# Eric

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bio website sites.google.com/site/… location Canada age member for 2 years, 4 months seen 7 hours ago profile views 5,957

I an interested in Number Theory, Analysis and Additive Combinatorics. My papers can be found here, or you can visit my website for more information.

# 2,687 Actions

 1d awarded Nice Answer May18 awarded Constituent May15 revised Proving two sequences identicaledited body May14 awarded Good Question May13 comment Closed form for $\int_0^1\log\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\mathrm dx$+1, Very nice solution. May13 comment Closed form for $\int_0^1\log\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\mathrm dx$It's not clear how to proceed from $$\int_{0}^{\infty}\log u\frac{\sinh(u)}{\cosh(u)^{2}}du,$$ although it is a rather nice form. May13 comment Closed form for $\int_0^1\log\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\mathrm dx$Here are some equivalent forms: Since $\text{sech}^{-1}(x)=\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^{2}}-1}\right),$ we are trying to evaluate $$\int_{0}^{1}\log\left(\text{sech}^{-1}(x)\right)dx.$$ Let $u=\text{sech}^{-1}x$ so that $x=\text{sech}(u)=\frac{1}{\cosh(u)}.$ Then $dx=d(\frac{1}{\cosh(u)})$ , and we are looking at $$-\int_{0}^{\infty}\log ud\left(\frac{1}{\cosh(u)}\right)$$ which equals $$\int_{0}^{\infty}\log u\frac{\sinh(u)}{\cosh(u)^{2}}du.$$ Using integration by parts, this becomes $$\lim_{a\rightarrow0}\left(\log a+\int_{a}^{\infty}\frac{\text{sech}(x)}{u}du\right).$$ May13 comment How many $p$-adic numbers are there?@PatrickDaSilva: You need to check that the map is well defined. It also gives two surjections, not two injections. May7 awarded Informed May6 comment Reflecting a golfball off a wall to a hole and compensating for the balls radiusI am deleting the comments regarding the inappropriate username on this thread. The original username, and description, were not appropriate at all, and have been changed to something less offensive. They have been recorded in a previous deleted comment. May6 awarded Caucus May6 revised To estimate $\sum_{m=1}^n \Big(d\big(m^2\big)\Big)^2$added 125 characters in body May6 answered To estimate $\sum_{m=1}^n \Big(d\big(m^2\big)\Big)^2$ May6 revised To estimate $\sum_{m=1}^n \Big(d\big(m^2\big)\Big)^2$added 104 characters in body May6 awarded Announcer May5 answered Two questions re: $\sum_{n=1}^{\infty}n^{-p_{n}}$ May5 answered To estimate $\sum_{m=1}^n \Big(d\big(m^2\big)\Big)^2$ May5 revised Divisor summatory function for squaresedited tags May5 revised analytic number theory, troubling bound on sum of $\varphi(n)$added 271 characters in body May5 revised analytic number theory, troubling bound on sum of $\varphi(n)$deleted 23 characters in body