2
votes
1answer
53 views

Integration of complex function with respect to complex variable

I was given as homework to calculate the complex integral limit $$\lim_{T\rightarrow \infty} \frac {1}{2\pi i}\int_{c-iT}^{c+iT}\frac {x^s}{s^{k+1}}ds $$ where $c>0$ and $k\geq1$ is an integer. ...
3
votes
0answers
53 views

Application of Dirichlet Theorem in AP to elementary number theory problems.

I have learnt this theorem in my class, however, "elementary" examples are very limited. (focusing more on analytic machinery) Are there any interesting applications to elementary number theory that ...
2
votes
1answer
44 views

On Dirichlet Theorem on primes in AP.

Let $A(h,k) = \{h + km: m = 0,1,2,\dots\}\;\;$ (EDIT: and $(h,k)=1$) Without using Dirichlet's Theorem, Prove that for every positive integer $n$, $A(h,k)$ contains infinitely numbers relatively ...
5
votes
1answer
119 views

Riemann Zeta Function Non-Vanishing on the Line $\mathrm{Re} \; z = 1$

The result quoted in the title is usually a stepping stone in the proof of the prime number theorem and I am familiar with the usual argument for this result. The other day my professor was telling ...
1
vote
0answers
31 views

Trouble Identifying a “Psi” Function in Number Theory

In these lecture notes on number theory I am reading I came across the notation $\Psi(e^t;a,q)$ in connection with the Dirichlet theorem on arithmetic progression. I was hoping someone could help me ...
2
votes
1answer
67 views

Continuation of the Riemann Zeta Function

I am actually aware of the argument showing $\zeta$ has a meromorphic extension to $\mathbb{C}$ with a single pole at $z = 1$. On a recent number theory exam, however, one of the questions asked to ...
1
vote
0answers
39 views

How to prove that $\sum_{p \leq x} {\log p \over p} = \log x + O(1)$? [duplicate]

Problem Prove that $$ \sum_{p \leq x} {\log p \over p} = \log x + O(1) $$ as $x \to \infty$. Notes: $p$ ranges over primes, $\log$ is natural Progress Using Riemann-Stieltjes integration and ...
5
votes
1answer
145 views

How to prove this inequality $\pi(x) > \log x - 1$ involving the prime counting function?

Problem Prove that $\pi(x) > \log x - 1$. Progress Based on a hint and very elementary methods, I got that $$ \prod_{p \leq x} (1-p^{-1})^{-1} \leq \prod_{k=2}^{\pi(x)+1} (1-k^{-1})^{-1}. $$ The ...
3
votes
1answer
74 views

How to prove $\sum_p {1 \over p^s} = \sum_{n=1}^\infty {\mu(n) \over n} \log \zeta(ns)$?

Problem Prove that for $\operatorname{Re}(s)> 0$, $$ \sum_p {1 \over p^s} = \sum_{n=1}^\infty {\mu(n) \over n} \log \zeta(ns), $$ where the sum extends over all primes $p$. Notes: $\log$ is ...
0
votes
1answer
124 views

Infinite series for Euler-Mascheroni constant

Problem Show that $$ \gamma = \tfrac 12 \log 2 + {1 \over \log 2} \sum_{n=2}^\infty (-1)^n {\log n \over n}. $$ Progress I tried writing the terms $1/k$ of the harmonic sum in the definition of ...
1
vote
1answer
48 views

Asymptotics of the logarithmic integral

Problem Given $$ \gamma = \int_0^1 {1-e^{-u} \over u} du - \int_1^\infty {e^{-u} \over u} du, $$ prove that $$ \int_0^x {dt \over \log t} = \gamma + \log \log x + \sum_{k=1}^\infty {\log^k x \over k ...
1
vote
0answers
44 views

Trouble computing a sum of Dirichlet characters.

Let $\chi(n)$ be a character mod $m$, and let $\rho$ be an $h$th root of unity. I am trying to compute the following sum \begin{equation} \sum_{\chi}(\rho^{-1}\chi(a) + \rho^{-2}\chi(a^2) + \cdots + ...
1
vote
1answer
51 views

Weak Version of Dirichlet's Theorem

I was asked to prove the following: For any given $(a,b) = 1$ and $m > 0$ there are infinitely many integers $x$ such that $(a+bx,m) = 1$. Now, I have a proof worked out that involves the ...
2
votes
2answers
160 views

Arithmetical functions summation

Problem (7.4.15) of Burton's Elementary Number Theory has been request that prove the following equalities. In this book isn't expressed Dirichlet multiplication and Riemann's zeta function before ...
1
vote
1answer
100 views

Don't understand a bound on Dirichlet L function for principal character

$s= \sigma + it$ is any complex number with real part $> 0$. This came up because $L(s,\chi) = \zeta(s)\prod_{p | q} (1-p^{-s})$ and I have a bound for zeta I want to change to a bound for $L$ ...
3
votes
0answers
263 views

Show that $n \sum\limits_{p \leq n} \frac{\log(p)}{p} = n \log(n) + \mathcal{O}(n)$

Using the fact that $\log(n!) = n \log(n) - n + \mathcal{O}(\log(n))$ I am asked to show that: $$ n \sum_{p \leq n} \frac{\log(p)}{p} = n \log(n) + \mathcal{O}(n) $$ Prior to this result it was ...
3
votes
1answer
339 views

Calculating a summation of a $\theta$ function

Let $ \theta_z(t) = \sum \limits_{m,n\in\mathbb{z}}e^{-\pi Q_z(m,n)t}$ where $Q_z(m,n)=y^{-1}|mz+n|^2$. I need to prove that $\theta_z(t)=t^{-1}\theta_z(t^{-1})$. Now, looking that up I know that ...
3
votes
1answer
101 views

$kN^{2}$ relatively prime pairs in $\mathbf{Z}^{2}$

If $k= 1- \sum _{p\in \mathbf{P}} \frac{1}{p^{2}}$, then there are at least $kN^{2}$ pairs $(n_{1},n_{2})$ in $\mathbf{Z}^{2}$, with $1\le n_{1}, n_{2} \le N$ and $gcd(n_{1},n_{2})=1$. Also since ...
8
votes
2answers
189 views

Complex integral with zeta

this is a homework problem I am stuck on: Compute the following integral for $\sigma > 1$ $$\displaystyle \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T}\left|\zeta{(\sigma + it)}\right|^2dt .$$ I ...
8
votes
1answer
177 views

Mean Value of a Multiplicative Function close to $n$ in Terms of the Zeta Function.

Let $f(n)$ be a multiplicative function defined by $f(p^a)=p^{a-1}(p+1)$, where $p$ is a prime number. How could I obtain a formula for $$\sum_{n\leq x} f(n)$$ with error term $O(x\log{x})$ and ...
7
votes
2answers
452 views

Ramanujan's Tau function, an arithmetic property

The problem: Let $\tau(n)$ denote the Ramanujan $\tau$-function and $\sigma(n)$ be the sum of the positive divisors of $n$. Show that $$ (1-n)\tau(n) = 24\sum_{j=1}^{n-1} \sigma(j)\tau(n-j).$$ ...
8
votes
1answer
180 views

Converting an infinite product to sum; Ramanujan $\tau$ function

I've gotten what seems most of the way, but I'm quite stuck at this point. Define $\tau(n)$ by \begin{align*} q\prod_{n=1}^\infty (1-q^n)^{24} = \sum_{n=1}^\infty\tau(n)q^n. \end{align*} ...
3
votes
1answer
159 views

Bounds for Fourier coefficients of cusp forms

I've asked the background question here, which still left unanswered. Now I have a more precise question. In my homework I've been asked to prove that $$\left| \sum_{1\leq n \leq N} a_f (n)e^{2\pi i ...