Growth of partial sums of divergent series

Abstract:

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