Problem 576

Irrational jumps

A bouncing point moves counterclockwise along a circle with circumference 1 with jumps of constant length $l<1$, until it hits a gap of length $g<1$, that is placed in a distance $d$ counterclockwise from the starting point. The gap does not include the starting point, that is $g+d<1$.

Let $S(l,g,d)$ be the sum of the length of all jumps, until the point falls into the gap. It can be shown that $S(l,g,d)$ is finite for any irrational jump size $l$, regardless of the values of $g$ and $d$.
Examples:
$S(\sqrt{\frac 1 2}, 0.06, 0.7)=0.7071 \dots$, $S(\sqrt{\frac 1 2}, 0.06, 0.3543)=1.4142 \dots$ and
$S(\sqrt{\frac 1 2}, 0.06, 0.2427)=16.2634 \dots$.

Let $M(n, g)$ be the maximum of $\sum S(\sqrt{\frac 1 p}, g, d)$ for all primes $p \le n$ and any valid value of $d$.
Examples:
$M(3, 0.06) =29.5425 \dots$, since $S(\sqrt{\frac 1 2}, 0.06, 0.2427)+S(\sqrt{\frac 1 3}, 0.06, 0.2427)=29.5425 \dots$ is the maximal reachable sum for $g=0.06$.
$M(10, 0.01)=266.9010 \dots$

Find $M(100, 0.00002)$, rounded to 4 decimal places.

无理数跳跃

一个点沿着周长为1的圆周逆时针跳跃，每次跳跃的长度为固定值$l<1$，直至其落入一个距离起始位置逆时针距离为$d$、长度为$g<1$的间隙中为止。这段间隙不会包括起始位置，也就是说$g+d<1$。

记$S(l,g,d)$为这个点掉入间隙之前所跳跃的总长度。可以证明,当跳跃的步长$l$是无理数时，无论$g$和$d$取什么值，$S(l,g,d)$都是有限的。
例如：
$S(\sqrt{\frac 1 2}, 0.06, 0.7)=0.7071 \dots$，$S(\sqrt{\frac 1 2}, 0.06, 0.3543)=1.4142 \dots$以及
$S(\sqrt{\frac 1 2}, 0.06, 0.2427)=16.2634 \dots$。

记$M(n, g)$为任取合理的$d$时，对所有素数$p \le n$求$\sum S(\sqrt{\frac 1 p}, g, d)$的最大值。
例如：
$M(3, 0.06) =29.5425 \dots$，因为$S(\sqrt{\frac 1 2}, 0.06, 0.2427)+S(\sqrt{\frac 1 3}, 0.06, 0.2427)=29.5425 \dots$是$g=0.06$时能达到的最大值。
$M(10, 0.01)=266.9010 \dots$

求$M(100, 0.00002)$，并保留4位小数。