things to remember

(based on the things i disliked being omitted from those i sat in)

for first-year teaching of sequences and series, convergence, etc., go on a detour upon getting to the example of harmonic numbers

Concrete Mathematics p.277 gives a cool formulation due to Euler

use \(\log\frac k{k-1}=\sum_{i=1}^\infty\frac1{ik^i}\) so (telescoping) \(\log\frac n1=\sum_{k=2}^n\sum_{i=1}^\infty\frac1{ik^i}\); transposing gives \(\log n=\sum_{i=1}^\infty{H^{(i)}_n-1\over i}\), so letting \(n\to\infty\), \(\log n-H_n=\sum_{i=2}^\infty{H^{(i)}_n-1\over i}\sim\sum_{i=2}^\infty{\zeta(i)-1\over i}\), we can then use that \(-H_{-x}=\sum_{i=2}^\infty\zeta(i)x^{i-1}\) (see this section for a discussion) to turn it into \[\log\infty-H_\infty=\lim_{x\to1}\left(\sum_{i=1}^\infty\frac{\zeta(i)x^i}i=\int_0^x-H_{-t}\,dt\overset{(*)}=\log((-x)!)-\gamma x\right)-\log\frac1{1-x}=\lim_{x\to0}\log\big(x\Gamma(x)\big)-\gamma\overset{(\dagger)}=-\gamma\], since \(\Gamma\)'s pole at \(0\) has residue \(1\).

the \((*)\) step follows directly from \(\gamma:=-\Gamma'(1),{d\over dx}\log\Gamma(x)=H_{x-1}-\gamma\) (from logarithmically differentiating \(\Gamma\)'s recurrence relation), and \((\dagger)\) from \(\underset{x=0}{\mathrm{res}}\,\Gamma(x)=1\)

show Euler's factorisation \(\sum_{n=1}^\infty n^{-s}=\prod_{p\in\mathbb P}\frac1{1-p^{-s}}\), and that if \(p_n\le c^n\) then the product is \(\le\frac1{(c^{-1};c^{-1})_\infty}=\frac1{\phi(c^{-1})}\) (the reciprocal of the )

the Euler function is finite within the unit disk and zero along its boundary, so the infinitude of \(H_\infty=\zeta(1)\) implies \(p_n\in o((1+\epsilon)^n)\) (primes grow slower than exponentially)

then show theorem 1.13 on p.18 of Apostol's Introduction to Analytic Number Theory (1976), on divergence of sum of reciprocals of primes

by assuming primes are bounded below by a polynomial (\(\rightarrow\) monomial as one considers arbitrarily distant tail sums) \(p_n\ge c_0n^{c_1}\) with \(c_1>1\), one can bound the sum above by \(\frac{\zeta(c_1)}{c_0}\); divergence contradicts this, so primes are \(p_n\in o(n^{1+\epsilon})\).

maybe then also give the weak prime number theorem, from my page on (much of which could also fit in such a course)

for second-year teaching of multivariable calculus, start with explanation of DFT, then Chebyshev polynomials as charpoly of path graph's adj. matrix (can be done quite easily via cofactor expansion!), then explain contents of (as a derivation of heat flow on a copper plane (surrounded by always-constant-temperature void) in the case that both space and time are discrete, before then covering the doubly-continuous case)

this would also fit in a linear algebra class, since (after finding the eigenvectors) it uses the spectral theorem to provide the coefficients that change from the basis of piece-amplitudes across squares to the eigenbasis

if ever i teach a statistics course (heaven forbid), this 1946 paper on the distribution of V-2 bombs across London could be a nice example for the effectiveness of the law of large numbers (and the tendency of humans to see incidental clustering in random distributions as evidence of nonrandomness)

for 2nd-to-3rd teaching of methods for asymptoterating functions via complex analysis, the matrix page's section coin survival is a good motivating example for why one would want to compute the inverse binomial transform of a rapidly-converging sequence (and thereby need to get around the numerical instability via eg. the )

get the University of Waterloo to digitise its old CORR research reports in a single open-access webpage (and fix all the old links that became defunct)

I have borrowed a physical copy (delivered across the Atlantic at £20 expense) of CORR 91-19, which i intend to write a review of (a very thorough one, in the manner of this wonderful review of On quaternions and octonions by John C. Baez)

get Stanford University to include scans of the research reports that it currently only offers mangled by destructive OCR

specifically, I found this report (a very good introduction to the \(r\)-Stirling numbers), which was also funded by the Office of Naval Research, which means it has an upload (as a direct scan!) at the Defense Technical Information Centre

I have seen an extremely similar effect (right down to the mismatched page widths) happen to the most common (and maybe only?) free copy of Bruce C. Berndt's Ramanujan's Notebooks accessible on the internet; some uploads' filenames seem to imply it was converted from .djvu, so perhaps the same happened inadvertently here

the same problem seems to persist across [between some and all] of the papers in the directory page

(who am i kidding; i am so procrastinatory as-is that it would probably not meaningfully alter my obligation-productivity)

make a page about all the fun corollaries that come out of viewing hypergeometric theorems (à la Gauss's) through multiple lenses (slightly similarly to ); this answer of mine employs Ruben's theorem, there are surely many more; also, it seems likely to have been already known before 1965

make a Wikipedia page about Euler's tree function as a more generating-function-focused counterpart of the one on the Lambert W function, with things like the expansion of \(\frac1{(1-T(x))^n}\) covered in this answer of mine partially done,

relatedly, make a page about Ramanujan's Q function; some of the history of its historical consideration (Cauchy, Ramanujan, Riordan, Knuth) is detailed in that answer, though not Knuth's An analysis of optimum caching (which is where I learned of the others); also, Flajolet et al.'s On Ramanujan's Q-function

make a wiki about maths theorems/results that lists (and gives some exposition of) corollaries of each one (or statements that are (perhaps with some manipulation) equivalent), focusing on little-known ones specifically

for instance,

the above examples with Stirling's approximation and the spectral theorem (being all you need to transform distributions of chess pieces on reflexive graphs into eigenbases) are pretty good!

how Fenwick trees are useful for tasks both combinatorial (see ) and numerical

if you have a list of floats all about the same size, adding them onto a single accumulator one-at-a-time while minimising roundoff error is hard (and requires storing additional bits to do perfectly), but making a Fenwick tree then extracting the total sum from that is of course much better than adding them onto an accumulator one at a time; it took me an embarrassingly long time to realise the latter application after only thinking about them for the former

an entry about the Riemann integral would contain (at least) the two genres of 'nontrivial' integral i know to be obtainable directly through it;

plugging in Gauss's multiplication theorem is the easiest route's to Binet's log-gamma theorem (covered in )

, in which a sum's closed form (a sum of nth powers) is calculable as a trace (interpreting powers' bases as eigenvalues of a circulant matrix, which is isomorphic to coeff extraction from polynomial exponentiation mod \(1-x^n\)); this converges to the integral's final value exactly (when the length of the cycle graph exceeds the length of the paths enumerated)

this is somewhat similar to Pólya and Szegő's Problems and Theorems in Analysis (1925) and the tricki (2008), whose intentions are to index problems by genre of solution type and solutions by genres of problem applicability, respectively

maybe a wiki oriented around 'frequently-asked questions' (ie. counterexamples demonstrating why things don't have properties that might be intuitively expected of them but aren't mentioned in textbooks/Wikipedia articles)

like "does the shape of the row echelon form \(R(M)\), as a tableaux, relate to a matrix \(M\)'s isomorphism class in any nontrivial way (beyond its height) like Jordan normal form?" "no, but the height of a column tells you the rank of \(M\)'s corresponding upper-left submatrix"

make a LifeWiki page on the bootstrap percolation von Neumann CA B234/S01234 (see also this paper analysing the rate of convergence (by the author of the webpage linked from that answer) and this paper on the higher-dimensional cases)

note that this is not the only famous CA in the von Neumann neighbourhood; there is also (where cells flip to state \(s\) if the cells above and right are both \(s\)) and the (covered in H-trees)

expound upon cool result (3.1.10) of this nice paper on Euler-Mascheroni and incorporate into

investigate \(r\)-(numbers ); this answer shows that \(B^+_n=-n\zeta(1-n)=\sum_{k=0}^n{k!\over k+1}(-1)^k{1+n\brace1+k}\) can be continued to negative \(n\), likewise for \(B(n,v)=-n\zeta(1-n,v)=\sum_{k=0}^n{k!\over k+1}(-1)^k{n+v\brace k+1}_v\)

update list of sequences via Q function in with (n) \(=n^n(n+Q(n)+1)\)

page about facts in number theory which have many proofs to collect; ie. divergence of sum of reciprocals of primes (of which i know three; Euler's (given on the Wikipedia page), the one Apostol aforementionedly presents in Introduction to Analytic Number Theory (1976), Monson's from Finite Fourier analysis and Dirichlet's theorem (2015))

essay on the factorial being better than (or rather, not worse than) the gamma function with reference to various identities (which i will try to remember to collect here until i split it off)

Ramanujan's integral (DLMF 5.13.4) is much nicer as \(\int_{-\infty}^\infty{dt\over(a+t)!(b+t)!(c-t)!(d-t)!}={(a+b+c+d)!\over(a+c)!(a+d)!(b+c)!(b+d)!}\), while Barnes' (DLMF 5.13.3) is nicest in the gammaful form given

page about inequalities (perhaps);

mostly this entry is here because i saw inequalities cheat sheet and thought "ooh ooh i know about this" when i saw the binomial one and thought writing it as a partial ordering of formulas would be cool; specifically,

\[ \begin{matrix} &&{n^k\over k!}\\ &\!\!\!\!\mathrel{\style{display:inline-block; transform: rotate(-45deg);}\le}\!&&\!\!\!\!\mathrel{\style{display:inline-block; transform: rotate(45deg);}\le}\!\!\!\!\!\!\\ \binom nk\!&\!\!\!\!\le\!\!&\!\!\sum_{j=0}^k\binom nj&\le&\!\left({en\over k}\right)^k\\ &\!\!\!\!\mathrel{\style{display:inline-block; transform: rotate(45deg);}\le}\!&\!\!\!\!\mathrel{\style{display:inline-block; transform: rotate(90deg);}{\underset{\tiny \text{iff}\,k\le\frac n2}\le}}&\!\!\!\!\mathrel{\style{display:inline-block; transform: rotate(-45deg);}\le}\!\!\!\!\!\!\\ &&{n^n\over k^k(n-k)^{n-k}}\\ &&&\mathrel{\style{display:inline-block; transform: rotate(45deg);}{\underset{\tiny k>1}\le}}&\binom nk2^k\\ \end{matrix} \]

where the \(k\le n/2\) inequality is a special case of the binomial distribution's Chernoff bound; see this old answer of mine for an extension of the last one into an enveloping series for \(\sum_{j=0}^k\binom nj\)

also, as discussed in this answer, we have the bounds \[\sqrt{x(x+c)}<(x+c)\frac{(1+\frac cx)^{\frac xc}}e<x+\frac c2\] (a refinement of AM-GM!)

(subset of preceding entry but large enough to deserve independence)

Ira M. Gessel, this paper on rook polynomials (and how to decompose products of Laguerre polynomials into the basis of Laguerre polynomials; maybe make a page one day showing how to do this for various polynomial families, giving this result and its corollaries alongside \(\frac{w^\underline rh^\underline r}{r!}=[x^\underline{w+h-r}]x^\underline wx^\underline h\)), and this one on Lagrange inversion

Sextus Empiricus, math.se answer (nice diagram) and this stats.se answer on the coupon collector's waiting time distribution being asymptotically Gumbel via Poissonisation (also this nice blogpost)

G. Cab, definition of r-subset numbers through Eulerians

Peter H. van der Kamp, On the Fourier transform of the greatest common divisor (2012), §3 'A historical remark, and generalised Ramanujan sums' mentions that J.C. Kluyver, Some formulae concerning the integers less than n and prime to n (1906) proved "Hölder’s relation" 30 years prior; the Wikipedia page does not mention this!

this chapter from a statistics textbook: was recommened for better understanding derivations of Sibuya distributions

Ocmo fangame (maybe entitled moOc, save for that already being a term; mo-Oc)

definitely to implement: replay system, n-body physics (already done), nested 2-body physics (planets form leafs of a binary tree)

maybe to implement: objects (based on either vicinity or touchedness) that slow down time followed by replay interpolating frames to show run at full speed

integrals of the family covered in

\(\int_0^\infty{t^s\over(e^t-1)^z}\,dt\), \(\int\limits_0^\infty\left(t\over e^t-1\right)^n\,dt\), \(\int_0^1{\log^nx\over(1-x)^m}\,dx=\int_0^\infty{(-t)^ne^{-t}\over(1-e^{-t})^m}\,dt\)