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Apr 18 |
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Non-normalized sinc function I had hoped there was a faster way but now that I have worked through your proof I doubt it--anything that works might be about this involved. This is a really nice answer and I hope you get more votes for it. |
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Apr 14 |
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Generalizing Ramanujan's proof of Bertrand's Postulate: Can Ramanujan's approach be used to show a prime between $4x$ and $5x$ for $x \ge 3$ You might look at the work of Bachraoui and Nagura if you haven't already. We also have a general proof that there is a prime on $(x,(1+1/k)x)$ for large enough x. |
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Apr 7 |
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Understanding Ramanujan's approach in his proof of Bertrand's Postulate +1. i didn't check the stirling approx but plotted the functions. if there's a mistake i don't see it. nice question either way. |
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Apr 7 |
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Understanding Ramanujan's approach in his proof of Bertrand's Postulate We have $\psi(2x)+\psi(2x/2)+\psi(2x/3)...-\psi(x)-\psi(x/2)-\psi(x/3)...$ So $\psi(2x)-\psi(x) < \psi(2x)$ and $\psi(2x/2)-\psi(x/2)< \psi(2x/2)...$ but the sum of these differences could exceed $\psi(2x)$? |
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Apr 5 |
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Dirichlets theorem on primes G.J.O. Jameson's 'The Prime Number Theorem' (Cambridge) is also a good one and it only takes 4 chapters to get the result. Long chapters, though. |
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Mar 31 |
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Discontinuities of $\sum \frac{x^{\rho}}{\rho}$ When you remind me that this expression equals the Chebyshev function it is clear. Thank you! |
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Mar 30 |
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Discontinuities of $\sum \frac{x^{\rho}}{\rho}$ Yes, according to Edwards they converge only conditionally even when paired. The increasing order of Im$|\rho|$ is crucial. That said... |
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Mar 23 |
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Question about Ramanujan's proof of Bertrand's Postulate Having cracked a book and taken out my magnifying glass, I feel that you may have mis-stated (7). I would be the last to down-vote for a typo... |
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Mar 15 |
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Interesting Integral $\int_{-\infty}^{\infty}\frac{e^{i nx}}{\Gamma(\alpha+x) \Gamma(\beta -x)}dx$ Just a comment really. Ramanujan derives several very similar equations at ch. 27 of Collected works. I think you can derive your integral from eq. (7.11). If I find time I will try to work it out and post as an answer. |
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Mar 14 |
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Fourier transform of $\cos(\sqrt{r^2-t^2})$ If you mention a few things you tried you're sure to get help. |
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Mar 11 |
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Ratio of primes $(x^2+x+(5+6m))$ to $(x^2+x+(3+6m))$ Thank you, the B-H conjecture is new to me. As you understood the question, any suggestions you might have for improving the first few lines would be appreciated. I added sums over m which I thought made things clearer but this question has been difficult for me to articulate clearly. |
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Mar 8 |
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Book Searching in Complex Analysis I wonder if it isn't a translation of the collection by Aramanovich et al., A Collection of Problems on Complex Analysis (Dover)? It has increasingly difficult problems taken from several well-known texts. |
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Feb 28 |
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On the number of integers with an even number of distinct prime factors @falang: edit in response to your comment. |
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Feb 28 |
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How was the Fourier Transform created? @WillieWong Have bookmarked the tool pending acquisition of keyboard. Thanks! |
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Feb 22 |
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$(1+\frac{1}{n\log n})^n-1=O(\frac{1}{n})$. I don't think it's true either. What is "it?" |
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Feb 22 |
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How was the Fourier Transform created? If someone could explain the best way to put in an umlaut I'd be grateful. |
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Feb 21 |
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Primes of form $x^2+x\pm k$ I upvoted the answer below because it showed work but it answers a question not asked. Hopefully my edited notation is finally right. Sum of logs of $x_i$ such that $x_i^2+x_i+k$ prime and sum of logs of primes $x_i^2+x_i+k$ differ by a factor of about 2. The former is what I am using. |
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Feb 18 |
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Showing that $ |\cos x|+|\cos 2x|+\cdots+|\cos 2^nx|\geq \dfrac{n}{2\sqrt{2}}$ Indices from k = 0 to n? |
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Feb 17 |
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Primes of form $x^2+x\pm k$ You seem to be talking about applying the PNT, which I did in the following line and which does not I think shed any light on the problem. |
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Feb 6 |
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Iteration of $x/\log x$ Will take some time to look at this. I follow your first thought beyond the edit but not your conclusion about the integral. |
