Intereting Posts

Is there a geometric meaning of the Frobenius norm?
Universal Covering Group of $SO(1,3)^{\uparrow}$
How many digits of accuracy will an answer have?
Is there a general algorithm to find a primitive element of a given finite extension (with a finite number of intermediate fields)?)
How do we know that certain concrete nonstandard models of the natural numbers satisfy the Peano axioms?
Proof that every repeating decimal is rational
Sign of determinant when using $det A^\top A$
Maximizing volume of a rectangular solid, given surface area
Is this reflexive?
Show that symmetric difference is $A \Delta B = (A \cup B)$ \ $(A\cap B)$
Difference between gradient and Jacobian
Is it mathematically wrong to prove the Intermediate Value Theorem informally?
Question regarding usage of absolute value within natural log in solution of differential equation
Some question of sheaf generated by sections
Are all continuous one one functions differentiable?

Suppose $f(n)$ is a positive real-valued arithmetic function such that

$$

\frac1n\sum_{k=1}^nf(k)\sim g(n)

$$

for $g(x)$ a monotonic increasing function. What can be said about the asymptotic behavior of

$$

\sum_{k=1}^n\frac1kf(k)

$$

? The reverse question is also of interest. In both cases it seems that the asymptotics should be fairly simple multiples of each other:

$$

\frac1n\sum_{k=1}^nk\sim\frac12n

$$

and

$$

\sum_{k=1}^n\frac1kk\sim n

$$

but I have seen results where this does not seem to hold and I want to know if they are right.

Perhaps the problem cannot be solved without further restrictions on $f$ or $g$. In the case of immediate interest, $f$ and $g$ are essentially linear, in that there exists a constant $k$ such that $x/(\log x)^k\ll h(x)\ll x(\log x)^k$ for $h\in\{f,g\}$ and $x$ in the appropriate domain.

- Riemann Hypothesis and the prime counting function
- Sum of squares of sum of squares function $r_2(n)$
- What's about the convergence of $\sum_{n=1}^\infty\frac{\mu(n)\pi(n)}{n^2}$?
- Values of the Riemann Zeta function and the Ramanujan Summation - How strong is the connection?
- Does Riemann Hypothesis imply strong Goldbach Conjecture?
- What is so interesting about the zeroes of the Riemann $\zeta$ function?

- Asymptotic formula for $\sum_{n\leq x}\mu(n)^2$ and the Totient summatory function $\sum_{n\leq x} \phi(n)$
- Calculating the Zeroes of the Riemann-Zeta function
- Efficiently calculating the logarithmic integral with complex argument
- What is known about the 'Double log Eulers constant', $\lim_{n \to \infty}{\sum_{k=2}^n\frac{1}{k\ln{k}}-\ln\ln{n}}$?
- Elements of finite order in the group of arithmetic functions under Dirichlet convolution.
- What is the probability that some number of the form $10223\cdot 2^n+1$ is prime?
- Counting squarefree numbers which have $k$ prime factors?
- Primes sum ratio
- Hardy-Ramanujan theorem's “purely elementary reasoning”
- Clarkson's Proof of the Divergence of Reciprocal of Primes

The asymptotics really do depend on the function $f$ in question.

**If $f(x)$ has a power function as its dominant term**, with, say, $f(x) = x^p + o(x^p)$, for $p > 0$, then you’re right about the two asymptotics being constant multiples of each other:

$$\begin{align}

\frac{1}{n} \sum_{k=1}^n f(k) &\sim \frac{1}{n}\frac{1}{p+1} n^{p+1} = \frac{1}{p+1} n^p, \\

\sum_{k=1}^n \frac{f(k)}{k} &= \sum_{k=1}^n \left(k^{p-1} + o(k^{p-1})\right) \sim \frac{1}{p}n^p,

\end{align}$$

where we use the formula for the sum of $k$th powers (or Euler-Maclaurin summation for $p$ not an integer). This, of course, includes the linear case.

However, **if $f(x)$ has $\log x$ as its dominant term**, we get the behavior noted by Antonio Vargas:

$$\begin{align}

\frac{1}{n} \sum_{k=1}^n f(k) &= \frac{1}{n} \sum_{k=1}^n \left(\log k + o(\log k)\right) = \frac{1}{n} \left(\log n! + o(\log n!)\right) \sim \frac{n \log n}{n} = \log n,\\

\sum_{k=1}^n \frac{f(k)}{k} &= \sum_{k=1}^n \left(\frac{\log k + o(\log k)}{k}\right) \sim \frac{1}{2} (\log n)^2,

\end{align}$$

where the first asymptotic follows from Stirling’s formula and the second is a consequence of Euler-Maclaurin (see, for example, Lemma 2.11 in my paper here).

And, of course, **if $f(x)$ has a constant $C$ as its dominant term**, we get

$$\begin{align}

\frac{1}{n} \sum_{k=1}^n f(k) &= \frac{1}{n} \sum_{k=1}^n (C + o(1)) \sim C, \\

\sum_{k=1}^n \frac{f(k)}{k} &= \sum_{k=1}^n \left(\frac{C + o(1)}{k}\right) \sim C \log n.

\end{align}$$

Based on this, I would conjecture that there’s some cutoff point on the growth rate of $f(x)$ such that if $f(x)$ grows at a sufficiently large rate then $\displaystyle \frac{1}{n} \sum_{k=1}^n f(k)$ and $\displaystyle \sum_{k=1}^n \frac{f(k)}{k}$ have asymptotic growth rates that are constant multiples. Otherwise, $\displaystyle \frac{1}{n} \sum_{k=1}^n f(k)$ is asymptotically smaller than $\displaystyle \sum_{k=1}^n \frac{f(k)}{k}$. Power growth would be above that cutoff point, and logarithmic growth would be below it.

- Area of the intersection of four circles of equal radius
- How to prove that $\lVert \Delta u \rVert_{L^2} + \lVert u \rVert_{L^2}$ and $\lVert u \rVert_{H^2}$ are equivalent norms?
- If one side of $\int f\ d\lambda = \int f\ d\mu – \int f\ d\nu$ exists, does the other one exist as well?
- Why is the Taylor expansion of $\cos$ decreasing?
- Confusion on how to solve this question about sequences.
- How to understand Weyl chambers?
- Hard Definite integral involving the Zeta function
- If $G$ is a group and $N$ is a nontrivial normal subgroup, can $G/N \cong G$?
- If $xy$ and $x+y$ are both even integers (with $x,y$ integers), then $x$ and $y$ are both even integers
- How to get nth derivative of $e^{x^2/2}$
- Galois theory: splitting field of cubic as a vector space
- Prove $e^x=1+\frac{1}{\sqrt{\pi }}{\int_0^x \frac{e^t \text{erf}\left(\sqrt{t}\right)}{\sqrt{x-t}} \, dt}$
- Number of distinct arrangements {$n_i$} $n_1<n_2<n_3<n_4<n_5$ such that $\sum n_i=20$
- Global maxima/minima of $f(x,y,z) = x+y+z$ in $A$
- How many words can be formed from 'alpha'?