Harmonic number and euler-maclaurin summation formula

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Reference no: EM132849768

Question 1: For a positive integer n the n-th Harmonic number H_n is defined to be

H_n: ∑_n=1ⁿ 1/n = 1+ 1/2 + 1/3 +..... + 1/n.

Show that

log n < H_n < log n + 1.

Question 2: Show that ∑^∞_n=1(-1)^n-1/n = log(2).

Question 3: For σ > 1, t ≥ 2, X ≥ 2, use Euler-Maclaurin summation formula to show that

ζ (s) = ∑_n≤X 1/n^s + 1/((s - 1)X^s-1) + (X - [X])/X^s - s _x∫^∞(x - [x])/x^s+1 dx.

Question 4. Define Λ₁(n) = 1/k, if m = p^k; 0, otherwise. Prove that

∑_m_≤X Λ₁(m) = Π(x) + O(x^1/2 log x).

Reference no: EM132849768

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