Questions on the use of the methods of real/complex analysis in the study of number theory.
Analytic number theory is a branch of mathematics that uses techniques in real and complex analysis to study the integers (including the primes).
Questions tagged [analytic-number-theory]
3923 questions
23
votes
3 answers
Motivation for Hecke characters
The context is the definition of Hecke Größencharakter:
http://en.wikipedia.org/wiki/Hecke_character
This is supposed to generalize the Dirichlet $L$-series for number fields. Dirichlet characters are characters of the multiplicative groups of…
angiquesophie
14
votes
3 answers
Proving $\sum\limits_{p \leq x} \frac{1}{\sqrt{p}} \geq \frac{1}{2}\log{x} -\log{\log{x}}$
How to prove this: $$\sum\limits_{p \leq x} \frac{1}{\sqrt{p}} \geq \frac{1}{2}\log{x} -\log{\log{x}}$$
From Apostol's number theory text i know that $$\sum\limits_{p \leq x} \frac{1}{p} = \log{\log{x}} + A +…
anonymous
12
votes
2 answers
Finding the integer $\le n$ with largest number of divisors
As mentioned in an answer to this question an integer less than $n$ with largest number of divisors can be found exploring the numbers $x$ of the form
$$ x = 2^{a_1} 3^{a_2} \dots p_k^{a_k} \dots $$
(where $p_k$ is the $k$-th prime number) with…
Esteban Crespi
- 4,007
10
votes
2 answers
Bounds for $\zeta$ function on the $1$-line
I was going over my notes from a class on analytical number theory and we use a bound for the $\zeta$ function on the $1$ line as $\vert \zeta(1+it) \vert \leq \log(\vert t \vert) + \mathcal{O}(1)$ for $t$ bounded away from $0$, say $\vert t \vert…
abhIta
- 301
9
votes
0 answers
Quadratic characters and Liouville's function
I'm working through the problems in Montgomery & Vaughan's Multiplicative Number Theory. In Section 11.2 'Exceptional Zeros', Exercise 9a says that for a quadratic character $\chi$, show that for all $k\ge 0, x\ge1…
stopple
- 1,739
8
votes
2 answers
Asymptotic density of numbers of the form $p_{1}^{\alpha_{1}^{2}}\cdot p_{2}^{\alpha_{2}^{2}}\cdots p_{k}^{\alpha_{k}^{2}}$
If $n$ is a number of the form $p_{1}^{\alpha_{1}^{2}}\cdot p_{2}^{\alpha_{2}^{2}}\cdots p_{k}^{\alpha_{k}^{2}}$ (OEIS A197680) and $T(x)$ counts how many of these numbers are between $1$ and $x$, what is
$$
\begin{equation*}
\lim_{x\rightarrow…
Neves
- 5,617
8
votes
1 answer
Understanding Zhang's result of bounded prime gaps
Here is a link on the internet:
https://www.dropbox.com/s/su3uak2a057yrqv/YitangZhang.pdf
Can someone teach me how to use trivial estimation to reach (6.1) on page 24?
Namely, how to impose $(d,P_0)
ericc
- 81
8
votes
1 answer
average order of $\sum\limits_{\substack{1\le k\le n \\ (k,n)=1}} \frac{1}{k}$
Introduce $$\varrho(n) = \sum\limits_{\substack{1\le k\le n \\ (k,n)=1}} \frac{1}{k}.$$ The following thread at math.stackexchange.com proposes to analyse the average order of $\varrho(n)$, i.e. $$\frac{1}{n} \sum_{k=1}^n \varrho(k).$$
I have tried…
Marko Riedel
- 61,317
8
votes
4 answers
Why are complex numbers necessary to prove the Prime Number Theorem?
The standard proof of the Prime Number Theorem requires taking into consideration that there are no zeroes of the Riemann Zeta function in which the real part equals one. But consider the following argument:
The probability that a number less than X…
Craig Feinstein
- 1,078
7
votes
1 answer
How will this equation imply PNT
So we have $$\sum_{n \leq x} \frac{\Lambda (n)}{n}=\log{x}+C+o(1)$$ where $C$ is a constant, its partial summation is $$\sum_{n \leq x} \frac{\Lambda (n)}{n}=\frac{\psi(x)}{x}+\int_1^x \frac{\psi (t)}{t^2} dt$$ How should I go from here to prove…
Rob
- 1,203
7
votes
2 answers
Mean value of arithmetic function
Suppose we define a mean value of arithmetic function $G(f)$ as $$ G(f)=\lim_{x \rightarrow \infty} \frac{1}{x \log{x}} \sum_{n \leq x} f(n) \log{n},$$ and suppose now for an arithmetic function $f$, $G(f)$ exist and is equal to $A$, how to use this…
Rob
- 1,203
7
votes
1 answer
Möbius function sum
If gcd(a,b)=1, $1\leq b\leq a$, and $\mu(k)$ is the Möbius function, what is $$\sum_{k=0}^\infty\frac{\mu(ak+b)}{(ak+b)^s}$$
Can it be expressed in terms of other functions? Can I get it in the form of an Euler product?
Ethan Splaver
- 10,613
7
votes
1 answer
Question on de la Vallee Poussin's simplified proof of Dirichlet's theorem on primes in arithmetic progressions
I've been trying to understand de la Vallee Poussin's "Demonstration Simplifiee du Theorem de Dirichlet sur la Progression Arithmetique" and I've got stuck at the following step on pg 18 where Poussin takes the logarithmic derivative…
Lea M
- 119
7
votes
1 answer
Von Mangoldt function estimate
In many treatments of Vinogradov's three prime theorem, one considers the summation $S(\alpha) = \sum_{k \leq N}\Lambda(k)e^{2\pi i\alpha k}$ in place of $T(\alpha) = \sum_{p \leq N}e^{2\pi i\alpha p}$ (where this sum runs over primes). I assume…
quant10
- 71
7
votes
1 answer
Why is width of critical strip what it is?
For Riemann zeta function and $L$-functions of number fields, the width of critical strip is $1$. For $L$-functions of modular forms of weight $k$, the width of the critical strip is $k$.
Why is there a variation in the width of the critical strip…
angiquesophie