Research proof index

Let $E_\pi(x)=|\pi(x)-\operatorname{Li}(x)|/(x/\log x)$. If $E_\pi$ obeys the classical bound $E_\pi(x)\le A\left(\frac{\log x}{R}\right)^B\exp\!\left(-C\sqrt{\frac{\log x}{R}}\right)$ for every $x\ge x_0$, with $A,B,C,R>0$, then beyond any $x_1\ge\max\{x_0,\exp(R(2B/C)^2)\}$ it obeys the uniform numerical bound obtained by evaluating that expression at $x_1$.

Let $E_\psi(x)=|\psi(x)-x|/x$. If $E_\psi$ obeys the classical bound $E_\psi(x)\le A\left(\frac{\log x}{R}\right)^B\exp\!\left(-C\sqrt{\frac{\log x}{R}}\right)$ for every $x\ge x_0$, with $A,B,C,R>0$, then beyond any $x_1\ge\max\{x_0,\exp(R(2B/C)^2)\}$ it obeys the uniform numerical bound obtained by evaluating that expression at $x_1$.

Running the monadic version of buildMerkleTree with an oracle function f is equivalent to running the functional version of buildMerkleTreeWithHash with the same oracle function.

Running the monadic getPutativeBatchRoot with an oracle function f is the same as running the functional getPutativeBatchRootWithHash with the corresponding hash function.

For nonzero natural exponents $p$ and $q$, taking a pointwise $q$-th power rescales the discrete normalized norm by $\|f^q\|_p=\|f\|_{pq}^{q}$.

When a quadratic form is equivalent to a weighted sum of squares, negative weights correspond to vectors where the form takes negative values. This is a concrete realization of a 1-dimensional negative definite subspace, contributing to O'Neill's index ν (Definition 18, p. 47).