Given the Zeta function Z(s)= the summation of (1/(n^s) n=1 one to infinity and s is a complex number of the form a+bi, how would you begin to solve the equation Z(s)=0
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douglasmkilmer
ANSWERS: 1
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Is this a trick question? I tell you the exact solutions and you win the million dollar prize for solving the Riemann hypothesis? You can get approximate solutions by using numerical techniques. A crude idea: pick a complex number where you think there is a zero, evaluate the zeta function at a couple of points near that number, and see which way you have to go to reduce the value of the zeta function. Other numerical techniques such as Newton-Raphson will converge quicker. One way to evaluate 1/(n^s) is to take logs. 1/(n^s) = e^(ln (1/n^s)) = e^(- ln (n^s)) = e^(- s ln n) ln n is a real number, so Now you only need to know how to do e^(complex number) e^(a + bi) = (e^a) (e^(bi)) = (e^a) (cos (b) + i sin (b))
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