Estimating Zeta

The Riemann zeta function is defined as <br />

ζ ( s ) = n = 1 1 n s = 1 1 s + 1 2 s + 1 3 s + . . . \zeta(s) = \sum_{n=1}^{\infty}\frac{1}{n ^ s} = \frac{1}{1 ^ s} + \frac{1}{2 ^s} + \frac{1}{3^s} + ... <br />

We can estimate Riemann zeta values by using large numbers in place of infinity as the upper limit of the series.

Let Z be an estimation of 100 6 ζ ( 2 ) 100\sqrt{6\zeta(2)} , calculated using 1000000 as infinity.

What is Z , rounded to the nearest integer?


The answer is 314.

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1 solution

The answer is 314 .

First, use iteration to calculate zeta(2). Then, multiply that value by 6, take the square root of the product, and finally multiply the result by 100.

# Python solution
def zeta(s): 
    sum = 0.0 

    # use 1000000 to approximate infinity
    for i in range(1000000): 
        sum+=1.0/(i + 1)**2
    return sum 

import math
# compute Z
print 100 * math.sqrt(6 * zeta(2))

Did the answer look familiar? It turns out that the sum of the squares of the reciprocals of the natural numbers is π 2 6 \dfrac{\pi^2}{6} .

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