Ze Chen

## Analytic Number Theory (I)

#### Some Elementary Results

##### $$\sum p^{-1}$$ Diverges

The infinitude of prime numbers had been well known since Euclid. A stronger result is that the following series diverges: $\sum_p \frac{1}{p} \rightarrow \infty.$

An Elementary Proof

Suppose the series above converges, i.e. there is a $$q$$ such that $\sum_{p>q} \frac{1}{p} < \frac{1}{2}.$ Then we let $$p_1,\cdots,p_m$$ denotes all prime numbers less than $$q$$, and let $Q = p_1 \cdots p_m.$ Now since $$\{1+nQ\}$$ are not divisible by $$p_1,\cdots,p_m$$, we find that $\sum_{n=1}^r \frac{1}{1+nQ} \le \sum_{i=1}^\infty (\sum_{p>q} \frac{1}{p})^i$ since every item on the LHS is included in the summation on the RHS. This yields a contradiction.

#### Arithmetic Functions

We define $$\operatorname{id}_k(n) = n^k$$, and $$\operatorname{id}(n) = n$$.

##### The Dirichlet Convolution and Derivative

Definition. $$\displaystyle (f * g)(n) = \sum_{d | n} f(d) g\qty(\frac{n}{d}).$$

Under this definition, the commutation law and the association law hold. $f*g = g*f,\quad (f*g)*h = f*(g*h).$ The Dirichlet inverse of $$f$$ is denoted by $$f^{-1}$$ and satisfies $$f^{-1}*f = I$$. If $$f(1) \neq 0$$, then the inverse exists and is unique.

Definition. The derivative of a arithmetic function is defined by $$f'(n) = f(n)\log n$$, for $$n\ge 1$$.

Many properties of the derivative in calculus hold for the arithmetic derivative as well.

• $$(f+g)' = f' + g'$$.
• $$(f*g)' = f'*g + f*g'$$.
• $$(f^{-1})' = -f' * (f * f)^{-1}$$.
##### Multiplicative Functions

Multiplicative functions make up a subgroup of arithmetic functions under the binary operation defined by the Dirichlet convolution, i.e. if $$f*g$$ is multiplicative if both $$f$$ and $$g$$ are, and $$f^{-1}$$ is multiplicative if $$f$$ is.

Examples of multiplcative functions includes

• $$1(n)$$ (completely multiplicative),
• $$\operatorname{id}_k(n)$$ (completely multiplicative),
• $$\lambda(n)$$ (complete multiplicative),
• $$\varphi(n)$$,
• $$\mu(n)$$,
• $$\sigma_k(n)$$.

Theorem. Let $$f$$ be multiplicative. Then $$f$$ is completely multiplicative if and only if $$f^{-1}(n) = \mu(n) f(n)$$.

Theorem. If $$f$$ is multiplicative, then $\sum_{d|n} \mu(d) f(d) = \prod_{p|n} (1-f(p)).$

##### The Number Theoretic $$\delta$$-Function

Definition. $$\displaystyle I(n) = \qty(n) = \begin{cases} 1, & \text{if } n = 1; \\ 0, & \text{if } n > 1.\end{cases}{}$$

This is the analogy of the $$\delta$$-function in number theory. $I * f = f * I = f.$

##### The Möbius Function $$\mu(n)$$

Definition. $$\mu(1) = 1$$, and for $$n = p_1^{\alpha_1}\cdots p_k^{\alpha_k}>1$$ we define $\mu(n) = \begin{cases} (-1)^k, & \text{if } \alpha_1 = \cdots = \alpha_k = 1; \\ 0, & \text{otherwise}. \end{cases}{}$

Theorem. If $$n\ge 1$$ we have $(1 * \mu)(n) = \sum_{d | n} \mu(d) = I(n) = \qty[\frac{1}{n}] = \begin{cases} 1, & \text{if } n = 1; \\ 0, & \text{if } n > 1. \end{cases}{}$

##### The Euler's Totient Function $$\varphi(n)$$

Definition. $$\varphi(n)$$ is defined to be the number of of integers in $$1,\cdots,n$$ that's coprime with $$n$$.

Theorem. If $$n\ge 1$$ we have $(1 * \varphi)(n) = \sum_{d | n} \varphi(d) = n.$

Theorem. If $$n\ge 1$$ we have $(\mu*\operatorname{id})(n) = \sum_{d | n}\mu(d)\frac{n}{d} = \varphi(n).$

Theorem. $$\varphi$$ could be written as the product form $\varphi(n) = n \prod_{p | n} \qty(1-\frac{1}{p}).$

Some properties of the totient function are given below.

• For a prime number $$p$$, $$\varphi(p^\alpha) = p^\alpha - p^{\alpha-1}{}$$.
• $$\displaystyle\varphi(mn) = \varphi(m)\varphi(n)\qty(\frac{d}{\varphi(d)})$$ where $$d = \gcd(m,n)$$.
• In particular, if $$\gcd(m,n) = 1$$ then $$\varphi(mn) = \varphi(m)\varphi(n)$$.
• If $$a|b$$ then $$\varphi(a)|\varphi(b)$$.
• For $$n\ge 3$$, $$\varphi(n)$$ is even. If $$n$$ has $$r$$ different prime factors, then $$2^r | \varphi(n)$$.
##### The Möbius Inversion Formula

Theorem. If $$f = 1 * g$$, then $$g = f * \mu$$.

##### The von Mangoldt Function $$\Lambda(n)$$

Definition. $\displaystyle \Lambda(n) = \begin{cases}\log p, & \text{if } n = p^m, \text{ where } p \text{ is a prime}, \text{ and } m\ge 1; \\ 0, &\text{otherwise}.\end{cases}{}$

Theorem. If $$n\ge 1$$, $$\displaystyle (1')(n) = \log n = 1 * \Lambda = \sum_{d|n} \Lambda(d)$$.

Theorem. If $$n\ge 1$$, $$\displaystyle \Lambda(n) = \mu * (1') = -\sum_{d|n} \mu(d) \log d$$.

##### The Liouville Function $$\lambda(n)$$

Definition. $$\lambda(1) = 1$$, and for $$n = p_1^{\alpha_1}\cdots p_k^{\alpha_k}>1$$ we define $\lambda(n) = (-1)^{\alpha_1 + \cdots + \alpha_k}.$

Theorem. For $$n>1$$, we have $(1 * \lambda)(n) = \sum_{d|n} \lambda(d) = \begin{cases} 1, &\text{if } n \text{ is a perfect square}; \\ 0, & \text{otherwise}. \end{cases}$ We have also $$\lambda^{-1}(n) = \mu^2(n)$$.

##### The Divisor Functions $$\sigma_k(n)$$

Definition. $$\sigma_k(n) = \operatorname{id}_k * 1 = \sum_{d|n} d^k$$.

In particular, we denote $$\sigma_0(n)$$ by $$d(n)$$, which counts the number of divisors, and $$\sigma_1(n)$$ by $$\sigma(n)$$, which is the sum of all divisors.

Theorem. The inversion of the divisor function is given by $\sigma_k^{-1} = (\mu \operatorname{id}_k)*\mu = \sum_{d|n} d^k \mu(d) \mu\qty(\frac{n}{d}).$

##### Generalized Convolution

In the following context, $$F$$ is a function on $$(0,+\infty)$$ that satisfies $$F(x) = 0$$ for $$x\in (0,1)$$.

Definition. $$\displaystyle (\alpha\circ F)(x) = \sum_{n\le x} \alpha(n) F\qty(\frac{x}{n})$$.

Theorem. Let $$\alpha$$ and $$\beta$$ be arithmetic functions. Then $\alpha\circ(\beta\circ F) = (\alpha*\beta)\circ F.$

Theorem. The generalized inversion formula is given by $G = \alpha \circ F \Leftrightarrow F = \alpha^{-1}\circ G.$

Theorem. For a completely multiplcative $$\alpha$$ we have $G = \alpha\circ F \Leftrightarrow F = (\mu\alpha)\circ G.$

##### Bell Series

Definition. Let $$f$$ be a arithmetic function. We define the Bell series of $$f$$ modulo $$p$$ by the formal power series $f_p(x) = \sum_{n=0}^\infty f(p^n) x^n.$

Examples of Bell series are listed below.

• $$\mu_p(x) = 1-x$$.
• $$\displaystyle\varphi_p(x) = \frac{1-x}{1-px}{}$$.
• If $$f$$ is completely multiplicative, we have $\displaystyle f_p(x) = \frac{1}{1-f(p)x}.$
• $$I_p(x) = 1$$.
• $$\displaystyle 1_p(x) = \frac{1}{1-x}$$.
• $$\displaystyle \operatorname{id}_p^\alpha(x) = \frac{1}{1-p^\alpha x}$$.
• $$\displaystyle \lambda_p(x) = \frac{1}{1+x}$$.

Theorem. Let $$h = f*g$$. Then $h_p(x) = f_p(x) g_p(x).$

##### Summary

We have the following table of convolution relations of various arithmetic functions.

 $$f*g$$ $$I$$ $$1$$ $$\operatorname{id}{}$$ $$\varphi$$ $$\mu$$ $$I$$ $$I$$ $$1$$ $$\operatorname{id}{}$$ $$\varphi$$ $$\mu$$ $$1$$ $$1$$ $$d$$ $$\sigma$$ $$\operatorname{id}{}$$ $$I$$ $$\operatorname{id}{}$$ $$\operatorname{id}{}$$ $$\sigma$$ $$\sigma\cdot \operatorname{id}{}$$ $$\varphi$$ $$\varphi$$ $$\varphi$$ $$\operatorname{id}{}$$ $$\mu$$ $$\mu$$ $$I$$ $$\varphi$$

2021/2/6 18:58:36

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