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  3. please prove number 1 and 2 with good clear notes...

Question: please prove number 1 and 2 with good clear notes...

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A point (a, b) on the xy-plane is called a lattice point if a, b E Z 1. Let f() be a continuous and non-negative function in the interval QsrsR Prove that the sum Σ If(n)] is equal to the number of lattice points in the plane region Q<sR, 0<ys f(x) 2. Suppose P and Q are positive odd integers such that (P,Q)-1. Prove that Pnl P-1 Q-1 0<m<P/2 3. Let X 1. Prove that VXI (Hint: First verify that the left side is equal to the number of lattice points in the plane region >0, y>0, ry s X.) 4. Prove that for any α E R and c E N [뷔-[2] Please prove number 1 and 2 with good clear notes. Please do not use other's work.

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