Let $N(\alpha, n)$ denote the number of integer points inside the origin-centered ball of radius $\alpha \sqrt n$ in $n$ dimensions, where $\alpha \in (0,\infty)$ is some fixed constant. Precisely:

$$N(\alpha, n) := |\{z\in \mathbb{Z}^n:\|z\|_2 \le \alpha \sqrt n\}|.$$

I am curious about what is known regarding explicit formulae for: $$\lim_{n \to \infty} \frac 1 n\log N(\alpha, n) $$

There is a classical paper of J.E. Mazo and A.M. Odlyzko (1990) called "lattice points in high dimensional spheres" (I will link in the comments; not sure if I am allowed to add link in the question). This paper gives an explicit asymptotic in equation 3.4, but it is given in terms of fixed points of some transcendental equations.

It seems sort of natural that this is the case. Nonetheless, I am wondering: are any alternate derivations or identities known for this asymptotic?

Some concrete examples of what I am looking for:

• Are there any specific (non-trivial) values of $\alpha$. for which this formula can be evaluated/understood?
• Are there any probablistic interpretations (since the discrete Gaussian seems to arise in the mentioned formula)?
• Is there some natural ``positive temperature'' relaxation of the criteria $\|z\|_2 \le \alpha \sqrt n$ that lends itself to some heuristic computation via statistical physics methods (cavity or replica)?

Let $N(\alpha, n)$ denote the number of integer points inside the origin-centered ball of radius $\alpha \sqrt n$ in $n$ dimensions, where $\alpha \in (0,\infty)$ is some fixed constant. Precisely:

$$N(\alpha, n) := |\{z\in \mathbb{Z}^n:\|z\|_2 \le \alpha \sqrt n\}|.$$

I am curious about what is known regarding explicit formulae for: $$\lim_{n \to \infty} \frac 1 n\log N(\alpha, n). $$

There is a classical paper of J.E. Mazo and A.M. Odlyzko (1990) called "Lattice points in high dimensional spheres". This paper gives an explicit asymptotic in equation 3.4, but it is given in terms of fixed points of some transcendental equations.

It seems sort of natural that this is the case. Nonetheless, I am wondering: are any alternate derivations or identities known for this asymptotic?

Some concrete examples of what I am looking for:

• Are there any specific (non-trivial) values of $\alpha$ for which this formula can be evaluated/understood?
• Are there any probablistic interpretations (since the discrete Gaussian seems to arise in the mentioned formula)?
• Is there some natural “positive temperature” relaxation of the criterion $\|z\|_2 \le \alpha \sqrt n$ that lends itself to some heuristic computation via statistical physics methods (cavity or replica)?

Let $N(\alpha, n)$ denote the number of integer points inside the origin-centered ball of radius $\alpha \sqrt n$ in $n$ dimensions, where $\alpha \in (0,\infty)$ is some fixed constant. Precisely:

$$N(\alpha, n) := |\{z\in \mathbb{Z}^n:\|z\|_2 \le \alpha \sqrt n\}|.$$

I am curious about what is known regarding explicit formulae for: $$\lim_{n \to \infty} \frac 1 n\log N(\alpha, n) $$

There is a classical paper of J.E. Mazo and A.M. Odlyzko (1990) called "lattice points in high dimensional spheres" (I will link in the comments; not sure if I am allowed to add link in the question). This paper gives an explicit asymptotic in equation 3.4, but it is given in terms of fixed points of some transcendental equations.

It seems sort of natural that this is the case. Nonetheless, I am wondering: are any alternate derivations or identities known for this asymptotic?

Some concrete examples of what I am looking for:

• Are there any specific (non-trivial) values of $\alpha$. for which this formula can be evaluated/understood?
• Are there any probablistic interpretations (since the discrete Gaussian seems to arise in the mentioned formula)?
• Is there some natural ``positive temperature'' relaxation of the criteria $\|z\|_2 = \alpha \sqrt n$ that lends itself to some heuristic computation via statistical physics methods (cavity or replica)?

Let $N(\alpha, n)$ denote the number of integer points inside the origin-centered ball of radius $\alpha \sqrt n$ in $n$ dimensions, where $\alpha \in (0,\infty)$ is some fixed constant. Precisely:

$$N(\alpha, n) := |\{z\in \mathbb{Z}^n:\|z\|_2 \le \alpha \sqrt n\}|.$$

I am curious about what is known regarding explicit formulae for: $$\lim_{n \to \infty} \frac 1 n\log N(\alpha, n) $$

There is a classical paper of J.E. Mazo and A.M. Odlyzko (1990) called "lattice points in high dimensional spheres" (I will link in the comments; not sure if I am allowed to add link in the question). This paper gives an explicit asymptotic in equation 3.4, but it is given in terms of fixed points of some transcendental equations.

It seems sort of natural that this is the case. Nonetheless, I am wondering: are any alternate derivations or identities known for this asymptotic?

Some concrete examples of what I am looking for:

• Are there any specific (non-trivial) values of $\alpha$. for which this formula can be evaluated/understood?
• Are there any probablistic interpretations (since the discrete Gaussian seems to arise in the mentioned formula)?
• Is there some natural ``positive temperature'' relaxation of the criteria $\|z\|_2 \le \alpha \sqrt n$ that lends itself to some heuristic computation via statistical physics methods (cavity or replica)?

Let $N(\alpha, n)$ denote the number of integer points on the surface of the origin-centered sphere of radius $\alpha \sqrt n$ in $n$ dimensions, where $\alpha \in (0,\infty)$ is some fixed constant. Precisely:

$$N(\alpha, n) := |\{z\in \mathbb{Z}^n:\|z\|_2 = \alpha \sqrt n\}|.$$

I am curious about what is known regarding explicit formulae for: $$\lim_{n \to \infty} \frac 1 n\log N(\alpha, n) $$

There is a classical paper of J.E. Mazo and A.M. Odlyzko (1990) called "lattice points in high dimensional spheres" (I will link in the comments; not sure if I am allowed to add link in the question). This paper gives an explicit asymptotic in equation 3.4, but it is given in terms of fixed points of some transcendental equations.

It seems sort of natural that this is the case. Nonetheless, I am wondering: are any alternate derivations or identities known for this asymptotic?

Some concrete examples of what I am looking for:

• Are there any specific (non-trivial) values of $\alpha$. for which this formula can be evaluated/understood?
• Are there any probablistic interpretations (since the discrete Gaussian seems to arise in the mentioned formula)?
• Is there some natural ``positive temperature'' relaxation of the criteria $\|z\|_2 = \alpha \sqrt n$ that lends itself to some heuristic computation via statistical physics methods (cavity or replica)?

Let $N(\alpha, n)$ denote the number of integer points inside the origin-centered ball of radius $\alpha \sqrt n$ in $n$ dimensions, where $\alpha \in (0,\infty)$ is some fixed constant. Precisely:

$$N(\alpha, n) := |\{z\in \mathbb{Z}^n:\|z\|_2 \le \alpha \sqrt n\}|.$$

I am curious about what is known regarding explicit formulae for: $$\lim_{n \to \infty} \frac 1 n\log N(\alpha, n) $$

There is a classical paper of J.E. Mazo and A.M. Odlyzko (1990) called "lattice points in high dimensional spheres" (I will link in the comments; not sure if I am allowed to add link in the question). This paper gives an explicit asymptotic in equation 3.4, but it is given in terms of fixed points of some transcendental equations.

It seems sort of natural that this is the case. Nonetheless, I am wondering: are any alternate derivations or identities known for this asymptotic?

Some concrete examples of what I am looking for:

• Are there any specific (non-trivial) values of $\alpha$. for which this formula can be evaluated/understood?
• Are there any probablistic interpretations (since the discrete Gaussian seems to arise in the mentioned formula)?
• Is there some natural ``positive temperature'' relaxation of the criteria $\|z\|_2 = \alpha \sqrt n$ that lends itself to some heuristic computation via statistical physics methods (cavity or replica)?