Partial summation and Abelian estimates

For coefficients $a_n$, define $A(t)=\sum_{n\leq t}a_n$ for real $t\geq1$. This is a right-continuous step function, not a smooth interpolation through the partial sums. If the weight is continuously differentiable on the closed interval from one to a real endpoint, integration by parts against the jumps of this function yields the displayed identity. The lower boundary before the first jump is zero. This explains why no separate subtraction of the first coefficient appears.

One can also prove the formula without Stieltjes integration. Replace the integral of the step function by a sum of ordinary integrals over successive unit intervals. Each integral is a partial sum multiplied by a difference of neighboring weight values. Expanding and rearranging the resulting finite sum makes all interior terms telescope. The endpoint term must be kept even when it eventually tends to zero; assuming its disappearance at the start can turn a correct finite identity into a false infinite one.

Suppose the cumulative sums are bounded by a constant. With the weight equal to the reciprocal of the variable, the boundary term tends to zero, and the integral converges absolutely because its integrand is bounded by a constant times the reciprocal square. The original weighted series therefore converges. This does not prove that the absolute values of the weighted coefficients are summable. Alternating signs illustrate the distinction: bounded partial sums create cancellation even though the absolute weighted series is harmonic and divergent.

More generally, a bound $A(t)=O(t^\theta)$ gives a useful integral whenever the derivative of the chosen weight decays sufficiently fast. For $f(t)=t^{-s}$, write the complex power using the real logarithm of positive $t$. The derivative is $-s t^{-s-1}$, and the resulting tail converges when $\operatorname{Re}s>\theta$, with constants on compact subsets controlled. This is an Abelian direction: known cumulative estimates imply analytic convergence. Recovering sharp cumulative estimates from a singularity requires additional Tauberian reasoning.

For coefficients all equal to one, the cumulative sum is the floor function. At a positive integer endpoint, partial summation for the reciprocal weight gives the harmonic sum as one plus the integral of the floor function divided by the square of the variable. Splitting the floor into the variable minus its fractional part produces a logarithm and a bounded correction. The correction has a convergent improper integral, so it determines a genuine limiting constant rather than an unspecified bounded error.

In detail, $H_N=\log N+\gamma+R_N$, where $\gamma=1-\int_1^\infty\{t\}t^{-2}\,dt$ and $R_N=\int_N^\infty\{t\}t^{-2}\,dt$. The fractional part lies between zero and one, giving $0\leq R_N\leq1/N$. This elementary error estimate retains the sign as well as the scale of the correction. When handling primes later, a similar separation of a cumulative main term from a controlled remainder converts logarithmically weighted counts into ordinary counts.