Articles of analytic number theory

Number of primitive characters modulo $m$.

Let $N(m)$ be the number of primitive Dirichlet characters modulo $m$. Could someone please explain me why it satisfies the following relation?: $$ \phi(m) = \sum_{d|m} N(m). $$ Thank you very much! PS here $\phi$ is the Euler totient function.

How can I calculate closed form of a sum?

As we know the closed form of $$ {1 \over k} {n \choose k – 1} \sum_{j = 0}^{n + 1 – k}\left(-1\right)^{\, j} {n + 1 – k \choose j}B_{j} = \bbox[10px,border:2px solid #00A000]{{n! \over k!}\, \left[\, z^{n + 1 – k}\,\,\,\right] {z \exp\left(z\right) \over 1 – \exp\left(-z\right)}} $$ where $B_{n}$ the Bernoulli sequence […]

Equivalent form of prime number theorem

In the book The elementary proof of Prime Number Theorem it says that The prime number theorem $\psi (x)\sim x$ is equivalent to $\int_1^\infty \frac{\psi (t)-t}{t^2} =-\gamma -1$, where $\psi (x)=\sum_{n\le x} \Lambda (n)$, $\gamma $ is the Euler constant. The book also gives a hint to prove $\sum_{n\le x}\frac{\Lambda (n)}{n}=\int_1^x \frac{\psi(t)}{t^2} dt+\frac{\psi (x)}{x}$ first, […]

What is the relationship between GRH and Goldbach Conjecture?

We know that we can prove weak Goldbach Conjecture (three prime theorem) if we assume GRH (Hardy-Littlewood had proved this). Can we also prove strong Goldbach Conjecture if we assume GRH ? Also, any results on reverse direction ? If we assume Goldbach Conjecture holds true, can we get any results about GRH ?

Primes and arithmetic progressions

In a book on complex analysis, the authors prove: Given finitely many (non-trivial) arithmetic progressions of natural numbers $$a_1, a_1+d_1, a_1+2d_1, \cdots $$ $$a_2, a_2+d_2, a_2+2d_2, \cdots, $$ $$a_k, a_k+d_k, a_k+2d_k, \cdots, $$ their totality (union) is never $\mathbb{N}$. Given: any $k$ (non-trivial) arithmetic progressions. Let $S$ denote the set of those numbers which are […]

Corollaries of Green-Tao Theorem?

there is already a good thread which discusses some corollaries of the Green-Tao Theorem, here: Constructing arithmetic progressions The question I was wondering about is of a similar flavor but isn’t mentioned, namely, does there exist a $c \in \mathbb{R}$ such that given any $k \in \mathbb{N}$, can you always find a progression of primes […]

Theorem 11.14, Apostol, pg 238 – need explanation

An excerpt from Introduction to Analytic Number Theory by Tom M. Apostol. I have three main concerns regarding this proof: What is the abscissa of convergence for $\frac{1}{F(s)}$ – why can we take the product of the two series. I only know this is true if the two series are absolutely convergent. Why can we […]

Euler Totient Issues

I was skimming again through Dummit & Foote’s Abstract Algebra and I came across this exercise: Prove that for any given positive integer $N$ there exist only finitely many integers $n$ with $\varphi(n)=N$, where $\varphi$ denotes Euler $\varphi$-function. Conclude in particular that $\varphi(n)$ tends to infinity as $n$ tends to infinity. I don’t doubt that […]

Zeros of the second derivative of the modular $j$-function

I am interested in the zeros of $j”(z)$, where $j:\mathbb{H}\rightarrow\mathbb{C}$ is the classic modular function. Specifically I am interested in knowing if the zeros of $j”$ are algebraic over $\mathbb{Q}$, or, even better, quadratic over $\mathbb{Q}$.

a formula involving order of Dirichlet characters, $\mu(n)$ and $\varphi(n)$

Let $p$ a prime number, ${q_{_1}}$,…, ${q_{_r}}$ are the distinct primes dividing $p-1$, ${\mu}$ is the Möbius function, ${\varphi}$ is Euler’s phi function, ${\chi}$ is Dirichlet character $\bmod{p}$ and ${o(\chi)}$ is the order of ${\chi}$. How can I show that: $$\sum\limits_{d|p – 1} {\frac{{\mu (d)}}{{\varphi (d)}}} \sum\limits_{o(\chi ) = d} {\chi (n)} = \prod\limits_{j = […]