Function Repository Resource:

QuadraticLatticeSum

Source Notebook

Sum an expression over integer lattice points inside a quadratic region

Contributed by: Arnoud Buzing

ResourceFunction["QuadraticLatticeSum"][expr,{x₁,x₂,…,x_n},Q,L,B]

sums expr over integer lattice points inside the quadratic region .

Details and Options

The lattice points x are defined by x∈ℤⁿ.

The argument Q must be a n⨯n symmetric positive definite numeric matrix.

The argument L is a numeric vector of length n.

The argument B is a numeric scalar constant term.

Examples

Basic Examples (3)

Sum 1 over all integers satisfying x²-4<=0:

In[1]:=

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Sum 1 over all points in or on the unit disk, x²+y²<=1:

In[2]:=

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There are 5 points, namely (0,0), (-1,0), (1,0), (0,-1), (0,1):

In[3]:=

$RegionPlot[x^2 + y^2 - 1 <= 0, {x, -2, 2}, {y, -2, 2}, GridLines -> {Range[-2, 2, 1], Range[-2, 2, 1]}]$

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Compute a lattice sum over an ellipsoid:

In[4]:=

$ResourceFunction["QuadraticLatticeSum"][ x^2 - y^2, {x, y}, {{2, 0}, {0, 1}}, {0, 2}, -7]$

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Plot the ellipsoid:

In[5]:=

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Scope (2)

Compute x²+y² over lattice points defined by an ellipsoid:

In[6]:=

$ResourceFunction["QuadraticLatticeSum"][ x^2 + 2 y, {x, y}, {{2, 0}, {0, 1}}, {0, -2}, -3]$

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Plot the corresponding ellipsoid:

In[7]:=

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Compute the number of lattice points in a sphere with radius 2:

In[8]:=

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Plot the region:

In[9]:=

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Possible Issues (4)

The number of variables must match the dimensionality of the region:

In[10]:=

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You must give a non-empty list of symbols:

In[11]:=

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Coefficients must have all numerical values:

In[12]:=

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Specified matrices must be positive-definite:

In[13]:=

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Publisher

Version History

1.0.0 – 25 March 2026

Related Resources

License Information

This work is licensed under a Creative Commons Attribution 4.0 International License