I recently discovered the following relation for arbitrary $d_r$:

$$ \lim_{k \to \infty} \lim_{n \to \infty}\ \sum_{r=1}^n d_r \left( f(\frac{k}{n}r)\frac{k}{n} \right) \sim \underbrace{\frac{1}{\zeta(s)} \sum_{r=1}^\infty \frac{d_r}{r^s}}_{\text{removable singularity}} \times \int_0^\infty f(x) \, dx $$

As $s \nearrow 1$ and $\int f(x) dx$ absolutely converges and $f(\infty)= f(0) = 0$

Some obvious choices are $d_r= 1$ , $d_r = s^r r^s$ or $d_r = \frac{\delta_{1,r}}{s-1}$ where the R.H.S $\neq 0$. Is there any possible use of this? (I was thinking along the lines of divergent series in QFT). I know this relation looks very bizzare so I've added a proof:

Proof

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Consider an integral such that $$ \int_0^\infty f(x) \, dx = C,$$where, $f(x)$ is a smooth and continuous function and absolutely converges.

Now we raise both sides to the power s:

$$\left(\int_0^\infty f(x) \, dx\right)^s = C^s $$

We substitute $x$ with $rx$ to get:

$$\left(\int_0^\infty f(rx) \, dx\right)^s = (C/r)^s $$

Multiplying both sides by an arbitrary coefficient:

$$ (b_r)\left(\int_0^\infty f(rx) \, dx\right)^s = (b_r)( C/r)^s $$

Taking their sum:

$$ \sum_{r=1}^\infty b_r \left(\int_0^\infty f(rx) \, dx\right)^s = C^s \underbrace{\sum_{r=1}^\infty \frac{b_r}{r^s}}_{\text{dirichlet series}} $$

We write the integral as a limit of a Riemann sum:

$$ \sum_{r=1}^\infty \lim_{k \to \infty} \lim_{n \to \infty}\ b_r \left( \sum_{x=1}^n f(\frac{kx}{n}r)\frac{k}{n} \right)^s = C^s \sum_{r=1}^\infty \frac{b_r}{r^s} $$

Using the mobius inversion formula:

$$ \sum_{r=1}^\infty \lim_{k \to \infty} \lim_{n \to \infty}\ b_r \left( \sum_{x=1}^n f(\frac{kx}{n}r)\frac{k}{n} \right)^s = C^s \frac{1}{\zeta(s)}\sum_{r=1}^\infty \frac{d_r}{r^s} $$

We define $ d_r = \sum_{e|r} b_e $

Note:

$$ (\frac{b_1}{1^s} + \frac{b_2}{2^s} + \frac{b_3}{3^s} + \frac{b_4}{4^s} + \dots) \times (\frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \frac{1}{4^s} + \dots) = \frac{b_1}{1^s} + \frac{b_1 + b_2}{2^s} + \frac{b_1 + b_3}{3^s} + \frac{b_1 + b_2 + b_4}{4^s} + \dots $$

Now focusing on the L.H.S ($s \nearrow 1 $):

$$ \sum_{r=1}^\infty \lim_{k \to \infty} \lim_{n \to \infty}\ b_r \left( \sum_{x=1}^n f(\frac{kx}{n}r)\frac{k}{n} \right)^s \sim \sum_{r=1}^\infty \lim_{k \to \infty} \lim_{n \to \infty}\ b_r \left( \sum_{x=1}^n f(\frac{kx}{n}r)\frac{k}{n} \right) $$

Focusing on the R.H.S of the asymptote:

$$ \lim_{n \to \infty} b_1 ((f(\frac{k}{n}) + f(2 \frac{k}{n}) + f(3 \frac{k}{n}) +f(4 \frac{k}{n}) + \cdots)\frac{k}{n} $$

$$+$$

$$ \lim_{n \to \infty} b_2 (0.f(\frac{k}{n}) + f(2 \frac{k}{n}) + 0.f(3 \frac{k}{n}) +f(4 \frac{k}{n}) +\cdots) \frac{k}{n}$$

$$+$$

$$ \vdots $$

$$ = \lim_{n \to \infty} (\underbrace{b_1}_{d_1} (f(\frac{k}{n}) + \underbrace{(b_1 + b_2)}_{d_2}f(2 \frac{k}{n}) + \underbrace{(b_1 + b_3)}_{d_3}f(3 \frac{k}{n}) +\underbrace{(b_1 + b_2 + b_4)}_{d_4}f(4 \frac{k}{n}) + \cdots)\frac{k}{n} $$

Hence, for special $d_r$ the R.H.S converges:

$$ \lim_{k \to \infty} \lim_{n \to \infty}\ \sum_{r=1}^n d_r \left( f(\frac{k}{n}r)\frac{k}{n} \right) \sim \lim_{s \to 1} \underbrace{\frac{1}{\zeta(s)} \sum_{r=1}^\infty \frac{d_r}{r^s}}_{\text{removable singularity}} \times \int_0^\infty f(x) \, dx $$