Growth of partial sums of divergent series

Source abstract

Let Σ f ( n ) \Sigma f(n) be a divergent series of decreasing positive terms, with partial sums s n {s_n} , where f decreases sufficiently smoothly; let φ ( x ) = ∫ 1 x f ( t ) d t \varphi (x) = \smallint _1^xf(t)dt and let ψ \psi be the inverse of φ \varphi . Let n A {n_A} be the smallest integer n such that s n ⩾ A {s_n} \geqslant A but s n − 1 > A ( A = 2 , 3 , … ) {s_{n - 1}} > A(A = 2,3, \ldots ) ; let γ = lim { Σ 1 n f ( k ) − φ ( n ) } \gamma = \lim \{ \Sigma _1^nf(k) - \varphi (n)\} be the analog of Euler’s constant; let m = [ ψ ( A − γ ) ] m = [\psi (A - \gamma )] . Call ω \omega a Comtet function for Σ f ( n ) \Sigma f(n) if n A = m {n_A} = m when the fractional part of ψ ( A − γ ) \psi (A - \gamma ) is less than ω ( A ) \omega (A) and n A = m + 1 {n_A} = m + 1 when the fractional part of ψ ( A − γ ) \psi (A - \gamma ) is greater than ω ( A ) \omega (A) . It has been conjectured that ω ( A ) = 1 / 2 \omega (A) = 1/2 is a Comtet function for Σ 1 / n \Sigma 1/n . It is shown that in general there is a Comtet function of the form $$ω ( A ) = 1 2 + 1 24 { | f ′ ( m ) | / f ( m ) } ( 1 + o ( 1 ) ) . \omega (A) = \frac {1}{2} + \frac {1}{24} \left \{ |f\prime (m)|/f(m) \right \} (1 + o(1)).$$ For Σ 1 / n \Sigma 1/n there is a Comtet function of the form 1 / 2 + 1 / ( 24 ) { 1 / ( 48 m 2 ) } ( 1 + o ( 1 ) ) 1/2 + 1/(24) \left \{ 1/(48m^2) \right \} (1 + o(1)) . Some numerical results are presented.