Gauss Sum Approximation
Evaluate finite quadratic Gauss sum Σ exp(2πi(ax²+bx)/m) over all residues, preserving complex output and exact BigInt phase reduction; distinct from Kloosterman inverse-residue phase.
Description
A quadratic Gauss sum adds complex phases over every residue class.
A quadratic Gauss sum adds complex phases over every residue class. It differs from an inverse-residue sum such as the Kloosterman sum.1
Evaluate finite quadratic Gauss sum Σ exp(2πi(ax²+bx)/m) over all residues, preserving complex output and exact BigInt phase reduction; distinct from Kloosterman inverse-residue phase.
When to use Gauss Sum Approximation
Use this tool to evaluate finite quadratic Gauss sum Σ exp(2πi(ax²+bx)/m) over all residues, preserving complex output and exact BigInt phase reduction; distinct from Kloosterman inverse-residue phase.
Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
How to use Gauss Sum Approximation and understand its results
- Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
- Enter quadraticCoefficient, linearCoefficient, modulus using the field table and calculation rules, then run the tool.
- Read quadraticGaussSumRealPart, quadraticGaussSumImaginaryPart. The explanation below defines the units, signs, ordering and any flags or placeholder values.
- If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.
| Input field | Meaning | Value type; see rules below | Requirement |
|---|---|---|---|
| quadraticCoefficient | Quadratic coefficient | number | Required |
| linearCoefficient | Linear coefficient | number | Required |
| modulus | Modulus | number | Required |
Enter integer modulus 1–10,000 and signed integer coefficients of magnitude ≤one million. Reduce each polynomial phase modulo m with BigInt, then accumulate floating cosine and sine values. Return both components; modulus one gives real value 1.
sum = Σ[r=0..m−1]exp(2πi(ar²+br)/m) Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.
| Output field | Meaning | Value type |
|---|---|---|
| quadraticGaussSumRealPart | Quadratic gauss sum real part | number |
| quadraticGaussSumImaginaryPart | Quadratic gauss sum imaginary part | number |
{
"input": {
"quadraticCoefficient": 1,
"linearCoefficient": 0,
"modulus": 4
},
"output": {
"quadraticGaussSumRealPart": 2,
"quadraticGaussSumImaginaryPart": 2
}
}Limitations and assumptions
Exact phase reduction does not make the trigonometric accumulation exact. Small imaginary roundoff may remain.
Numeric inputs and returned numeric values must be finite; an unrepresentable returned value produces an error. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.
Alternative or Complementary methods
Use Kloosterman Sum Approximation: Evaluate finite Kloosterman exponential sum over invertible residues modulo caller m, using exact modular inverses and floating complex phases; nonunits are omitted, not assigned inverses. For a complementary task, use Jacobi Theta Approximation: Evaluate specifically real Jacobi θ3(z,q)=1+2Σ q^(n²)cos(2nz) with a geometric upper tail bound; theta index/real-nome convention explicit, not all theta functions or complex continuation.
References
-
Gauss Sum Approximation — tool.io
Similar or alternative tools
- Kloosterman Sum Approximation
Evaluate finite Kloosterman exponential sum over invertible residues modulo caller m, using exact modular inverses and floating complex phases; nonunits are omitted, not assigned inverses.
- Jacobi Theta Approximation
Evaluate specifically real Jacobi θ3(z,q)=1+2Σ q^(n²)cos(2nz) with a geometric upper tail bound; theta index/real-nome convention explicit, not all theta functions or complex continuation.