# Tagged Questions

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

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 ...
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),...
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, ...
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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43 views

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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. ...
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 ...
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 ...
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 ...
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.
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".
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 ...
2answers
78 views

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### 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 ...
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 ...
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'...
0answers
26 views

1answer
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1answer
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40 views