仕事の部屋: 160619

Ruby

Glaisher's chi_4(n)(1)

ラマヌジャン《ゼータ関数論文集》
を見ていると、chi_4(n) について以下の式が載っていた。

chi_4(n) = f(x^2)^14 / f(-x)^4
（ただし、f(x) = x^(1 / 24) (1 - x)(1 - x^2)(1 - x^3)…）

整理すれば、
φ(x) φ(-x) φ(x^4) = φ(x^2)^3
（ただし、φ(x) はオイラー関数）
を示せばいいだけだが、
あえて
φ(x)^4 φ(x^2)^2 φ(x^4)^4 φ(-x)^4 = φ(x^2)^14
を確認することにした。

# m次以下を取り出す
def mul(f_ary, b_ary, m)
  s1, s2 = f_ary.size, b_ary.size
  ary = Array.new(s1 + s2 - 1, 0)
  (0..s1 - 1).each{|i|
    (0..s2 - 1).each{|j|
      ary[i + j] += f_ary[i] * b_ary[j]
    }
  }
  ary[0..m]
end

def p0(n)
  (3 * n * n - n) / 2
end

def p1(n)
  (3 * n * n + n) / 2
end

def A010815(n)
  ary = Array.new(n + 1, 0)
  ary[0] = 1
  i = 1
  j = p0(i)
  while j <= n
    ary[j] = (-1) ** i
    i += 1
    j = p0(i)
  end
  i = 1
  j = p1(i)
  while j <= n
    ary[j] = (-1) ** i
    i += 1
    j = p1(i)
  end
  ary
end

# m次以下を取り出す
def power(ary, n, m)
  return [1] if n == 0
  k = power(ary, n >> 1, m)
  k = mul(k, k, m)
  return k if n & 1 == 0
  return mul(k, ary, m)
end

def A(n)
  ary = A010815(n)
  ary0 = (0..n).map{|i| (-1) ** i * ary[i]}
  ary2 = Array.new(n + 1, 0)
  (0..n / 2).each{|i| ary2[2 * i] = ary[i]}
  ary4 = Array.new(n + 1, 0)
  (0..n / 4).each{|i| ary4[4 * i] = ary[i]}
  ary04 = power(ary0, 4, n)
  ary14 = power(ary, 4, n)
  ary22 = mul(ary2, ary2, n)
  ary44 = power(ary4, 4, n)
  ary = mul(ary14, ary22, n)
  ary = mul(ary, ary44, n)
  mul(ary, ary04, n)
end

def B(n)
  ary = A010815(n)
  ary2 = Array.new(n + 1, 0)
  (0..n / 2).each{|i| ary2[2 * i] = ary[i]}
  power(ary2, 14, n)
end

n = 1000
p a = A(n)
p b = B(n)
p a == b

出力結果
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true

仕事の部屋

2016年6月19日日曜日

160619

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