Pattern in Bessel J_0 zeros divided by pi $x_{0n}/\pi$.

Question

I began by wondering if any zeros of Bessel function of the first kind $J_m(x)$ were multiples of $\pi$. Having no real idea on how to answer that, I conjectured that the first several zeros wouldn't be (or else I would have heard about it), and so decided to do a numerical investigation.

Table[N[BesselJZero[0, n]/Pi, 100], {n, 1, 100}]

0.76547974956201235684194840406615
1.75709543501086983306413279034849
2.75456714702419669134233661494223
3.75336198521488396584959236737820
.
.
.
99.7501269680668222195941884899656

Here I truncated the results. Out of the first hundred zeros, none appeared to be a candidate multiple of $\pi$, but there was an interesting pattern.

For all but the first zero, I noticed that the mantissa began with the decimal digits $75$. More importantly, the zeros, when normalized by $pi$, appear almost evenly spaced. Any reason?

Answer 1

I think that you could be interested by this paper where the author propose, as an approximation of the zeros of $J_0(x)$

$$\color{red} {x_m=p+\frac{1}{8 p}-\frac{31}{384 p^3}+\frac{3779}{15360 p^5}-\frac{6277237}{3440640 p^7}+\cdots}$$ where $\color{blue} {p=\pi \left(m-\frac{1}{4}\right)}$.

The next table compares the "exact" values and the approximated ones $$\left( \begin{array}{ccc} m & \text{approx } x_m & \text{exact } x_m\\ 1 & 2.401937102 & 2.404825558 \\ 2 & 5.520074720 & 5.520078112 \\ 3 & 8.653727839 & 8.653727913 \\ 4 & 11.79153443 & 11.79153444 \\ 5 & 14.93091771 & 14.93091771 \\ 6 & 18.07106397 & 18.07106397 \\ 7 & 21.21163663 & 21.21163663 \\ 8 & 24.35247153 & 24.35247153 \\ 9 & 27.49347913 & 27.49347913 \\ 10 & 30.63460647 & 30.63460647 \end{array} \right)$$

Setting $y_m=\frac{x_m} \pi$, the table becomes $$\left( \begin{array}{ccc} m & \text{approx } y_m & \text{exact } y_m\\ 1 & 0.764560326 & 0.765479750 \\ 2 & 1.757094356 & 1.757095436 \\ 3 & 2.754567123 & 2.754567147 \\ 4 & 3.753361984 & 3.753361985 \\ 5 & 4.752658716 & 4.752658716 \\ 6 & 5.752198315 & 5.752198315 \\ 7 & 6.751873641 & 6.751873641 \\ 8 & 7.751632441 & 7.751632441 \\ 9 & 8.751446213 & 8.751446213 \\ 10 & 9.751298098 & 9.751298098 \end{array} \right)$$

which confirms your interesting observations.

If you have a look here, on page 371, you could find similar things for other Bessel functions.