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

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2
votes
1answer
22 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" ...
1
vote
0answers
15 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 ...
-1
votes
0answers
19 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?
13
votes
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. ...
0
votes
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 ...
1
vote
2answers
75 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 \...
1
vote
0answers
64 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 ...
3
votes
1answer
119 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.
0
votes
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)^{-...
1
vote
0answers
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 ...
1
vote
1answer
39 views

Alternative proofs that Dirichlet products are associative?

Is there alternative proof of the following fact: Dirichlet product on arithmetic function is associative. I'm looking for something different than that given in Dirichlet's product with ...
3
votes
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/...
1
vote
1answer
56 views

On the sum of the reciprocals of the zeros of $\zeta(s)$

It is well known that whenever $\rho$ is a nontrivial zero of the Riemann zeta function $\zeta(s)$, then $1-\rho$ is also a zero. But does the equality $\Re \sum_{\rho} \dfrac{1}{\rho} = \Re \sum_{\...
3
votes
0answers
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 ...
2
votes
1answer
35 views

Can the Von-Mangoldt function and the Chebyshev function be defined for entire complex plane?

Can the von-Mangoldt function and the Chebyshev function be defined for the entire complex plane ? I assume so, but I had not seen the definition. Can anyone provide some links for this? Thank you.
8
votes
1answer
1k views

Divisor summatory function for squares

The Divisor summatory function is a function that is a sum over the divisor function. $$D(x)=\sum_{n\le x} d(n) = 2 \sum_{k=1}^u \lfloor\frac{x}{k}\rfloor - u^2, \;\;\text{with}\; u = \lfloor \sqrt{x}...
0
votes
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/...
0
votes
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 ...
0
votes
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} \...
2
votes
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, ...
0
votes
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| \...
0
votes
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)=...
0
votes
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)\...
1
vote
0answers
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\...
2
votes
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'...
1
vote
1answer
1k views

Euler totient function sum of divisors. Theorem 2.2 Apostol

Prove that : If $ n\ge{1} $ then $ \sum_{d|n}\phi(d)=n $ Let $S$ denote the set {1,2,...,n}. We distribute the integers of $S$ into disjoint sets as follows. For each divisor $d$ of $n$, let $A(d) ...
8
votes
1answer
185 views

Riemann zeta function and the volume of the unit $n$-ball

The volume of a unit $n$-dimensional ball (in Euclidean space) is $$V_n = \frac{\pi^{n/2}}{\frac{n}{2}\Gamma(\frac{n}{2})}$$ The completed Riemann zeta function, or Riemann xi function, is $$\xi(s) ...
0
votes
0answers
26 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(...
2
votes
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 ...
-1
votes
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 {...
-1
votes
0answers
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 ...
1
vote
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(\...
2
votes
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 ...
2
votes
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, ...
0
votes
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),...
3
votes
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 ...
0
votes
0answers
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, ...
0
votes
0answers
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+...
0
votes
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
votes
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 ...
6
votes
3answers
333 views

Is $\eta^{24}(\tau)\,j(\tau) = {E_4}^3(q)$?

Given the j-function $j(\tau)$, $j(\tau) = 1728J(\tau)$, where $J(\tau)$ is Klein’s absolute invariant, the Dedekind eta function $\eta(\tau)$, and the following Eisenstein series, $\begin{align} ...
0
votes
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(...
7
votes
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
votes
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 ...
0
votes
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 ...
1
vote
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 ...
3
votes
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.
1
vote
1answer
44 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".
0
votes
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)-\...
4
votes
1answer
60 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,...