partial_sum_of_the_alternating_sum_of_divisors.sf

#!/usr/bin/ruby

# Author: Trizen
# Date: 11 May 2024
# https://github.com/trizen

# A sublinear formula for computing the partial sums alternating sum of divisors function (A206369), using Dirichlet's hyperbola method.

# Formula:
#
#   a(n) = Sum_{k=1..n} Sum_{d|k} (-1)^bigomega(k/d) * d
#        = Sum_{k=1..n} Sum_{d|k, d is a perfect square} phi(k/d)
#        = (1/2) * Sum_{k=1..n} (-1)^bigomega(k) * floor(n/k) * floor(n/k + 1)
#        = (Sum_{k=1..floor(n^(1/4))} S(floor(n/k^2))) + (Sum_{k=1..floor(sqrt(n))} phi(k) * floor(sqrt(floor(n/k)))) - S(floor(sqrt(n))) * floor(n^(1/4))
#
# where S(n) = Sum_{k=1..n} phi(k), where `phi` is the Euler totient function.

# Example:
#   a(10^1)  = 35
#   a(10^2)  = 3311
#   a(10^3)  = 329283
#   a(10^4)  = 32900704
#   a(10^5)  = 3289890783
#   a(10^6)  = 328986877961
#   a(10^7)  = 32898683779520
#   a(10^8)  = 3289868149892131
#   a(10^9)  = 328986813691207320
#   a(10^10) = 32898681337806276406

# Asymptotic formula due to Tóth, 2013:
#   a(n) ~ (Pi^2/30) * n^2.

# OEIS sequences:
#   https://oeis.org/A370905 -- Partial sums of the alternating sum of divisors function (A206369).
#   https://oeis.org/A002088 -- Sum of totient function: a(n) = Sum_{k=1..n} phi(k), cf. A000010.

# See also:
#   https://en.wikipedia.org/wiki/Dirichlet_hyperbola_method
#   https://trizenx.blogspot.com/2018/11/partial-sums-of-arithmetical-functions.html

func euler_totient_partial_sum (n) {      # using Dirichlet's hyperbola method

    var total = 0
    var s = n.isqrt

    for k in (1..s) {
        total += k*mertens(idiv(n,k))
    }

    s.each_squarefree {|k|
        total += moebius(k)*faulhaber(idiv(n,k), 1)
    }

    total -= mertens(s)*faulhaber(s, 1)

    return total
}

func partial_sums_of_asod_1(n) {   # sublinear complexity

    func S(n) {
        return euler_totient_partial_sum(n)
    }

    sum(1..n.iroot(4), {|k|
        S(floor(n / k**2))
    }) + sum(1..n.isqrt, {|k|
        phi(k) * isqrt(floor(n/k))
    }) - S(n.isqrt)*n.iroot(4)
}

func partial_sums_of_asod_2(n) {    # sublinear time (slow-ish)
    sum(1..n.isqrt, {|k|
        k*liouville_sum(idiv(n,k)) + liouville(k)*faulhaber_sum(idiv(n,k), 1)
    }) - liouville_sum(n.isqrt)*faulhaber_sum(n.isqrt, 1)
}

func partial_sums_of_asod_test(n) {           # just for testing
    sum(1..n, {|k| liouville(k) * faulhaber(idiv(n,k), 1) })
}

for m in (0..10) {

    var n = 1000.irand

    var t1 = partial_sums_of_asod_1(n)
    var t2 = partial_sums_of_asod_2(n)
    var t3 = partial_sums_of_asod_test(n)

    assert_eq(t1, t2, [n, t1, t2])
    assert_eq(t1, t3, [n, t1, t3])

    say "Sum_{k=1..#{n}} Sum_{d|k} λ(k/d) * d = #{t2}"
}