[Note] Totient summatory function in sublinear time

Original post: https://math.stackexchange.com/questions/316376/how-to-calculate-these-totient-summation-sums-efficiently

Background:

• Totient summatory function: Wolfram

Python code snippet:

class TotientSummatoryFunction():
  """Totient summatory function class."""

  def __init__(self, bound, mod):
    """Initialize.

    Args:
      bound: An integer.
      mod: An integer.
    """
    self.mod = mod
    self.small_values = None
    self.small_value_bound = None
    self.big_values = {}
    self.InitSmallValues(bound)

  def InitSmallValues(self, bound):
    """Initialize small values for totient summatory function.

    Args:
      bound: An integer.
    """
    n = 3 * int(math.pow(bound, 2 / 3))
    assert(n**3 >= bound**2)
    print('InitSmallValues({n})'.format(n=n))

    phi = [i for i in range(n + 1)]
    for i in range(2, n + 1):
      if phi[i] != i:
        continue
      for j in range(i, n + 1, i):
        phi[j] = phi[j] // i * (i - 1)

    self.small_values = [0 for i in range(n + 1)]
    self.small_values[1] = phi[1] % self.mod
    for i in range(2, n + 1):
      self.small_values[i] = (self.small_values[i - 1] + phi[i]) % self.mod
    self.small_value_bound = n

  def Get(self, n):
    """Get totient summatory function value.

    Args:
      n: An integer.

    Returns:
      An integer.
    """
    if n <= self.small_value_bound:
      return self.small_values[n]

    if n not in self.big_values:
      result = n * (n + 1) // 2
      m = self.IntegerSqrt(n)
      for d in range(1, m + 1):
        result -= self.Get(d) * (n // d - n // (d + 1))
      for k in range(2, n // (m + 1) + 1):
        result -= self.Get(n // k)
      self.big_values[n] = result % self.mod
    return self.big_values[n]

  def IntegerSqrt(self, n):
    """
    Get the integer part of Sqrt[n].

    Args:
      n: An integer.

    Returns:
      An integer.
    """
    guess = (n >> n.bit_length() // 2) + 1
    result = (guess + n // guess) // 2
    while abs(result - guess) > 1:
      guess = result
      result = (guess + n // guess) // 2
    while result * result > n:
      result -= 1
    return result