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

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Justify $\lim_{n\to\infty}n^p\int_0^1\sum_{k=n}^\infty\frac{\sigma(k)e^{x/k}}{k^{p+2}\log\log k} dx=\frac{e^\gamma\int_0^1f(x)dx}{p}$

Inspired in PROBLEM 207, La Gaceta de la Real Sociedad Matemática Española, Vol. 16, N0. 3 (page 507 in spanish, proposed and solved by Furdui), I've tried write examples of this new statement ...
2
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1answer
31 views

Explaining an integral involving the divisor function

In a 1973 paper by Martinet, Deshouilliers and Cohen, $A(x)$ is defined as $$A(x)=\lim_{N\to\infty}\frac{\#\{n\leq N\mid \frac{\sigma(n)}{n}≥x \}}{N}$$ where $\sigma(n)$ is the "sum-of-divisors" ...
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0answers
18 views

Dirichlet character to prime power modulus

Let $p$ be an odd prime number and let $\alpha \geq 1$ be an integer. Let $\chi$ be a real, primitive Dirichlet-character mod $p^{\alpha}$. How does one show that $\alpha = 1$? If we choose an ...
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0answers
21 views

Does existence of mean of arithmetic function imply it is bounded except on a set of density zero? [on hold]

Let $f(n)$ be an arithmetic function whose mean is finite. Is $f$ bounded outside a set of density zero?
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0answers
65 views

Are complex numbers complete in every way?

I was told many times a story. Indeed a fascinating one to me as a student learning mathematics. First there were natural numbers. People started adding things and finding solutions to finding the ...
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1answer
37 views

How Changing the order of integration(Elementary proof of the prime number theorem)?

I'm studying the exchange of integration order, I need help, any hint? For every real number $\rho \geq 0$, write $V(\rho)=e^{-\rho}R(e^{\rho})=e^{-\rho}\psi(e^{\rho})-1$ where $\psi(x)$ is the ...
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1answer
55 views

Contour Integration, Riemann Zeta (-n)

I was reading Riemann's Zeta Function by H. Edwards, and could not understand the equation on the page 12. \begin{align*} \zeta(-n) &= \frac{\prod(n)}{2\pi i}\int_{+\infty}^{+\infty} \frac{(-x)^{-...
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26 views

The asymptotic behaviour of $\sum_{1\leq k\leq N-1}\int_{p_k}^{p_{k+1}}\log x d[x]$, where $p_n$ is the nth prime number

Let $p_k$ is the kth prime number and consider for $N\geq 2$ the arithmetic function $$f(N)=\sum_{k=1}^{N-1}\int_{p_k}^{p_{k+1}}\log(x) d[x]$$ where $[x]$ is the integer part function (provide us in ...
3
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1answer
120 views

Is the infinite decimal fraction $1.23456…n$ irrational?

How to prove that the number $ 1.23456\dots n$ is an irrational number? The number consist, of course, of natural numbers in increasing sequence.
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69 views

When does $\sum_{p\in\mathbb{P}} \frac{1}{|p|^2}$ diverges?

We know $\sum_{p\in\mathbb{P}} \frac{1}{|p|^2}$ diverges where $\mathbb{P}$ denotes set of all primes in $\mathbb{Z}[i]$ (because that sum is greater that $\sum_{p \equiv 3 \mod 4} \frac{1}{p}$, which ...
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0answers
54 views

Next book in in learning Analytic Number Theory

I have just finished the book "Tom M. Apostol - Introduction to Analytic Number Theory". My aim is to reach to graduate level to do research, especially on Rationality/Irrationality and Algebraic/...
3
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2answers
46 views

Estimate for $\sum_{q=1}^{M}\frac{\varphi(q)}{q^{2}}$ Related to Bourgain Paper [duplicate]

Let $N\gg 1$ be a large parameter, which I ultimately want to let tend to infinity. I am reading an old paper of Bourgain, where he claims the lower bound (Equation 2.50, pg. 118) $$\sum_{q=1}^{N^{1/...
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1answer
70 views

about integral logarithm

I would to ask for a logarithm integral, used for Gauss. I read that he uses it to calculate the number of primes, less than a given natural number. It is like: $Li= \int_{0}^x(dt/lnt)$ I read that he ...
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2answers
66 views

Theorem 3.16. in Analytic Number Theory by Apostol

The below texts are from the book Introduction to Analytic Number Theory by Apostol: I have two questions which I couldn't find solutions for them: $1-$ According to Thm 3.16., $\sum_{n\le x} \...
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1answer
95 views

Prove $\sum_{d\le x} \sum_{q\le x/d} \dfrac{1}{q^{\beta}} = \sum_{d\le x} \dfrac{1}{d^{\beta}} \sum_{q\le x/d} 1$?

The below texts are from the book Introduction to Analytic Number Theory by Apostol: Trying to calculate $\sum_{n\le x} \sigma_{\alpha} (n)$ for negative $\alpha$ I followed the advice of the book, ...
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0answers
29 views

Probability distribution of $\omega'(n)$. [duplicate]

$\omega(n)$ is the number of distinct prime factors of $n$ and $\omega'(n)$ is the number of distinct prime factors of $n$ with multiplicity. For example if $p,q$ are prime numbers then $\omega(p^2q)=...
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0answers
22 views

Get an upper bound of $\left| F(1+it) \right|$ in an example of Perron type formula

From Proposition 3 of Tao, A cheap version of Halasz’s inequality, I know how get for example upper bounds for $x,T\geq 1$ $$\frac{1}{x}\sum_{n\leq x}\frac{\mu(n)\log n}{n}\ll\int_{-T}^{T} \left| \...
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76 views

Theorem 3.3 Apostol's Analytic Number Theory

The below texts are from the book Introduction to Analytic Number Theory by Apostol: Note. Part (d) Thm 3.2 [green-underlined] is $$\sum_{n\le x}n^a=\dfrac{x^{a+1}}{a+1} +O(x^a) \ \ \text{if} \ a\...
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1answer
20 views

An upper bound of $ \left| \frac{1}{s}\log\zeta(s) \right| $ for $\Re s=\sigma>1$, from this integral formula and a related comparison

For $\Re s=\sigma>1$ one has the following known formula $$\frac{1}{s}\log\zeta(s)=\int_1^\infty \Pi(x)x^{-s-1}dx,$$ then if we take the derivative we can write $$\frac{1}{s}\log\zeta(s)=s(s+1)\...
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1answer
56 views

Given $N$ find the number of natural numbers less than $N$ that may be written in the form $\frac{(k)(k+1)}{2}$

Given $N$, find the number of natural numbers less than $N$ that may be written in the form $$\frac{k(k+1)}{2},$$ where $k\in \Bbb N$. I know that the answer to this problem is approximately $\sqrt {...
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28 views

Confusion between Sequences and Number theoretic functions.

I've just started learning Number Theoretic function,the definition of ,Number Theoretic function,which i've just read created some confusion b/w Number Theoretic function & Sequences. The ...
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1answer
121 views

What is number theory today? [closed]

I try to explaine my problem and I hope do not disturb or annoy; I know that number theory is very vast but essentially it is divided into two parts: analytic number theory and algebraic number ...
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1answer
49 views

On computations related with $\lim_{x\to\infty} e^{-x}\sum_{\rho}\frac{(e^x)^\rho}{\rho}=0$

When I've reproduced the shape of the function $\sigma(x)$ of Apostol's section 4.10, a view of the page 98 is avaible as a Google Book (Apostol, Introduction to Analytic Number Theory, Springer 1976),...
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29 views

On inequalities related with $f(s):=-(1-\frac{2}{2^s})^{-1}$

My Question. a) How can you prove easily that the multivariable function in LHS is positive on $x^2+y^2<1$ $$2^{1-x}\cos(y\log 2)-1>0?$$ b) Let $s=\sigma+it$ the complex variable, ...
3
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0answers
40 views

Good approximation to zeta function in the critical strip by smoothed sum

I'm self-studying analytic number theory from terry tao's blog, there is an exercise (Exercise 33) from the blog that I cannot solve: Let ${\eta: {\bf R} \rightarrow {\bf C}}$ be a smooth ...
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42 views

BBP formula for $e$

For the number $\pi$ we can use the BBP formula to find a sequence of digit starting from the digit $n$, simply using the formula: $$\displaystyle\pi=\sum_{k=0}^\infty\dfrac{1}{16^k}\left(\dfrac{4}{8k+...
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1answer
61 views

I have difficulties in solving problems in analytic number theory.

My problem consists of 3 parts. Let $\alpha,\beta>0$ and $\alpha\beta=\pi^2$ (1) Let $f(\alpha)=\sum_{k=0}^{\infty}\frac{1}{(2k+1)(e^{(2k+1)\alpha}+1)}+\frac{1}{8}\log\alpha$ Then $f(\alpha)=f(\...
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1answer
36 views

Equidistributed problem about polynomial with irrational coefficient

This problem is from Stein, Fourier Analysis,Chapter 4,problem 2(d). Problem:Suppose that $P(x)=c_n x^n+……+c_0$ is a polynomial with real coefficients, where at least one of $c_1,……,c_n$ is ...
2
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1answer
56 views

Does satisfy $f(n)=\frac{\sigma(n)}{n^2}$ the hypothesis of Halasz’s inequality?

Let $\sigma(n)=\sum_{d\mid n}d$ the sum of divisor function. I would like to know if I can write an example of some of the following Theorem 1 or Theorem 2 from $$f(n)=\frac{\sigma(n)}{n^2}$$ in Tao, ...
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1answer
78 views

Modular transformations of $\eta(\tau)$

Under a modular transformation the Dedekind $\eta$ function transforms as $$\eta(-1/\tau) = \sqrt{-i}\eta(\tau).\tag*{$(*)$}$$Siegel gives a proof in this paper here that uses complex analytic ...
2
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0answers
25 views

Convergence of the sum $\sum\limits_{p}^{}\sum\limits_{k=1}^{\infty}\frac{\log p}{p^{ks}}$

How can I prove the following sum converges, where $s>1$ and the sum is over all primes. $$\displaystyle\sum_{p}^{}\displaystyle\sum_{k=1}^{\infty}\frac{\log p}{p^{ks}}$$ I've tried grouping terms ...
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0answers
27 views

Legendre's Conjecture Theme (Part II)

This is a continuation of this question. My main question is that, in the previous question we were mainly concerned about the sign of, $$f_{2}(n)=\pi\left((n+1)^2\right)+\pi\left(n^2\right)-2\pi\left(...
13
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6answers
303 views

Is $ \sin: \mathbb{N} \to \mathbb{R}$ injective?

I was trying to show that $\sin(x)$ is non-zero for integers $x$ other than zero and I thought that this result might emerge as a corollary if I managed to show that the result in question is true. ...
2
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0answers
66 views

Legendre's Conjecture Theme (Part I)

Main Question Recently I have been thinking about the Legendre's Conjecture. I noticed that a proof of the conjecture can be obtained if we can prove any one of the following, Conjecture 1. For ...
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1answer
56 views

Characters on rings of residue classes modulo polynomials over finite fields

First recall the following orthogonality relation on $\mathbb{Z}/n\mathbb{Z}$. Fix $n \in \mathbb{Z}$, $n \neq 0$. For $r \in \mathbb{Q}$, let $e(r) := e^{2 \pi i r}$. Let $x \in \mathbb{Z}$. Then ...
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1answer
39 views

Positive integral solutions of $\pi(x)+\pi(y)=2\pi\left(\dfrac{x+y}{2}\right)$

Recently I was reading one of my earlier posts. There it has been conjectured that, For all sufficiently large $x,y$ we have, $$\pi(x)+\pi(y)\le 2\pi\left(\dfrac{x+y}{2}\right)$$ But it turned ...
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1answer
30 views

Primes with $p^9\pm1 = q^4r$

Are there distinct primes $p,q,r$ with $$ p^9\pm1 = q^4r $$ ? This is related to a series of conjectures going back to Erdos regarding $d(n)=d(n+1)$. Of course either $q$ or $r$ is 2.
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1answer
45 views

If the value of Mertens function follow normal distribution, does this imply Riemann Hypothesis?

If the value of Mertens function follows normal distribution, does this imply Riemann Hypothesis ? I thought the answer shall be NO, because normal distribution still has "long tail".
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1answer
28 views

How to get values of Summatory Liouville function from Mertens function?

All: For Liouville function λ(n), we can define summatory Liouville as the accumulated sum of of λ(n). Mertens function is the accumulated sum of Mobius function. Is there any ways to get the value ...
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2answers
76 views

Integral solution of separable differential equation

On page 524 of Tenenbaum's Introduction to Analytic and Probabilistic Number Theory (3rd edition) it is essentially stated that the solution to the first-order differential equation $$y' = e^{-x}y/x \...
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0answers
14 views

Doubts and computations about Dirichlet series and aliquot sequences II

From previous post* dedicated to aliquot sequences I believe that I can state that for $\Re s>2$, on assumption that the Catalan-Dickson conjecture is false $$\sum_{n=1}^{\infty}\frac{s^{k+1}(n)-\...
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0answers
24 views

Doubts and computations about Dirichlet series and aliquot sequences I

Perhaps the more easier statement that one can deduce for aliquot sequences (which is the Wikipedia's Page) is the following Lemma. For an integer $n\geq 1$, let $s^0(n)\equiv n$, $s(n)\equiv s^1(...
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1answer
61 views

Asymptotic density of Zhang's primes

By this point, it is well known that Yitang Zhang's result implies for some $c$, there are infinitely many primes $p$ such that $p+c$ is also prime, and that the smallest such $c$ is less than $70,000,...
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61 views

Proving this formula for the Zeta function?

Could some one link me to a proof of this integral? $\zeta{(s)} = \frac{1}{\Gamma{(s)}}\int_{0}^{\infty} \frac{x^{s-1}}{e^x - 1} dx$ All the sites I've seen so far just introduce with the definition ...
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0answers
31 views

How to prove an inequality in number theory without induction?

$\sum_{n=1,(n,m)=1 }^{km} \frac{1}{n} \leq (\gamma + \log(km) + \sum_{p|m}\frac{\log p}{p-1})\prod_{p|m}(1-\frac{1}{p})+\frac{2^{\pi(m)}}{km}$ where $\pi(m)$ is the number of distinct prime ...
2
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1answer
56 views

Hardy- Littlewood Circle Method

I'm currently trying to get to grips with the Hardy Littlewood circle method so I'm working through Vaughan's book. In the past I've been very bad for leaving a point behind if I don't follow it so I'...
3
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0answers
25 views

Sum of integers and zêta functions

I am working on generalizing some works from the usual rational case to general number fields. That implies some technical changes I am not really at ease with. For instance: $$\sum_{m \leqslant X} m ...
1
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1answer
46 views

doesn't exist an $N$ s.t. all $n \ge N$ satisfy an equation.

I came across this problem on my own and i'm asking for any potential techniques/strategies/hints for attacking it. Prove that there does not exist an $N$ such that for every natural number $n \...
1
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0answers
28 views

$p$-adic Fourier transforms and orthogonality relations

In $\mathbb{C}$, we have the following orthogonality relation $$ \int_{0}^{1} e^{2\pi i (m-n)x} dx = \begin{cases} 1 & \mbox{ if } m = n;\\ 0 & \mbox{ otherwise.} \end{cases} $$ Do we have ...
4
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1answer
70 views

Is a strong form of Goldbach conjecture equivalent of Generlized Riemann Hypothesis?

In Andrew Granville's paper: REFINEMENTS OF GOLDBACH’S CONJECTURE, AND THE GENERALIZED RIEMANN HYPOTHESIS He said that: "we show that if a strong form of Goldbach's conjecture is true then every ...