Can we compute the exact value of the sum S = <munderover> &#x2211;<!-- ∑ --> <mrow

hornejada1c

hornejada1c

Answered question

2022-07-06

Can we compute the exact value of the sum
S = n = 1 1 n sinh ( π n / 4 ) .

Answer & Explanation

sniokd

sniokd

Beginner2022-07-07Added 22 answers

For ( x ) > 0
f ( x ) = n 1 1 n ( e n x e n x ) = n 1 1 n k 0 e ( 2 k + 1 ) n x = k 0 log ( 1 e ( 2 k + 1 ) x )
For ( z ) > 0 we consider the modular discriminant
Δ ( z ) = ( 2 π ) 12 e 2 i π z k 1 ( 1 e 2 i π k z ) 24
log Δ ( z ) Δ ( 2 z ) = 2 i π z + 24 k 0 log ( 1 e 2 i π ( 2 k + 1 ) z ) = 2 i π z 24 f ( 2 i π z )
So that
S = 2 f ( π / 4 ) = 1 12 ( log Δ ( i / 8 ) Δ ( i / 4 ) π / 4 )
Using Δ ( z ) = z 12 Δ ( 1 / z ) (weight 12 cusp form) we get
S = 1 12 ( log 2 12 Δ ( 8 i ) Δ ( 4 i ) π / 4 ) = 1 12 ( 24 log η ( 8 i ) η ( 4 i ) + 12 log 2 π / 4 )

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