Number of Lattice Points in a Hyperball

Let $T(r)$ be the number of integer quadruplets $x, y, z, t$ such that $x^2 + y^2 + z^2 + t^2 \le r^2$. In other words, $T(r)$ is the number of lattice points in the four-dimensional hyperball of radius $r$.
You are given that $T(2) = 89$, $T(5) = 3121$, $T(100) = 493490641$ and $T(10^4) = 49348022079085897$.
Find $T(10^8) \bmod 1000000007$.

To find the value of $T(10^8) \bmod 1000000007$, we can use the given values of $T(r)$ for $r = 2, 5, 100$ and $10^4$ to estimate the value of $T(10^8)$.

The problem can be solved using a modified version of the principle of inclusion-exclusion applied to the four-dimensional hyperball. We will count the number of lattice points in the hyperball centered at the origin with radius $r$, and subtract the number of lattice points outside the hyperball. We will use the formula for the number of lattice points in a hyperball given by:

$$T(r) = (2r+1) \cdot \text{vol}(r)$$

where $\text{vol}(r)$ is the volume of the four-dimensional hyperball of radius $r$. The volume of a four-dimensional hyperball can be calculated as:

$$\text{vol}(r) = \frac{1}{2} \times \frac{\pi^2}{2} \times \frac{\pi^2}{2} \times r^4$$

Now, let’s write Python code to calculate the value of $T(10^8) \bmod 1000000007$:

“`python
import math

def T(r):
vol = 0.5 * (math.pi ** 2 / 2) ** 2 * (r ** 4)
num_points = (2 * r + 1) * int(vol)
return num_points

T_2 = 89
T_5 = 3121
T_100 = 493490641
T_10_4 = 49348022079085897

T_10_8 = T(10**8) % 1000000007

print(T_10_8)
“`

Running this code will output the value of $T(10^8) \bmod 1000000007$.

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