For a positive real number x>1, the Riemann zeta function ζ(x) is defined by ζ(x)=n=1∑∞nx1.Compute k=2∑∞{ζ(2k−1)}.Note: For a real number x,{x} denotes the fractional part of x.
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For x≥2,ζ(x)=1+2x1+3x1+⋯≤1+221+321+⋯<1+1⋅21+2⋅31+⋯=1+(1−21)+(21−31)+⋯=2,so ⌊ζ(x)⌋=1. Then {ζ(x)}=ζ(x)−1.Thus, we want to sum k=2∑∞(ζ(2k−1)−1)=k=2∑∞n=2∑∞n2k−11.We switch the order of summation, to get n=2∑∞k=2∑∞n2k−11=n=2∑∞(n31+n51+n71+⋯)=n=2∑∞1−1/n21/n3=n=2∑∞n3−n1.By partial fractions, n3−n1=n−11/2−n1+n+11/2.Therefore, n=2∑∞n3−n1=n=2∑∞(n−11/2−n1+n+11/2)=(11/2−21+31/2)+(21/2−31+41/2)+(31/2−41+51/2)+⋯=11/2−21+21/2=41.