Landau 3 / Legendre, third computation

Outcome

Assume RH. For every real $x\geq1$ and every

\delta\geq\frac{4504880398387}{20000000000000} =0.22524401991935,

the interval $[x^{2+\delta},(x+1)^{2+\delta}]$ contains a prime. This tightens q1's certified threshold $0.22525$ . It does not prove Legendre's conjecture and does not extend the finite Oppermann record.

Replay

From any directory, run

problems/landau-legendre/compute/q3/run_all.sh

The replay uses Python 3 and a standard C compiler, needs no network, and writes only to a temporary directory. It first rechecks q1's exact certificate for the previous threshold, regenerates the q3 certificate, byte-compares it with the committed copy, reconstructs every rational enclosure independently, and runs a separate long double calculation.

Why the extra digits are real

Put

N=70{,}500{,}000{,}000{,}000,\qquad d=0.22524401991935,\qquad \alpha=2+d,\qquad X=N^{2/\alpha}.

For $2<\alpha<3$ , Taylor's theorem gives, for $x>0$ ,

(x+1)^\alpha-x^\alpha >\alpha x^{\alpha-1} +\frac{\alpha(\alpha-1)}2x^{\alpha-2}.

Consequently, Chamberland--Straub inequality (6) follows if

S(x)=x^{d/2}+\frac{\alpha-1}{2}x^{d/2-1} -\frac1\alpha x^{-\alpha/2}-\frac{22}{25}\log x>0.

Their simpler condition (5) drops the last two power terms. At the splice $x=X$ , its certified logarithmic margin is negative: its exact upper bound is less than $-2.03\cdot10^{-13}$ . Thus q1's one-term test does not certify this new value.

Write $A=X^{d/2}=N^{d/\alpha}$ and $b=(\alpha-1)/2$ . At $X$ , the ratio of the retained correction to $A$ is

t=\frac bX-\frac1{\alpha N A}.

The rational certificate proves $2\cdot10^{12}<X<2.8\cdot10^{12}$ and $25<A<26$ , then obtains an exact lower bound for $t$ . Using $\log(1+t)\geq t/(1+t)$ gives a positive strengthened overlap margin greater than $1.51\cdot10^{-14}$ .

For $x\geq X$ , differentiation gives

xS'(x)=\frac d2x^{d/2}-\frac{22}{25} -b\left(1-\frac d2\right)x^{d/2-1} +\frac12x^{-\alpha/2}.

The same coarse bounds prove this is greater than $1.93$ . Hence $S$ is increasing after the splice. The analytic argument covers $x\geq X$ , while the published finite Oppermann result and Chamberland--Straub Proposition 3.4 cover $x<X$ . The exponent-transfer argument from q1 covers every larger $\delta$ .

Certificate

certs/rh_delta_taylor.json contains normalized rational enclosures for all logarithms and every sign used above. No decimal approximation is trusted by the authoritative verifier.