# How to derive the Asymptotic series of sums related to Euler's Summatory Function?

I need techniques to solve the density of $T$, a subset of $\mathbb{Q}$ in form of an albegraic expression with relatively prime numerator and denominator values. The best way of doing this is by deriving an asymptotic series of sums related to Euler's Summatory Function.

Many are familiar with Euler's Totient Function or $\phi(n)$; however, the function has an alternate form

$$\phi(n)=\left|\left\{\left.\frac{m}{n}\in[0,1]\right|\gcd{\left(m,n\right)}=1\right\}\right|$$

where $m,n\in\mathbb{Z}$

Similarly, Euler's Summatory Function or $\Phi(t)$ can be defined as

$$\Phi(t)=\sum_{0<n<t}\left|\left\{\left.\frac{m}{n}\in[0,1]\right|\gcd{\left(m,n\right)}=1\right\}\right|$$

which has an asymptotic series of

$$\frac{3}{\pi^2}t^2+O\left(t\left(\log t\right)^{2/3}\left(\log \log t\right)^{4/3}\right)$$

In general, I want to find the asymptotic series of sums that contain the subset of $\left\{\left.\frac{m}{n}\in[0,1]\right|\gcd{\left(m,n\right)}=1\right\}$. This helps me derive an approximation. The sums are in the form

$$\sum_{0<D(n)<t}\left|\left\{\left.\frac{N(m)}{D(n)}\in[0,1]\right|\gcd{\left(N(m),D(n)\right)}=1\right\}\right|$$

Where $N$ and $D$ are functions that allow the set inside the sum to be a subset of $\left\{\left.\frac{m}{n}\in[0,1]\right|\gcd{\left(m,n\right)}=1\right\}$.

I was unable to mathematically derive the asymtotic series for sums with specific functions of $N$ and $D$. Instead I found approximations using Computer Programming. Unfortunately most approximations were inaccurate and for those that were, I could not solve certain variables.

For example, in equation

$$\sum_{0<D_1 n+ D_0<t}\left|\left\{\left.\frac{N_1m+N_0}{D_1n+D_0}\in[0,1]\right|\gcd{\left(N_1 m +N_0, D_1 n +D_0\right)}=1\right\}\right|\approx A\Phi(t)\approx \frac{3A}{\pi^2}t^2$$

I am unable to determine $A$ in terms of integers $N_0, N_1, D_0$ and $D_1$.

In equation

$$\sum_{0<D_c n^c+D_0<t}\left|\left\{\left.\frac{{N_p}m^p+N_0}{D_{c}n^c+D_0}\in[0,1]\right| \gcd{\left({N_p}m^p+{N_0},D_{c}n^c+D_0\right)}=1\right\}\right|\approx \frac{R}{t^{(p-1)/p} t^{(c-1)/c}}\Phi(t)$$

I am unable to solve $R$ in terms of integers $D_c,D_0,N_p,$ and $N_0$. Morover, the approximation is poor since the relative error is less than $.01$.

And in equation

$$\sum_{0<{\left(D_1\right)}^{n}+D_0<t}\left|\left\{\left.\frac{N_1m+N_0}{{\left(D_1 \right)}^{n}+D_0}\in[0,1]\right|\gcd{\left(N_1 m+N_0, \left(D_1\right)^{n}+D_0\right)}=1\right\}\right|$$

I am unable to find an approximation

## Questions

In conclusion:

How does one mathematically derive the asymptotic series of the sums listed above?

Are there better approximations that can be used?

Are there research papers on this subject? I have searched but found nothing. The closest I've got is the Totient function being related to factor rings

• Any ideas, hints, comment....? – Arbuja Jul 28 '17 at 22:02
• In general $\sum_{n \le x} \phi(n)$ is estimated using this method. Then you can twist it by Dirichlet characters and estimate sums of the form $\sum_{n \le x, n \equiv a \bmod b} \phi(n)$ and $\sum_{n \le x, n \equiv a \bmod b} \phi_\chi(n)$. More generally, look at the poles and residue of the Dirichlet series generating function. – reuns Jul 28 '17 at 23:57
• @reuns Thank you but what does ${\phi}_{X} (n)$ stand for? – Arbuja Jul 29 '17 at 2:51
• @reuns Can I convert the sums listed in my post to the sums listed in your comment? – Arbuja Jul 29 '17 at 14:55
• $\displaystyle\phi(n) = \sum_{m \le n, gcd(m,n)=1} 1$. I told you to look at the derivation of the asymptotic for $\sum_{n \le x} \phi(n)$ – reuns Jul 30 '17 at 0:35

For a Dirichlet character $\chi$ let $$F_\chi(s) = \sum_{n=1}^\infty n^{-s} (\sum_{m \le n}\chi(m)), \qquad L(s,\chi) = \sum_{n=1}^\infty n^{-s} \chi(n),\qquad G_\chi(s) = \frac{F_\chi(s)}{L(s,\chi)}$$ Then if $\gcd(a,b)=1$ $$H(s) =\frac{1}{\phi(a)}\sum_{\chi \bmod a} \overline{\chi(b)} G_\chi(s) = \sum_{n=1}^\infty c_n n^{-s}, \qquad c_n = \sum_{m \le n, m \equiv b \bmod a, \gcd(n,m)=1} 1$$
You'll obtain that $$\sum_{n \le x} c_n = \frac{1}{2i\pi} \int_{3-i\infty}^{3+i\infty} H(s) \frac{x^s}{s}ds \sim \text{Res}(H(s) \frac{x^s}{s},2) \\= \frac{1}{\phi(a)}\sum_{\chi \bmod a} \overline{\chi(b)}\frac{x^2}{2 L(2,\chi)} \text{Res}(F_\chi(s) ,2) =\frac{1}{\phi(a)} \overline{\chi_0(b)}\frac{x^2}{2 L(2,\chi_0)} \text{Res}(F_{\chi_0}(s) ,2)= \frac{x^2}{2a \zeta(2)}\prod_{p | a} (1-p^{-2})$$ (where $\text{Res}$ is the residue of a meromorphic function and $\chi_0(m) = 1_{\gcd(m,a)=1}$ is the trivial character so that $L(s,\chi_0) = \zeta(s)\prod\limits_{p | a} (1-p^{-s})$ and $F_{\chi_0}(s) \sim \sum\limits_{n=1}^\infty n^{-s} \frac{n\phi(a)}{a}$)
Following the same lines, if $gcd(d,e)=1$ you should get $$\sum_{n \le x, n \equiv e \bmod d} c_n \sim \frac{x^2}{2a \phi(d) \zeta(2)}\prod_{p | a} (1-p^{-2})$$
• I still do not understand your answer. How is it related to a sum such as $\sum\limits_{0<D_1n+D_0<t}\left|\left\{\left.\frac{N_1 m+N_0}{D_1 n+D_0}\right|m,n\in\mathbb{Z}\right\}\right|$, where all variables are integers? – Arbuja Jul 30 '17 at 22:11
• @Arbuja $$\displaystyle\sum_{l < t, l \equiv D_0 \bmod D_1} \quad\sum_{m\le l, gcd(m,l)=1, m \equiv N_0 \bmod N_1} 1$$ Now instead of searching for more complicated cases, why don't you work on the proof that $\sum_{n \le x} \frac{\phi(n)}{n} \sim \frac{x}{ \zeta(2)}, \qquad \sum_{n \le x} \phi(n) \sim \frac{x^2}{2 \zeta(2)}$ ? – reuns Jul 30 '17 at 22:21