Questions on the use of the methods of real/complex analysis in the study of number theory.

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1
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1answer
35 views

Twin primes : prove the convergence of $ \lim_{N\sim\infty} \frac{1}{N} \sum^{N}_{p\in T} (\log(p)+\frac{1}{p})²$

let $T$ be the twin primes set : $p \in T $ if and only if $ p$ and $p+2$ are primes. Can you help me establish the convergence of : $$ \lim_{N\sim\infty} \frac{1}{N} \sum^{N}_{p\in T} ...
5
votes
2answers
185 views

least common multiple $\lim\sqrt[n]{[1,2,\dotsc,n]}=e$

The least common multiple of $1,2,\dotsc,n$ is $[1,2,\dotsc,n]$, then $$\lim_{n\to\infty}\sqrt[n]{[1,2,\dotsc,n]}=e$$ we can show this by prime number theorem, but I don't know how to start I ...
2
votes
0answers
33 views

Growth rate of arithmetical function

I'm interested in how one would estimate the growth rate of $$f(n)=\sum_{k\le n}\mu^2(k)\log(k)$$ I.e. sum of logarithms of square free integers. I can think of some trivial methods in my head ...
0
votes
0answers
24 views

A question on the big-O value of the complex integral especially in the number theory

My question is quite simple and elementary. Let $A(x)=\sum_{1}^{x}a(n)$ and $\alpha(s)=\sum_{1}^{\infty}a(n)n^{-s}$. Then, as we know, $$ A(x)= ...
1
vote
1answer
70 views

Sum of reciprocals of primes for known primes.

I was reading through some old analytic number theory notes earlier and found the interesting fact that even though $\sum\frac{1}{p}$ diverges: $\sum_{\text{known primes}}\frac{1}{p} < 4$. ...
1
vote
1answer
48 views

Problem from Apostol's analytic number theory book

Im trying to solve the exercise 13.2 in Apostol's analytic number theory book: Let $A(x)=\sum_{n\leq x}a(n)$, where $a(n)$ is zero unless $n=p^k$ for some prime $p$, in that case $a(n)=1/k$. Prove ...
0
votes
0answers
25 views

Riemann's hypotesis and some equivalences [duplicate]

Use that $\psi(x)=x+O(\sqrt{x}\:log^{2}x)$ to show that $\pi(x)=li(x)+O(\sqrt{x}\:logx)$ where $li(x)=\int_{2}^{x}\frac{dt}{logt}$ I tried but I get confused. Many pdf's say it's very easy to show. I ...
1
vote
1answer
52 views

On Newman/Zagier's proof of PNT

I have just got this paper: http://people.mpim-bonn.mpg.de/zagier/files/doi/10.2307/2975232/fulltext.pdf and I have a serious doubt: When proving that soft Tauberian theorem he explicitly uses ...
0
votes
0answers
26 views

Zeta Riemann Function

Use that $\zeta(s)=1+\frac{1}{s-1}-s\int_{1}^{\infty}\frac{\left\{u\right\}}{u^{s+1}}du$ if $Re(s)>0$ to show that. 1) $\zeta(s)=s\int_{1}^{\infty}\frac{\left[ u\right]}{u^{s+1}}du$ for ...
1
vote
1answer
38 views

Based on prime number theorem

I have a problem... If $A(x)=\sum_{n\leq x}a(n)$ where $a(n)=\frac{1}{k}$ if $n=p^{k}$ and $a(n)=0$ in other case, show that $A(x)=\pi(x)+O(\sqrt{x}\:log\:logx)$ I think I should use the theorem of ...
0
votes
1answer
34 views

Summation formula in dimension 2

One of the most common tools in analytic number theory is the summation by parts, my question is what is the similar formula when we are, for example, in dimension two and we have the sum $$ ...
0
votes
1answer
21 views

bounding gaps between points in an interval

I've been reading Davenport's Multiplicative Number Theory and came across something that I didn't understand. On p. 108, there is an argument for finding a lower bound on the imaginary parts $\gamma$ ...
1
vote
2answers
108 views

A set with zero density

Let $a>1$ be a positive integer and $f\in \mathbb{Z}[x]$ with positive leading coefficient. Let $S$ be the set of integers $n$ such that $$n \mid a^{f(n)}-1.$$ Prove that $S$ has density $0$; that ...
0
votes
0answers
54 views

Finding an asymptotic formula for $f(m,n)=\sum_{\substack{d\mid m \\ d\leq n}}1$?

$$f(m,n)=\sum_{\substack{d\mid m \\ d\leq n}}1$$ Here $n<m$ and $m$, $n$ are positive integers.
3
votes
0answers
49 views

Prove that the set has zero density

Let $a>1$ be a positive integer and $f\in \mathbb{Z}[x]$ with positive leading coefficient. Let $S$ be the set of integers $n$ such that $$n \mid a^{f(n)}-1.$$ Prove that $S$ has density $0$; that ...
4
votes
1answer
41 views

Identities of Hecke operators

While studying, I recently came across the following interesting problem. Let's say that the (level one) weight $k$ modular forms $M_k(\Gamma(1))$ have dimension $d$. We know by the ring structure ...
1
vote
1answer
29 views

How to show $\sum_{d\mid k}\frac{\mu (d)}{d}\left(\log\left(\frac{x}{d}\right)+O(1)\right)=\left(\sum_{d\mid k}\frac{\mu (d)}{d}\right)\log x+O(1)$

How to show this is true. $$\sum_{d\mid k}\frac{\mu (d)}{d}\left(\log\left(\frac{x}{d}\right)+O(1)\right)=\left(\sum_{d\mid k}\frac{\mu (d)}{d}\right)\log x+O(1)$$ I'm studying the book which is ...
5
votes
2answers
59 views

How to find an upper bound for $f(n)=\sum_{k=1}^{n}\frac{1}{d^{9}(k)}$?

How to find an upper bound for $$f(n)=\sum_{k=1}^{n}\frac{1}{d^{9}(k)}$$ where $d(n)$ is the divisor function?
2
votes
1answer
49 views

Confused About Step in Proof of Divergence of $\sum \frac{1}{p}$

I was going through the number theory text by Ireland and Rosen, and was following the proof of the divergence of the sum of reciprocal primes. But I came across a step unclear to me. The proof so ...
0
votes
1answer
34 views

What am I doing wrong with Möbius inversion?

Let $p(n)$ be $1$ if $n$ is a prime, and $0$ otherwise. Recall the prime divisor function. $$w(n)=\sum_{d\mid n}p(d)$$ By the Möbius inversion formula, we have $$p(n)=\sum_{d\mid n}w(d)\mu ...
2
votes
1answer
224 views

Is there a formula for Merten's function $M(x)=\sum_{n\leq x}\mu (n)$? [closed]

Is there formula for sum of the Möbius function, $$M(x)=\sum_{n\leq x}\mu (n)?$$
1
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2answers
27 views

How to show $\sum_{n\leq x}d(n)=\sum_{ab\leq x}1$?

How to show this equation below is true. $$\sum_{n\leq x}d(n)=\sum_{ab\leq x}1$$ $d(n)$ is the divisior function. It seems easy but i just can't see it.
2
votes
1answer
36 views

Apostol, Introduction to Analytic Number Theory, Chapter 1, Ex. 29

Given $n > 0$, let $S$ be a set whose elements are positive integers $\leq 2n$ such that if $a$ and $b$ are in $S$ and $a\neq b$ then $a\nmid b$ . What is the maximum number of integers that $S$ ...
0
votes
3answers
117 views

Calculating the class group of $\mathcal{O}_K$, for $K=\mathbb{Q}(\sqrt{7})$?

How to calculate the class group of $\mathcal{O}_K$, for $K=\mathbb{Q}(\sqrt{7})$ without using the Minkowski bound?
1
vote
1answer
70 views

Prove or disprove that $\forall k\in\mathbb N$ there exist tree consecutive primes such that $p_i-p_{i-1}\gt k$ and $p_{i+1}-p_{i}\gt k$

Prove or disprove that for every positive integer $k$, there exist tree consecutive primes $p_{i-1}, p_i, p_{i+1}$ such that $p_i-p_{i-1}\gt k$ and $p_{i+1}-p_{i}\gt k$. It's well known that ...
4
votes
1answer
51 views

What is $\varlimsup \frac{\omega(n)}{\log n}$?

$\omega(n)$ is the number of distinct prime divisors of $n$. How to figure out? $$\varlimsup_{n\to\infty} \frac{\omega(n)}{\log n}$$ or $ \dfrac{\omega(n)}{\log n}$ is convergent, so ...
10
votes
1answer
123 views

The probability that $\dfrac{p-1}2$ is square-free

Let $Q(x)$ denote the number of square-free integers between $1$ and $x$, we obtain the approximation $$\eqalign{ &Q(x)\approx x\prod_{p\,{\rm prime}}\left(1-\dfrac1{p^2}\right)=x\prod_{p\,{\rm ...
2
votes
1answer
59 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. ...
1
vote
1answer
85 views

Bessel function and upper bound

I'm stuck on this following problem: Let $G$ a function such that $0\leq G(t)\leq 1$, and $G(t)=1$ if $B^2\leq t\leq 4B^2$, with $\operatorname{supp}G\subset [\frac{1}{4}B^2, 9B^2]$ and $G^{(j)}\ll ...
0
votes
1answer
41 views

How do you generate results for various n in the following formula:

Let f be the arithmetic function defined by $f(n)$ = $3^{w(n)}$, where $w(n)$ is the number of distinct prime factors of n. Let $f^{-1}$ be the inverse of f with respect to the convolution product. ...
1
vote
1answer
33 views

Positive Integer points of $f(x)=\frac{1}{c-\frac{1}{x}}$, where c is fixed

So I am looking for the integer solutions of $f(x)=\frac{1}{c-\frac{1}{x}}$ for fixed $c\in \mathbb{Q}$ i.e. points $(x,f(x))\in \mathbb{N}\times \mathbb{N}$. (The c equals $\frac{4}{n}-\frac{1}{k}$ ...
3
votes
2answers
54 views

$\pi(x)\leq \frac x{f(x)}$ for some unbounded function $f(x)$

Let $\pi(x)$ denote the number of primes $\le x$. Can one prove $$\pi(x)\leq \frac x{f(x)}$$ for some function $f(x)(x\gt0)$, and $f(x)$ is unbounded? Please do not refer to prime number ...
2
votes
1answer
164 views

How do we prove $p_n\sim n\log(n\log(n))$ from the Prime Number Theorem?

Let $p_n$ be the $n$th prime. Could someone please help me with the steps between $\pi(n)\sim\dfrac{n}{\log(n)}$ and $n=\pi(p_n)$, to the statement $p_n\sim n\log(n\log(n))$?
0
votes
1answer
46 views

How would one find a) All the primitive characters modulo 8, b) All the non-primitive characters modulo 8?

Preferably explained in novice terms! I can start it off by having the multiplicative group modulo 8 with elements $[1], [3], [5], [7]$ and not sure where to go now. I see there is a similar question ...
2
votes
2answers
91 views

Derivative of $\Gamma$ at $1$

I've been given two definitions of the Gamma function, the integral defintion: $\Gamma(z) = \int_0^\infty t^{z-1}e^{-t}dt$ (for $Re(z)>0$) and the product definition (for $1/\Gamma$): ...
4
votes
1answer
96 views

Is this sequence monotonically decreasing?

Let $a_n = \frac{p_n - p_{n-1}}{p_n \log p_n}$ where $p_n$ denotes the $n$-th prime. Is this sequence decreasing (or decreasing after some $N$)?
3
votes
2answers
71 views

Number of algebraic integer divisors of an algebraic integer

Let $\alpha$ be an algebraic integer of degree $d$. Let $\tau(\alpha)$ be the number algebraic integers $\beta$ of degree $d$ such that $\alpha/\beta \in \mathbb{Z}$. What is a good upper bound on ...
3
votes
3answers
67 views

Inverse of Dirichlet series equality

I stumbled across a formula in here and tried to prove it for myself: $$\frac{1}{L(s,\chi)}=\sum\limits_{n=1}^{\infty}\frac{\mu(n)\chi(n)}{n^s}$$ However I got stuck. In my attempt I tried to show ...
2
votes
1answer
42 views

In how many ways can a number be factorized over the field $\mathbb{Z}_p$ into two numbers?

For example, over the field $\mathbb{Z}_5$, we can factor number 4 into two numbers in three different ways, i.e. 4=4$\times$1, 4=2$\times$2, and 4=3$\times$3. I am looking for a general formula of ...
-1
votes
1answer
64 views

Fermat Last theorem on Poly-Euler numbers

The poly-Euler numbers, denoted as $E_{n}^{(k)}$, are defined by the following generating functions :$${2\operatorname{Li}_k(1-e^{-x}) \over 1+e^{-x}}=\sum_{n=0}^\infty E_n^{(k)}{x^n\over n!}$$ The ...
1
vote
2answers
153 views

one square and seven cubes, circle method

I'm trying to solve the exercise 4 of chapter 2 of vaughan's book, i want to show that every large positive integer is the sum of one square and seven cubes. Can somebody give me the solution orm at ...
3
votes
0answers
75 views

Major arcs in the proof that every odd number is the sum of at most 5 primes

In his proof that all odd numbers greater than 1 are the sum of at most 5 primes, Terence Tao uses one large major arc around 0 rather than small ones around the rationals, which I am more accustomed ...
3
votes
0answers
55 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
47 views

Product of zeta and its conjugate

Suppose we have the zeta function $\zeta(s)$, and we want to multiply it by its complex conjugate $\zeta(s)^*$. Since $\zeta(s)^* = \zeta(s^*)$, we get $\displaystyle \zeta(s)\cdot\zeta(s)^* = ...
0
votes
1answer
32 views

Evaluation of Riemann-Stieltjes integral in Laurent expansion of zeta function

I'm probably being really stupid but in a proof of the Laurent expansion of the Riemann zeta function the quantity \begin{equation} S_r(t) = \sum_{n \leq t} \frac{(\log (x/n))^r}{n} \end{equation} is ...
2
votes
1answer
48 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 ...
4
votes
1answer
44 views

Motivation for using $L(1,\chi)$ in the proof of Dirichlet's Theorem

Having read the proof of Dirichlet's Theorem on the infinitude of primes in arithmetic progressions, I am left wondering what his motivation for studying $L(1,\chi)$ was and why it is reasonable that ...
0
votes
0answers
61 views

A partial sum involving Euler's function

This is Exercise 2.1.17 of the book "H. Montgomery and R. Vaughan. Multiplicative Number Theory— I. Classical Theory". For $x\ge 2$, $\sum_{n\le x}\frac{\mu(n)^2}{\varphi(n)}=\log ...
6
votes
1answer
105 views

2014 USAMO #6, analytic number theory

Prove that there is a constant $c > 0$ with the following property: If $a, b, n$ are positive integers such that $ \gcd(a+i, b+j)>1 $ for all $ i, j\in\{0, 1,\ldots, n\} $ then $$ \min\{a, ...
1
vote
0answers
26 views

Dirichlet L function

The function is defined here - http://en.wikipedia.org/wiki/Dirichlet_L-function If $\chi$ is primitive and $\chi(-1)=1$ how do I show that $L$ has infinite number of zeros in the critical strip