Eric Naslund
Reputation
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151/100 score
 1d reviewed Leave Open Limit of the succession $\lim_{n \to{+}\infty}{n\sin(n\pi)}$ 1d reviewed Leave Open Combinatorics problems involving permutations 1d reviewed Close What's the difference between $\mathbb{R}^2$ and the complex plane? 2d answered How to maximize the product xy? 2d comment How to maximize the product xy? The gradient of the constraint is $(2,3)$ and so $xy$ is maximized when $\nabla xy=(y,x)$ is proportional to that. Hence you get $(x,y)=(5,10/3)$ as the solution, and a product of $50/3$. Apr 30 revised An infinite series of a product of three logarithms deleted 23 characters in body Apr 30 revised An infinite series of a product of three logarithms deleted 18 characters in body Apr 27 comment $\sum_{p} \chi(p)/p$ is conditionally convergent for non-principal character +1, The result you are trying to prove is equivalent to the non-vanishing of $L(1,\chi)$/Dirichlet's theorem. Note however that proving the estimate you refer to from Tao's blog (Theorem 70) is the heart of this question, and Dirichlet's theorem. Quoting that proof from elsewhere skips over the important details about why this result is true. Apr 26 comment Prove that $\prod\limits_{2 < p \leq y}\left(1-\frac{2}{p}\right)\sim\frac{D}{\log ^2 y}$ The exact constant $D=e^A$ is given by $$D=4\Pi_2 e^{-2\gamma}$$ where $\Pi_2$ is the twin prime constant and $\gamma$ is the Euler-Mascheroni constant. This is proven in the accepted answer here: math.stackexchange.com/questions/22411/… Apr 23 awarded linear-algebra Apr 20 comment $\int_{-\infty}^{+\infty} e^{-x^2} dx$ with complex analysis @user1952009 You are critizing me because you don't understand why these proofs work, or why the underlying theorems are true. Weierstrass Factorization is a very natural theorem - it roughly says that a meromorphic function is given by the location of it's zeros, poles and it's growth rate. The proof for Poisson summation on the real line is quite short and just uses basic Fourier analysis on the torus. Could the proof that $\Gamma(1/2)=\sqrt{pi}$ using Poisson and zeta be streamlined? Certainly, but you'd lose the lovely connection in doing so, and where is the fun in that? Apr 20 comment $\int_{-\infty}^{+\infty} e^{-x^2} dx$ with complex analysis @user1952009 You shouldn't critize my answer just because you don't know how to prove the identities I used nicely. For (1) The functions $\Gamma(s)\Gamma(1-s)$ and $1/\sin(\pi s)$ both have no zeros and poles at every integer, so the first identity follows directly from the Weierstrass Factorization Theorem. It in no way relies on $\Gamma(1/2)$ as you incorrectly state. As for (4), the functional equation comes from an application of the Poisson summation on $\psi(x)=\sum_{n=1}^\infty e^{-\pi n^2 x}$, and (5) has little to do with (2). Apr 19 comment Can the real vector space of all real sequences be normed so that it is complete ? $\ell^p$ for any $p$ is a complete metric space, but this requires taking the subspace of vectors for which the norm is not infinite. Is that what you are looking for? Mar 21 comment Lower bound for $\phi(n)$: Is $n/5 < \phi (n) < n$ for all $n > 1$? @M.S.Dousti: It does capture a negative quantity Mar 20 comment Lower bound for $\phi(n)$: Is $n/5 < \phi (n) < n$ for all $n > 1$? @M.S.Dousti: That's capture by the big-$O$ term in the expression I give above. Mar 7 comment A Lagrange multiplier It's a local minimum Mar 7 answered The last two digits of $13^{1010}$. Feb 27 awarded Good Question Feb 24 awarded Nice Question Feb 8 awarded Nice Answer