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
80 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
54 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 ...
11
votes
1answer
142 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
66 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
102 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
42 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
37 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
55 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
183 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
56 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
99 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
102 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
78 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
78 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
70 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
162 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
76 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 ...
4
votes
0answers
60 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
50 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
35 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
50 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
58 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
65 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
117 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
32 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
2
votes
1answer
96 views

$\sum_{p\le x} \frac{1}{pq}$

I was given that $\sum_{p\le x} \frac{1}{p}$ = $\log\log x$+O(1). I need to show that $\sum_{pq\le x} \frac{1}{pq} = (\log \log x)^2 + O(\log \log x)$. Here we go: Break the sum into two sums: ...
0
votes
0answers
55 views

Generalized Riemann Hypothesis : Zeros of Dirichlet L function and its functional equation

Let $\chi$ pe a primitive character modulo q with $\chi(-1)=1$ ; L is the Dirichlet - L function Define, $\xi(z,\chi)=(q/\pi)^{z/2}\Gamma(z/2)L(z,\chi)$ Show that $L(z,\chi)$ has infinitely many ...
1
vote
1answer
46 views

Changing order of summation with Mobius function

Let $\mu(d)$ be the Mobius function, and $\mu_r(d)$ be the modified Mobius function which satisfies $\mu_r(d)=0$ if $d$ has strictly more than $r$ distinct prime factors. Let $\psi_r(n)=\sum_{d\mid ...
1
vote
0answers
32 views

Generalized Logarithmic Integral

Euler's logarithmic integral (of particular application in the Prime Number Theorem, for instance) is of the form \begin{equation*} \text{li}(x) := \int_0^{x} \frac{dt}{\log t} \end{equation*} and it ...
2
votes
1answer
66 views

Residues of $\frac{x^s}{s}\frac{\zeta'(s)}{\zeta\phantom{'}(s)}$

Going through a proof in Analyti number theory, the calculation of the residues of $$ f(s) = \frac{x^s}{s}\frac{\zeta'(s)}{\zeta\phantom{'}(s)} $$ came up. I do have some experience with complex ...
1
vote
1answer
34 views

Low bound of Dirichlet eta function

every one. Suppose that $\eta(s)$ is Dirichlet eta function, I may find a low bound of this function, namely $\eta(2n)>\frac{2^{2n-1}-2}{2^{2n-1}-1}$ with $n>1$ and $n$ is a integer. But is ...
4
votes
2answers
72 views

Squares modulo 2^n

How many squares are there modulo $2^n$? If we would deal with $p^n$, where p an odd prime, then we could use Hensel's Lemma, which clearly doesn't work with $2^n$.
4
votes
2answers
136 views

What does this $\asymp$ symbol mean? (subject: analytic number theory)

I'm reading a survey article by Andrew Granville on analytic number theory. On page 22 of the paper, there appears a strange looking symbol, undefined. I've circled it in red in the screenshot ...
2
votes
1answer
97 views

Error term of the prime number theorem in arithmetic progressions

It is known that if $(a, q)$ and $q\le (\ln x)^N$, then the following is true $$\sum_{k\le x, k\equiv a\pmod{q}}\Lambda(k) = \frac{x}{\phi(q)} + O(x\exp(-C\sqrt{\ln x}))$$ where $C$ depends only on ...
2
votes
1answer
65 views

Rewriting $\tau(p)\Delta(\tau)$ when $p$ is prime

$p$ is a prime, and $\tau$ is Ramanujan's tau function: $$p^{11}\Delta(p\tau)+\frac{1}{p}\sum_{k=0}^{p-1}\Delta\bigg(\frac{\tau + ...
2
votes
2answers
123 views

Apostol - Analytic Number Theory, Chapter 3 problem 4a

The problem comes from "Introduction to Analytic Number Theory" by Tom M. Apostol, Chapter 3, Problem 4a: Question: Prove $\sum_{n \le x} \mu(n)[ \frac xn]^2 = \frac{x^2}{\zeta(2)} + O(x \log(x))$ ...
1
vote
1answer
56 views

Asymptotic result about analytic number theory

I don't know if there is any done work done about ehis matter, and I don't have access to research news. I'm interested in this question (I haen't tried to answer it myself, but it seems very ...
1
vote
1answer
53 views

Rational vs irrational

If two points on a number line is shown, are rational numbers between the two points is more or irrational number is more ? I have tried using probability , my collegue who was like my teacher also ...
1
vote
0answers
58 views

Generalizations of results on the sum of divisors function over $\mathbb{Q}$ to number fields

Consider the sum of divisor function $$ \sigma_0(n) = \sum_{d\mid n} 1. $$ This is known to satisfy $\sum_{n\leq x} \sigma_0(n) = (x\log x)+2\gamma x+\mathcal{O}(\sqrt{x})$. If, instead, we shift the ...
2
votes
2answers
98 views

Riemann zeta, why are the residues either zero or one?

One more question, probably equally simple to answer but I don't know how this is true either: Why is the residue of Riemann zeta zero - trivial or non-trivial: $$\text{residue}\left(\zeta ...
1
vote
1answer
85 views

Prime Counting: Relationship between Chebyshev's function and the Prime counting function

How do I show that if $\psi(x)=x+O(x^{1/2}\log^2(x))$ then $\pi(x)=\int_2^x \frac{dt}{logt} + O(x^{1/2}\log x)$ Where $\psi(x)$ is Chebyshev's second function and $\pi(x)$ is the prime counting ...
2
votes
1answer
76 views

Pole of Riemann zeta and Riemann zeta zeros, prove this relation.

Prove this relation: $$\displaystyle \lim_{s\to 1} \, \left(\zeta (s)-\frac{\zeta '(s-1+\rho _n)}{\zeta \left(s-1+\rho _n\right)}\right)=\gamma -\frac{\zeta ''(\rho _n)}{2 \zeta '(\rho ...
0
votes
1answer
51 views

Equation with a sum for the prime-counting function involving the Mobius function

I have come across the statement that $$ \sum_{n\leq x}\sum_{d\mid(n,P_z)}\mu(d) = \sum_{d\mid P_z}\mu(d) \left[\frac{x}{d}\right], $$ where $P_z=\prod_{p\leq z}p$ where $p$ is prime, $\mu(d)$ is the ...
5
votes
2answers
395 views

Applications of generating functions to number theory

I am familiar (at least at a cursory level) with the extensive role generating functions play in the theory of partitions. What are some other prominent applications of generating functions to number ...
1
vote
0answers
40 views

Relation between Dirichlet convolution and Bell series and convolution of functions and the Fourier transform?

We define the Dirichlet convolution of two arithmetical functions $a,b:\mathbb{N}\to\mathbb{C}$ to be $$ (a*b)(n)=\sum_{d\mid n}a(d)b\left(\frac{n}{d}\right). $$ Given a prime $p$, we define the Bell ...
2
votes
1answer
45 views

Sums Involving the Mobius Function

Are there any good approximations for the following sums in terms of $n$? $$\sum_{k=1}^{n}\mu(k)$$ $$\sum_{k=1}^{n}\mu(k)\log^m(k)$$ $$\sum_{k=1}^{n}\frac{\mu(k)}{k}.$$ I realize that the third sum ...
3
votes
1answer
94 views

Can the exact value of the product over the Riemann zeta function at even arguments be evaluated?

According to wolframalpha, the product over the Riemann zeta function at even arguments converges : $$\prod_{n=1}^{\infty} \zeta(2n) \approx 1.82 $$ Q1: Can it be proved that this product actually ...
2
votes
1answer
39 views

Numerical verification of the ternary Goldbach conjecture

In his proof of the ternary Goldbach conjecture, H.A. Helfgott says that it has been verified that every odd number less than $N_0 = 10^{30}$ is the sum of at most 3 primes. How would one verify this ...
2
votes
0answers
32 views

Zero Free Regions of Zeta'

I'm interested in calculating all of the zeroes of the first derivative of the Riemann $\zeta$ function. I know that (on the RH), all of these zeroes will have real part $\geq \frac{1}{2}$. I am ...