Let H n ( 3 ) = k = 1 ∑ n k 3 1 . If n = 1 ∑ ∞ n 2 H n ( 3 )
is in the form c a ζ ( b ) − g π d ζ ( f )
for positive integers a , b , c , d , f and g where a and c are coprime, find a + b + c + d + f + g .
Notation : ζ ( ⋅ ) denotes the Riemann zeta function .
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The sum is n = 1 ∑ ∞ n 2 H n ( 3 ) = 1 + n = 1 ∑ ∞ ( n + 1 ) 2 H n + 1 ( 3 ) = 1 + n = 1 ∑ ∞ ( n + 1 ) 2 H n ( 3 ) + n = 1 ∑ ∞ ( n + 1 ) 5 1 = ζ ( 5 ) + σ h ( 3 , 2 ) using the standard notation for the Euler sum σ h ( m , n ) = k = 1 ∑ ∞ ( j = 1 ∑ k j m 1 ) ( k + 1 ) n 1 = k = 1 ∑ ∞ ( k + 1 ) n H k ( m ) . Now standard results tell us that σ h ( 3 , 2 ) = 2 9 ζ ( 5 ) − 2 ζ ( 2 ) ζ ( 3 ) and hence the sum is n = 1 ∑ ∞ n 2 H n ( 3 ) = 2 1 1 ζ ( 5 ) − 2 ζ ( 2 ) ζ ( 3 ) = 2 1 1 ζ ( 5 ) − 3 1 π 2 ζ ( 3 ) making the answer 1 1 + 5 + 2 + 2 + 3 + 3 = 2 6 .