log-2-x-1-2015-y-1-2015-1-e-2-ixy-2015-

Question Number 7159 by Master Moon last updated on 14/Aug/16

{l o g}_{2} \underset{x = 1}{\prod^{2015}} \underset{y = 1}{\prod^{2015}} (1 + e^{\frac{2 π i x y}{2015}}) = ?

Commented by FilupSmith last updated on 14/Aug/16

S = \log_{2} (\underset{x = 1}{\prod^{2015}} \underset{y = 1}{\prod^{2015}} (1 + e^{\frac{2 π i x y}{2015}})) = \log_{2} (P)

P = \underset{x = 1}{\prod^{2015}} (\underset{y = 1}{\prod^{2015}} (1 + e^{\frac{2 π i x y}{2015}}))

{(- 1)}^{k} = e^{i π k} = {(e^{i π})}^{k}

P = \underset{x = 1}{\prod^{2015}} (\underset{y = 1}{\prod^{2015}} (1 + {(- 1)}^{\frac{2 x y}{2015}}))

S \in C \Leftrightarrow P \notin Z

trying to think how to continue

Commented by Yozzia last updated on 14/Aug/16

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L e t θ = \frac{x y π}{2015} \Rightarrow u (x, y) = 1 + e^{\frac{2 π i x y}{2015}} = 1 + c o s 2 θ + i s i n 2 θ

u (x, y) = 2 {c o s}^{2} θ + 2 i s i n θ c o s θ

u (x, y) = 2 c o s θ (c o s θ + i s i n θ)

u (x, y) = (2 c o s θ) e^{i θ} = {2 c o s \frac{x y π}{2015}} e^{\frac{π x y i}{2015}}

\Rightarrow \underset{y = 1}{\prod^{2015}} u (x, y) = 2^{2015} (\underset{y = 1}{\prod^{2015}} c o s \frac{x y π}{2015}) e^{\frac{x π i}{2015} (1 + 2 + 3 + 4 + \dots + 2015)}

\underset{y = 1}{\prod^{2015}} u (x, y) = 2^{2015} [\underset{y = 1}{\prod^{2015}} c o s \frac{x y π}{2015}] e^{\frac{x π i}{2015} \times \frac{2015 \times 2016}{2}}

\underset{y = 1}{\prod^{2015}} u (x, y) = 2^{2015} [\underset{y = 1}{\prod^{2015}} c o s \frac{x y π}{2015}] e^{1008 π i x}

x \in N \Rightarrow e^{1008 π i x} = c o s 1008 π x + 0 = c o s 1008 π x = 1

\Rightarrow \underset{y = 1}{\prod^{2015}} u (x, y) = 2^{2015} [\underset{y = 1}{\prod^{2015}} c o s \frac{x y π}{2015}]

\underset{x = 1}{\prod^{2015}} (\underset{y = 1}{\prod^{2015}} u (x, y)) = \underset{x = 1}{\prod^{2015}} [2^{2015} \underset{y = 1}{\prod^{2015}} c o s \frac{π x y}{2015}]

\underset{x = 1}{\prod^{2015}} (\underset{y = 1}{\prod^{2015}} u (x, y)) = 2^{2015^{2}} \underset{x = 1}{\prod^{2015}} [\underset{y = 1}{\prod^{2015}} c o s \frac{π x y}{2015}]

\underset{x = 1}{\prod^{2015}} (\underset{y = 1}{\prod^{2015}} u (x, y)) = 2^{2015^{2}} \underset{y = 1}{\prod^{2015}} [\underset{x = 1}{\prod^{2015}} c o s \frac{π x y}{2015}]

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