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Fixed Point Theory and Algorithms for Sciences and Engineering

Table 1 Identities of divisor functions

From: Convolution identities for twisted Eisenstein series and twisted divisor functions

Identities of convolution sums

Reference

∑ s = 0 k − 1 ( 2 k 2 s + 1 ) ( ∑ m = 1 N − 1 σ 2 k − 2 s − 1 ( m ) σ 2 s + 1 ( N − m ) ) = 2 k + 3 4 k + 2 σ 2 k + 1 ( N ) + ( k 6 − N ) σ 2 k − 1 ( N ) + 1 2 k + 1 ∑ j = 2 k ( 2 k + 1 2 j ) B 2 j σ 2 k + 1 − 2 j ( N )

[[2], Theorem 12.3]

∑ s = 0 k − 1 ( 2 k 2 s + 1 ) ( ∑ m = 1 N − 1 σ 2 k − 2 s − 1 ∗ ( m ) σ 2 s + 1 ∗ ( N − m ) ) = 1 2 { σ 2 k + 1 ∗ ( N ) − N σ 2 k − 1 ∗ ( N ) }

(10)

∑ s = 0 k − 1 ( 2 k 2 s + 1 ) ( ∑ m = 1 N σ 2 k − 2 s − 1 ( 2 m − 1 ) σ 2 s + 1 ( 2 N − 2 m + 1 ) ) = 1 4 σ 2 k + 1 ∗ ( 2 N )

(15)

∑ s = 0 k − 1 ( 2 k 2 s + 1 ) ∑ m = 1 2 N − 1 ( − 1 ) m + 1 σ 2 k − 2 s − 1 ∗ ( m ) σ 2 s + 1 ∗ ( 2 N − m ) = 1 2 ( 2 N ⋅ σ 2 k − 1 ∗ ( 2 N ) )

(16)

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