Exact Dedekind Sum

Compute s(h,k)=Σ(r=1…k−1)((r/k))((hr/k)) as a reduced fraction and finite value, using the zero-at-integers sawtooth convention and k≤10,000.

Description

A Dedekind sum combines two periodic sawtooth functions.

A Dedekind sum combines two periodic sawtooth functions. The convention that the sawtooth is zero at integers changes which summands contribute.^[1]

The tool returns numerator, denominator, value. The indexing and output conventions below determine how to read these values.

When to use Exact Dedekind Sum

Use this for the following task: Compute s(h,k)=Σ(r=1…k−1)((r/k))((hr/k)) as a reduced fraction and finite value, using the zero-at-integers sawtooth convention and k≤10,000. Use the input and output conventions below to reproduce the intended mathematical quantity.

How to use Exact Dedekind Sum and understand its results

Choose the required indexes, integers or sequence entries. Preserve decimal-string encodings where the field table specifies strings.
Enter h, k according to the field table, then run the tool.
Read numerator, denominator, value. Preserve exact decimal strings and the stated indexing or coefficient order.
Keep exact decimal strings for subsequent integer arithmetic; distinguish them from any explicitly approximate numeric result.

Input field	Meaning	Value type and bounds	Requirement
h	Signed integer multiplier	string (integer/arbitrary); maximum text length 200; pattern ^-?(0\|[1-9][0-9]*)$	Required
k	Positive denominator (summation resource limit 10000)	integer; minimum 1; maximum 10000	Required

Normalize signed h modulo positive k, sum scaled BigInt numerators for nonzero residues and reduce the fraction by gcd. Return exact strings and a floating quotient.

s(h,k)=Σ[r=1..k−1]((r/k))((h×r/k)); ((x))=fractionalPart(x)−1/2 unless x is integral

The symbols and boundary conditions refer to the supplied fields and the implementation described above.

Output field	Meaning	Value type
numerator	Reduced Dedekind sum numerator	string (integer/arbitrary)
denominator	Reduced Dedekind sum denominator	string (integer/arbitrary)
value	Dedekind sum value	number

json

{
  "input": {
    "h": "1",
    "k": 5
  },
  "output": {
    "numerator": "1",
    "denominator": "5",
    "value": 0.2
  }
}

Limitations and assumptions

The denominator k is at most 10000 and need not be coprime with h. The floating value is approximate; the reduced fraction preserves the chosen convention.

Use the declared integer bounds and string formats. Large exact outputs remain decimal strings; numerical approximations use ordinary floating-point arithmetic. A rejected input is different from a valid zero count or false membership flag.

Alternative or Complementary methods

Choose Jacobi Symbol when you need its operation: Return (a/n) for a signed decimal numerator and positive odd decimal denominator, including (a/1)=1; a value of one does not prove a quadratic residue for composite n. Choose Modular Inverse for the complementary operation: Find the least nonnegative inverse of a signed decimal integer modulo a decimal modulus of at least two, rejecting noncoprime pairs.

Background reference: Arithmetic functions: dedekind sum.

References

Arithmetic functions: dedekind sum — SageMath developers

Similar or alternative tools

Jacobi Symbol
Return (a/n) for a signed decimal numerator and positive odd decimal denominator, including (a/1)=1; a value of one does not prove a quadratic residue for composite n.
Modular Inverse
Find the least nonnegative inverse of a signed decimal integer modulo a decimal modulus of at least two, rejecting noncoprime pairs.