partial_sums_of_euler_totient_function.sf

#!/usr/bin/ruby

# Author: Daniel "Trizen" Șuteu
# Date: 20 November 2018
# Edit: 06 June 2021
# https://github.com/trizen

# A new algorithm for computing the partial-sums of `ϕ(k)`, for `1 <= k <= n`:
#
#   Sum_{k=1..n} phi(k)
#
# where phi(k) is the Euler totient function.

# Based on the formula:
#   Sum_{k=1..n} phi(k) = (1/2)*Sum_{k=1..n} moebius(k) * floor(n/k) * floor(1+n/k)

# Example:
#   a(10^1) = 32
#   a(10^2) = 3044
#   a(10^3) = 304192
#   a(10^4) = 30397486
#   a(10^5) = 3039650754
#   a(10^6) = 303963552392
#   a(10^7) = 30396356427242
#   a(10^8) = 3039635516365908
#   a(10^9) = 303963551173008414

# This algorithm can be improved.

# See also:
#   https://oeis.org/A002088
#   https://oeis.org/A064018
#   https://en.wikipedia.org/wiki/Mertens_function
#   https://en.wikipedia.org/wiki/M%C3%B6bius_function
#   https://en.wikipedia.org/wiki/Euler%27s_totient_function
#   https://en.wikipedia.org/wiki/Dirichlet_hyperbola_method
#   https://trizenx.blogspot.com/2018/08/interesting-formulas-and-exercises-in.html

func euler_totient_partial_sum (n) {

    var total = 0

    var s = n.isqrt
    var u = idiv(n, s+1)

    var prev = mertens(n)

    for k in (1..s) {
        with (mertens(idiv(n, k+1))) {|curr|
            total += ((prev - curr) * faulhaber(k, 1))
            prev = curr
        }
    }

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

    return total
}

func euler_totient_partial_sum_2 (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 euler_totient_partial_sum_test (n) {    # just for testing
    1..n -> sum { .euler_phi }
}

for m in (0 .. 10) {

    var n = 10000.irand

    var t1 = euler_totient_partial_sum(n)
    var t2 = euler_totient_partial_sum_2(n)
    var t3 = euler_totient_partial_sum_test(n)

    assert_eq(t1, t2)
    assert_eq(t1, t3)

    say "Sum_{k=1..#{n}} phi(k) = #{t1}"
}

__END__
Sum_{k=1..6000} phi(k) = 10943164
Sum_{k=1..9647} phi(k) = 28292296
Sum_{k=1..1767} phi(k) = 949684
Sum_{k=1..4643} phi(k) = 6554816
Sum_{k=1..5814} phi(k) = 10276132
Sum_{k=1..3120} phi(k) = 2959328
Sum_{k=1..1998} phi(k) = 1213790
Sum_{k=1..1269} phi(k) = 489854
Sum_{k=1..1298} phi(k) = 512500
Sum_{k=1..6555} phi(k) = 13062444
Sum_{k=1..1101} phi(k) = 368850