Intereting Posts

Question about permutation cycles
Differential Geometry without General Topology
Fast(est) and intuitive ways to look at matrix multiplication?
How many combinations of $3$ natural numbers are there that add up to $30$?
Is ${\mathbb Z} \times {\mathbb Z}$ cyclic?
Is there any motivation for Zorn's Lemma?
Is there a function $f: \mathbb R \to \mathbb R$ that has only one point differentiable?
DTFT of a triangle function in closed form
Relationship between very ample divisors and hyperplane sections
point on line to form angle
What is the correct answer to this answered combinatorics problem?
Radical of an ideal using Macaulay2 software.
For $F$ closed in a metric space $(X,d)$, is the map $d(x,F) = \inf\limits_{y \in F} d(x,y)$ continuous?
Show that the curve $\dfrac{x^2}{a^2} +\dfrac{ y^2}{b^2} = 1$ form an ellipse
Normal bundle of twisted cubic.

I am interesting in bounding the arithmetic sum

$$ \sum_{n \leq x} \frac{\mu(n)^2}{\varphi(n)}$$

(The motivation is that this is a sum that comes up a lot in sieving primes, in particular in the Selberg sieve.) It is not too hard to show that this is $\geq \sum_{n \leq x} \frac{1}{n} \approx \log x$, and in fact the difference be described succinctly as follows: if $s(m)$ denotes the largest squarefree factor of $m$, then

- Analytic Continuation of Zeta Function using Bernoulli Numbers
- Theorem 11.14, Apostol, pg 238 - need explanation
- Does dividing by zero ever make sense?
- Evaluating $\sum\limits_{n=1}^{\infty} \frac{1}{n\operatorname{ GPF}(n)}$, where $\operatorname{ GPF}(n)$ is the greatest prime factor
- Are these zeros equal to the imaginary parts of the Riemann zeta zeros?
- Average order of $\mathrm{rad}(n)$

$$\sum_{n \leq x} \frac{\mu(n)^2}{\varphi(n)} = \sum_{s(m) \leq x} \frac{1}{m}$$.

In particular, one sees that the difference

$$ \sum_{n \leq x} \frac{\mu(n)^2}{\varphi(n)} – \sum_{n \leq x} \frac{1}{n} = \sum_{m > x, s(m) \leq x} \frac{1}{m}. $$

I would like to be able to bound how small this actually is. Montgomery-Vaughan cites an unpublished result of De Bruijn that is is actually $O(1)$, so I would certainly like to see that.

The closeness of these sums is rather unintuitive to me (apart from the vague comment that $\varphi(n) \approx n$ unless $n$ has a lot of distinct prime factors, which should be relatively rare), so I would also appreciate any insight into why we could expect this.

- (Non-)Canonicity of using zeta function to assign values to divergent series
- Sign of Ramanujan $\tau$ function
- Small primes attract large primes
- Intervals that are free of primes
- How to show that the Laurent series of the Riemann Zeta function has $\gamma$ as its constant term?
- What do following asymptotic symbols mean?
- On the mean value of a multiplicative function: Prove that $\sum\limits_{n\leq x} \frac{n}{\phi(n)} =O(x) $
- A $\frac{1}{3}$ Conjecture?
- Proving that $\pi(2x) < 2 \pi(x) $
- A number-theory question on the deficiency function $2x - \sigma(x)$

Take a look at Montgomery and Vaughn chapter 2.1 exercise 17:

17.(cf. Ward 1927) Show that for $x\geq2$, $$\sum_{n\leq x }\frac{\mu^2(n)}{\phi(n)}=\log x+\gamma+\sum_{p}\frac{\log p}{p(p-1)}+O\left(x^{-1/2}\log x\right).$$

Here $\gamma$ denotes the Euler-Mascheroni constant.

*Sketch:* Let $f(n)/n=\mu^2(n)/\phi(n)$. Then $f(n)$ will be a multiplicative function close to $1$. Setting $g(n)=(\mu*f)(n)$ so that $f(n)=(1*g)(n)$, it follows that $g(n)$ will be very close to zero. Indeed, at the primes $$f\left(p^{k}\right)=\begin{cases}

\frac{p}{p-1} & \text{ if }k=1\\

0 & \text{ if }k\geq2

\end{cases} $$

and

$$g\left(p^{k}\right)=\begin{cases}

\frac{1}{p-1} & \text{ if }k=1\\

\frac{-p}{p-1} & \text{ if }k=2\\

0 & \text{ if }k\geq3

\end{cases} .$$

Let $G(s)=\sum_{n=1}^\infty g(n)n^{-s}$. Then $G(1)=1$ and $$G'(1)=\frac{G'(1)}{G(1)}=\sum_{p}\frac{\log p}{p(p-1)}.$$

Then by carefully splitting up the sum we can show that $$\sum_{n\leq x }\frac{f(n)}{n}=G(1)\log x+G(1)*\gamma+G'(1)+O(Error)$$ where the error term depends on the smallest $\sigma$ such that $G(\sigma)$ converges absolutely.

- Number of solutions of $x^2=1$ in $\mathbb{Z}/n\mathbb{Z}$
- definition of the constant $e$
- Probability that 2 appears at an earlier position than any other even number in a permutation of 1-20
- Examples of categories where epimorphism does not have a right inverse, not surjective
- Has knot theory led to the development of better knots?
- Evaluating $\int_0^\infty \sin x^2\, dx$ with real methods?
- Notation to work with vector-valued differential forms
- How do you describe your mathematical research in layman's terms?
- When can we interchange the derivative with an expectation?
- are any two vector spaces with the same (infinite) dimension isomorphic?
- Is it true that $E = X$?
- A question about $\prod_{x\in \mathbb{R}^{*}}{x}$
- Importance of Representation Theory
- What is a physical interpretation of a skew symmetric bilinear form?
- Fourier transform of $\mathrm{sinc}(4t)$