Dirichlet Hyperbola Calculator

Compute exact Σ_(ab≤N) f(a)g(b) via Dirichlet hyperbola split using caller complete arithmetic sequences; unlike existing Dirichlet convolution at a single n, this sums the entire lower hyperbola.

Description

The Dirichlet hyperbola method sums products over all positive integer pairs with ab≤N.

The Dirichlet hyperbola method sums products over all positive integer pairs with ab≤N. This is a summatory convolution, rather than convolution evaluated at one index.^[1]

Compute exact Σ_(ab≤N) f(a)g(b) via Dirichlet hyperbola split using caller complete arithmetic sequences; unlike existing Dirichlet convolution at a single n, this sums the entire lower hyperbola.

When to use Dirichlet Hyperbola Calculator

Use this tool to compute exact Σ_(ab≤N) f(a)g(b) via Dirichlet hyperbola split using caller complete arithmetic sequences; unlike existing Dirichlet convolution at a single n, this sums the entire lower hyperbola.

Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.

How to use Dirichlet Hyperbola Calculator and understand its results

Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
Enter leftArithmeticValues, rightArithmeticValues, summationLimit using the field table and calculation rules, then run the tool.
Read exactSummatoryDirichletConvolution. 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
leftArithmeticValues	Left arithmetic values	list of number	Required
rightArithmeticValues	Right arithmetic values	list of number	Required
summationLimit	Summation limit	number	Required

Use integer N from 1 to 100,000 and exactly N signed integer values for each arithmetic sequence, |value|≤one million. Build exact BigInt prefix sums and evaluate the hyperbola split, returning a decimal integer string.

sum = Σ[i≤floor sqrt(N)](f_i G(floor N/i)+g_i F(floor N/i))−F(h)G(h)

Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.

Output field	Meaning	Value type
exactSummatoryDirichletConvolution	Exact summatory dirichlet convolution	string

json

{
  "input": {
    "leftArithmeticValues": [
      1,
      1,
      1,
      1
    ],
    "rightArithmeticValues": [
      1,
      1,
      1,
      1
    ],
    "summationLimit": 4
  },
  "output": {
    "exactSummatoryDirichletConvolution": "8"
  }
}

Limitations and assumptions

Both complete finite input sequences must be supplied. No values outside their declared window are inferred.

Numeric inputs and returned numeric values must be finite; an unrepresentable returned value produces an error. Numeric lists are limited to 100,000 entries; the rules above may require smaller limits. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.

Alternative or Complementary methods

Use Partition Transform Calculator: Compute exact coefficients through N of formal product ∏(1−x^k)^(-a_k), allowing signed caller exponent sequence; not ordinary unrestricted partition count or finite q product evaluation. For a complementary task, use Bell Transform Calculator: Compute complete exponential Bell-polynomial transform with b0=1 and b_n=Σ C(n−1,k−1)a_k b_(n−k); caller sequence differs materially from a single Bell number.

References

Dirichlet Hyperbola Calculator — tool.io

Similar or alternative tools

Partition Transform Calculator
Compute exact coefficients through N of formal product ∏(1−x^k)^(-a_k), allowing signed caller exponent sequence; not ordinary unrestricted partition count or finite q product evaluation.
Bell Transform Calculator
Compute complete exponential Bell-polynomial transform with b0=1 and b_n=Σ C(n−1,k−1)a_k b_(n−k); caller sequence differs materially from a single Bell number.