log T}$$</p> <p>Is it possible to determine $A$ and $T$ without assuming the Riemann hypothesis?</p> <p>Or</p> <p>Are there any other known results (with explicit) around this question?</p> http://mathoverflow.net/questions/84405/graph-of-the-size-of-a-complex-function graph of the size of a complex function asd 2011-12-27T19:17:02Z 2011-12-28T13:00:52Z <p>Hi Here there are two graphs for two functions from $R^2\mapsto R$.</p> <p>Is there similar graph for the absolute value of a complex variable function $f:C\mapsto C$ that has the same point (like saddle point or transition). I know some functions that have the point $(x,y,|f(x+iy)|)$ on that such that in one direction it is maximum, and in the other direction it is minimum. </p> <p>My question here is that: is there any such point such that in one direction it is maximum (or minimum) but in the other direction it is not maximum nor minimum (similar to $(0,0)$ in $y=x^3$ in the real case).</p> <p>Thanks</p> <p><a href="http://postimage.org/image/v2ig8ycx7" rel="nofollow">link text</a></p> <p><a href="http://postimage.org/image/57zgroy4t" rel="nofollow">link text</a></p> http://mathoverflow.net/questions/80973/lower-bound-for-re-zeta1it lower bound for $\Re\zeta(1+it)$ asd 2011-11-15T11:35:03Z 2011-11-15T12:38:35Z <p>Hi</p> <p>is there any lower bound for $\Re\zeta(1+it)$. </p> <p>I did try with computer until some ordinate and I saw $\Re\zeta(1+it)>0$.</p> <p>If it is true, is there any reference to prove it. thanks</p> http://mathoverflow.net/questions/65801/sum-of-the-series-sum-n1-infty-frac-1n-1-sqrtn sum of the series $\sum_{n=1}^\infty \frac{(-1)^{(n-1)}}{\sqrt{n}}$? asd 2011-05-23T22:20:00Z 2011-05-23T22:25:30Z <p>Hi,</p> <p>We know that the series $\sum_{n=1}^\infty \frac{(-1)^{(n-1)}}{\sqrt{n}}$ is convergent and it is oscillating. and numerically it is almost 0.6048986434.</p> <p>I want to know what is the exact limit of this series and how we can find that analytically.</p> <p>Now if we let $A(t):=\sum_{n=1}^t\frac{(-1)^{(n-1)}}{\sqrt{n}}$, is there any simple formula without summation that is equal to $A(t)$.</p> http://mathoverflow.net/questions/125958/product-11-p-in-terms-of-chebyshevs-theta-or-psi-function Comment by asd asd 2013-04-04T23:07:42Z 2013-04-04T23:07:42Z @Greg Martin, you are right. Asymptotically the product is logθ(y)/logθ(x). but I would like more sharper, since if I write it as the formula in your Answer, I lose some accuracy to invert to terms including only $\theta$ and it is not appropriate for my question. thanks! http://mathoverflow.net/questions/125958/product-11-p-in-terms-of-chebyshevs-theta-or-psi-function/125974#125974 Comment by asd asd 2013-04-02T18:53:18Z 2013-04-02T18:53:18Z @Greg Martin. Thank you very much. Actually, I should have clarified my question. So I did now. Sorry I am late. http://mathoverflow.net/questions/110679/integrate-of-functions-involving-floor Comment by asd asd 2012-10-25T19:19:05Z 2012-10-25T19:19:05Z @Davide Giraudo, @Gerhard Paseman, I added some phrases. http://mathoverflow.net/questions/99366/explicit-large-gap-for-consecutive-zeros-of-the-riemann-zeta-function/99368#99368 Comment by asd asd 2012-06-12T15:08:38Z 2012-06-12T15:08:38Z @Dierk. If so, it would be better, if not just to know if it is not large. for example if $A=1$ and $T&lt;10^{10}$, it's OK. http://mathoverflow.net/questions/84405/graph-of-the-size-of-a-complex-function/84410#84410 Comment by asd asd 2011-12-29T10:57:31Z 2011-12-29T10:57:31Z @Robert Israel: thanks http://mathoverflow.net/questions/84405/graph-of-the-size-of-a-complex-function/84410#84410 Comment by asd asd 2011-12-28T14:46:06Z 2011-12-28T14:46:06Z Is it possible $f'(z)\neq0$ and $f(z)\neq0$, but $z$ is saddle point of the above type. http://mathoverflow.net/questions/84405/graph-of-the-size-of-a-complex-function/84410#84410 Comment by asd asd 2011-12-28T14:42:50Z 2011-12-28T14:42:50Z @Robert Israel: Yes, it is. but I was meaning that in one direction similar to $x^2$ and in the other direction similar to $x^3$. For example: $f(z)=cos(z)$ in $z=0$ has that condition, $|f|$ in one direction is like $x^2$ and in the other is like $(−x^2)$ http://mathoverflow.net/questions/84405/graph-of-the-size-of-a-complex-function Comment by asd asd 2011-12-28T14:40:23Z 2011-12-28T14:40:23Z $f$ is analytic http://mathoverflow.net/questions/50368/series-for-log-log-n/50370#50370 Comment by asd asd 2010-12-25T21:24:36Z 2010-12-25T21:24:36Z Actually I need simpler, for example by powers of x not again some logarithm! http://mathoverflow.net/questions/50368/series-for-log-log-n Comment by asd asd 2010-12-25T20:04:48Z 2010-12-25T20:04:48Z If the tag is not true, please let me know.