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We have $$(|i|\cdot2^j)^{-1} \int_0^1 \big(x^l(1-x)^{2(j+2)}(k+(i+k)x^2)\big)/(1+x^2)\; dx $$ $$= (\operatorname{sgn}(i) \cdot2^j)^{-1} \int_0^1 \big(x^{l+2} (1-x)^{2(j+2)}\big)/(1+x^2)\; dx +\frac{ …

There's two parts of this: $\sigma(n) - 2n$ being congruent to $2$ mod $4$ and being divisible by $3$. Since $n$ is odd, $2n$ is congruent to $2$ mod $4$, so the first part is explained by $\sigma(n)$ …

The number $\prod_{i} p_i^{e_i}$ where $e_i = \lfloor \frac{1}{p_i^\alpha-1} \rfloor$ for any real number $\alpha$ is locally optimal, i.e. no smaller number has a larger product of factors. (This use …

Generally speaking, formulas like the trace formula admit an uncertainty principle: To obtain an identity where one side is highly concentrated (e.g. a sum over a small number of class numbers, or eve …

Mahler proved(1) that for any $p,q$, $\left | \pi -\frac{p}{q} \right| > \frac{1}{q^{42}}$. It follows that if one can compute $2^{ 42n} \pi$ to within an error of at most $1$, one can compute the fi …