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

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2
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1answer
129 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 ...
0
votes
1answer
51 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),...
0
votes
0answers
30 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
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
43 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+...
1
vote
1answer
62 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(\...
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
57 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, ...
7
votes
1answer
84 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
26 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
0answers
28 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
votes
6answers
315 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
votes
0answers
72 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 ...
2
votes
1answer
58 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 ...
1
vote
1answer
40 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
33 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
50 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
1answer
31 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
2answers
78 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 \...
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)-\...
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(...
4
votes
1answer
67 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,...
1
vote
0answers
62 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 ...
0
votes
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
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'...
3
votes
0answers
26 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
vote
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
vote
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
votes
1answer
73 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 ...
2
votes
0answers
90 views

What are some practical attempts to disprove Riemann Hypothesis?

Most people believe Riemann Hypothesis is true. Since RH has not been proved yet, so it is not completely insane to disprove RH. Among the ways to disprove RH, straightforward ways, such as: try to ...
1
vote
0answers
32 views

Estimate of the log derivative of zeta function in the classic zero-free region

We know that the Riemann zeta function $\zeta$ has no zeros in the region $\{\beta+it:\beta>1-\frac{c}{\log(2+|t|)}\}$, where $c>0$ is an absolute constant. This is known as the classical zero-...
0
votes
0answers
12 views

Factorization of Dirichlet characters

A Dirichlet character $\chi$ of modulus $q$ is called primitive if it cannot be factored as $\chi=\chi'\chi_0^{''}$, where $\chi_0^{''}$ is a principal character and $\chi'$ is a character of modulus ...
1
vote
3answers
71 views

What are the applications of Sigma Function?

I read about the Sigma Function today.It tells that- The $\sigma(n)$ is the sum of all the positive divisors of $n$. But I had no idea how they can be useful.What are the practical applications ...
1
vote
3answers
39 views

Why does Dirichlet Series of Mangoldt Function has simple pole of order 1 at s = 1

Could someone explain why $\sum_{n=1}^\infty \frac{\Lambda(n)}{n^s} = -\frac{\zeta'(s)}{\zeta(s)}$ has a first order pole at $s=1$ with residue 1? That's what I found from Apostol's Introduction to ...
2
votes
1answer
90 views

How to find the bound of this sum?

Let $t>0,a(t)=\arg(\Gamma(1/4+it))$,$\kappa(n)=\frac{1}{2}x\pi n^2$,we need to calculate the bound,$A(x)$, of the following finite sum: $$ S(x)=\sum_{1\le n\le x}e^{\kappa(n)}\left(e^{ia(t)}(\kappa(...
0
votes
1answer
37 views

Finding an upper bound for a sum over primes

Fix $X>\geq 1$ a real number and let $1\leq y<X.$ Is there a positive constant $B$ such that $$\prod_{y<p\leq X} \left(1+\frac{3}{p}+ \sum_{\nu \geq 2} \frac{(\nu+1)^2}{p^{\nu}}\right)\leq B....
0
votes
1answer
16 views

Is Hurwitz contour integration the same as the Fourier series of the Hurwitz Zeta function?

Whittaker and Watson show a derivation of the Hurwitz representation of the Hurwitz Zeta function as a trigonometrical series. This represenation is achived by doing a countour integration. The ...
0
votes
0answers
46 views

Some inequalities with Mangoldt function

Let $\Lambda$ be the Mangoldt function defined by $$-\dfrac{\zeta'(s)}{\zeta(s)}=\sum_{n=1}^{+\infty}\frac{\Lambda(n)}{n^s}$$ then $$\Lambda(n)=\left\{% \begin{array}{cc} \log p & \text{if}\;n=p^...
2
votes
1answer
46 views

Is a modular form on $\text{SL}_2(\mathbb{Z})$ also a modular form on congruence subgroups?

Is a modular form $f$ of weight $k$ with respect to $\text{SL}_2(\mathbb{Z})$ always a modular form to a congruence subgroup $\Gamma$ (for example $\Gamma_1(N)$)? If the transformation law $f|_k\...
1
vote
1answer
43 views

Understanding the proof of $\phi(n)=\sum_{d\mid n}\mu (d) \left(\frac{n}{d}\right)$.

A proof for the identity $\phi(n)=\sum_{d\mid n}\mu (d) \left(\frac{n}{d}\right)$ several times in this website. After studying the book Apostle's Analytic Number Theory and failed to understand I ...
0
votes
0answers
9 views

What's the function that it is neccesary to show being bounded locally integrable in the Wiener-Ikehara Theorem?

When I am reading in (this video from an official channel in You$\color{red}{\text{Tube}})$ mathscienciechannel, that has the most high quality, in my attempt to understand the facts that currently I ...
0
votes
0answers
41 views

Characters with values on the $p$-adic complex field $\mathbb{C}_p$?

Characters $\psi : G \to \mathbb{C}$ from abelian groups $G$ to the complex field $\mathbb{C}$ are well-known and appear all over. Is there an analogue for the $p$-adic complex numbers $\mathbb{C}_p$, ...
-2
votes
1answer
30 views

Effective upper bound for a sum over prime numbers

Fix $y$ a positive real number. Is there an effective bound for the following sum i.e a positive constant B such that $$\sum_{p>y}\sum_{\nu \geq 4} \frac{1}{p^{9\nu/32}} \leq B.$$ Many thanks.
2
votes
1answer
49 views

Generalised Gauss sums

Let $\chi$ be a non-trivial Dirichlet character modulo an odd prime $p$ and let $f(x) \in \mathbb{Z}[x]$ be a polynomial. We define the generalised Gauss sum $$ G(\chi, f):=\sum_{y \in \mathbb{F}_p^*} ...
4
votes
1answer
67 views

My attempt to follow Tatuzawa and Iseki strategy to get a bound for $\int_2^x \frac{dt}{\log t}-\pi(x)$, where $\pi(x)$ is the prime counting function

I don't know if this exercise is in the literature, where $Li(x)=\int_2^x\frac{dt}{\log t}$ is the logarithmic integral and $\pi(x)$ is the prime counting function Question. Compute a good bound ...
1
vote
1answer
20 views

On computations around $\sum_{n=1}^N\frac{n\Lambda(n)}{n+N}$, where $\Lambda(n)$ is von Mangoldt function

By specialization with $F(x)=\frac{1}{1+x}$ in Apostol's Theorem 4.17 (Apostol, Introduction to Analytic Number Theory (Springer)), for intergers $N\geq 1$ one has $$\frac{\log N}{1+N}+\sum_{n=1}^N\...
1
vote
0answers
12 views

Bound on smallest $n$ for consistency of a system of equations?

Given small $\epsilon>0$ how small should $n\in\Bbb N$ be such that if $a,b,c,d,q,r,u,v,x,y,m,m'\in\Bbb N$ with $gcd(a,b)=gcd(a,x)=gcd(b,y)=1$ the following relations can hold with constraints $c,d=...
2
votes
0answers
40 views

derivative of riemann zeta function

I try to find a represantation for the derivative of the riemann zeta function. I do have for $\Re(s)>0$ and $s \neq 1$ $\zeta(s)=\dfrac{1}{s-1}+1-s\int_{1}^{\infty} \dfrac{x-\lfloor x \rfloor}{x^{...
5
votes
1answer
71 views

Average prime value in n factorial.

I was wondering about the (weighted) average prime value in the factorisation of $n!$. $\\$ If we call $f(n)$ the average prime value in $n!$, then $f$ seems to increase rather linear. Is there a ...
1
vote
0answers
70 views

Is this a new twin prime sieve method? Any information or comments is very appreciated.

I'm studying the twin prime numbers. Instead of sieving prime numbers, I found this method to sieve $\{x: x \neq \pm 1 \text{( mod $p$)}, x \in \mathbb{N}, p \le p_i\}$, so that $(x-1,x+1)$ will be ...