bio  website  math.lsa.umich.edu/~speyer 

location  Ann Arbor  
age  34  
visits  member for  4 years, 11 months 
seen  1 hour ago  
stats  profile views  29,892 
Associate Professor of Mathematics at the University of Michigan. My research interests are in combinatorial algebraic geometry, particularly Schubert calculus, matroids and cluster algebras. I also enjoy thinking about number theory and computational mathematics.
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Go I Know Not Whither and Fetch I Know Not What
@FelipeVoloch Nonetheless, sorry about that. 
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Go I Know Not Whither and Fetch I Know Not What
No, I didn't do that one. The primes $2$, $3$ and $5$ are going to be wonky since (as GH from MO points out) you aren't working with an integer basis for the ring of integers, and I didn't want to work that hard. 
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answered  Go I Know Not Whither and Fetch I Know Not What 
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Prove that the Dirichlet eta function is monotonic
Whoops! Undergrad, not underaged! 
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Prove that the Dirichlet eta function is monotonic
@Alexander You can group terms of a conditionally convergent sum, you just can't rearrange them. Write out the partial sums of $(1/2) + 1/2 \sum (1/(2k1)^s  2/(2k)^s + 1/(2k+1)^s)$ and check that they approach the same limit as the partial sums of $\sum (1)^{n1}/n^s$. 
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Prove that the Dirichlet eta function is monotonic
@Alexander Agreed. I think what I wrote is at a good level for communicating with other professionals but you are right, there are subtleties here, since $\sum (1)^{n1} e^{nx}$ isn't uniformly convergent on $[0, \infty)$. If I were presenting this to an underaged analysis class, there would have to be a discussion about cutting the integral into $[a, \infty)$, where the convergence is uniform so we can interchange $\sum$ and $\int$, and $[0,a)$, where the integral is at most $\int_0^a x^s \frac{dx}{x} = a^s/s$. Then send $a \to 0$ to complete the computation. 
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Prove that the Dirichlet eta function is monotonic
@GHfromMO Agreed, rearrangement is the case $g(x)=1$, $s(x) = t(x)$ (or $s(x) = t(x^{1})$ if we are using $dx/x$). Chebyshev's inequality is $g(x) = t(x) = 1$. 
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Prove that the Dirichlet eta function is monotonic
added 542 characters in body 
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awarded  Nice Answer 
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answered  Prove that the Dirichlet eta function is monotonic 
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Prove that the Dirichlet eta function is monotonic
There is an easy proof that $\eta(p) \geq \eta(0) = 1/2$. We have $\eta(p) = (1/2) + \sum_{n=0}^{\infty} (1/2) (2n)^{p}  (2n+1)^{p} + (1/2) (2n+2)^{p} \geq (1/2) \sum_{n=0}^{\infty} \sqrt{(2n)(2n+2)}^{p}  (2n+1)^{p} \geq 1/2$ where the first inequality is AMGM. I've been trying to adapt this to show $\eta'(p)>0$, but no luck yet. 
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awarded  Enlightened 
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awarded  Nice Answer 
Sep 12 
awarded  Enlightened 
Sep 12 
awarded  Nice Answer 
Sep 12 
answered  What is the Implicit Function Theorem good for? 
Sep 10 
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Why does the Gammafunction complete the Riemann Zeta function?
If I'm thinking correctly, $\zeta(s)$ is formally $\Gamma(g,1s)$ where $g$ is a sum of $\delta$ functions at the positive integers. Is there some sense in which $g = \hat{g}$ here? 
Sep 10 
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Inequality of arithmetic, geometric and harmonic means
Over at math.SE, we have had two questions about for which $\theta$ (as a function of $n$), we have $GM \leq (1\theta) AM + \theta HM$. (Your question asks whether $\theta=1/2$ works.) We know that the optimal $\theta$ goes to $0$ as $n \to \infty$, and $\theta=1/n$ works, but there is still some room for improvement. math.SE links: math.stackexchange.com/questions/92935 math.stackexchange.com/questions/803960 
Sep 8 
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Solving polynomials of arbitrary degree
Other previous questions mathoverflow.net/questions/89144/… mathoverflow.net/questions/61409/… . 
Sep 6 
awarded  Popular Question 