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

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6
votes
0answers
34 views

Books to read to understand Terence Tao's Analytic Number Theory Papers

I tried to understand Terence Tao's Analytic Number Theory Papers. For example, this paper, Every Odd Number Greater Than 1 is The Sum of at Most Five Primes. Which books shall I read to prepare ...
0
votes
0answers
9 views

Questions on Heath-Brown's paper “Kummer’s Conjecture for Cubic Gauss Sums”

On page 21 in Heath-Brown's paper "Kummer’s Conjecture for Cubic Gauss Sums" (http://eprints.maths.ox.ac.uk/158/1/kummer.pdf), a formula says $$\sum_{j\in \mathbb{Z}[\omega]}f(j)=\sum_{k\in ...
1
vote
0answers
12 views

a question on upper bound for Bessel function $K_{2it}(x)$

Can we have $$K_{2it}(x)\sinh(t)\ll_{x} 1$$ for $1<x< (1+|t|)^3,$ where $K_{2it}(x)$ deotes the ordinary K-bessel function and $t>1$. This is true when $x\ge (1+|t|)^3$ from some references. ...
1
vote
0answers
15 views

Solution of Pell equation over field of p-adic numbers

Right now I am studying Pell equation. Using continued fractions, we can find the solution of Pell equation. Now my question, is it possible for me to find a solution in the field of p-adic numbers ...
1
vote
0answers
27 views

Looking for proof of formula in WolframMathWorld article [duplicate]

I came across the formula (24) in the WolframMathWorld article on Web page http://mathworld.wolfram.com/TrigonometryAngles.html where no source of the proof could be identified by me. The formula is ...
0
votes
0answers
18 views

Estimation with logarithm

I have to prove: $q$ an integer $\geq 2$, $\tau$ a real number $\geq 2$ and $\sigma \leq 1- \frac{1}{\log q\tau}$, $M\geq 0$. Then $M^{1-\sigma }\leq (q\tau )^{-\sigma }$ and $(M+1)^{-\sigma }\leq ...
0
votes
0answers
26 views

Dirichlet L series estimation

let $\chi$ be a non-principal character modulo $q$, $M\geq 1$. I have to prove that, if $\vert \sigma - 1 \vert \leq \frac{1}{\log M}$, then $\vert \sum_{n=1}^M \chi (n)n^{-s}\vert\leq 1+e\log M$ and ...
-3
votes
0answers
13 views

How can y be used to factor N [duplicate]

Let x be the given muliple of φ(N). Then for any g in Z*[sub N] we have g^x=1 in Z[sub N]. Alice chooses a random g in Z∗[sub N] and computes the sequence ...
2
votes
1answer
20 views

Dirichlet character modulo p

How can I prove that if $\chi$ is a non-principal character modulo $p$ prime, then $\chi (-1) = \overline{\chi} (-1)= \pm 1$ and $\sum_{x=1}^p \chi (x) e^{2\pi i x}=0$? For the first question, I just ...
0
votes
1answer
15 views

Any books on Trigonometrical Sums (for the Theory of Numbers )?

All: Can anyone recommend good books on Trigonometrical Sums ? The only book I found is Vinogradov's book: Method of Trigonometrical Sums in the Theory of Numbers. but it is really old. I am ...
0
votes
1answer
26 views

Partial sums of powers of the divisor function

It is easy to establish that $$\sum_{n\le x}\tau(n) \sim n\log n$$ How would one find good bounds on $$\sum_{n\le x} \tau(n)^k $$ for some $k > 0$
2
votes
0answers
53 views

Mellin transform on $\mathbb{Z}[\omega]$

Let $\omega=\frac{-1+i\sqrt{3}}{2}$ be a complex cube root of unity. The Eisenstein integers $\mathbb{Z}[\omega]$ (a unique factorization domain) are of the forms $a+b\omega$ where $a$ and $b$ are ...
0
votes
0answers
41 views

sth interesting about $\gamma$ the euler's Constant

$\gamma $ is one of interesting constants in mathematics, like $\pi$ or $e$ ... But even more strange and mysterious!We even don't know about $\gamma$ whether it is rational or irrational, algebraic ...
0
votes
1answer
17 views

Why does $f(x) \asymp g(x) \implies log(f(x)) = log(g(x)) + O(1)$?

Why does $f(x) \asymp g(x) \implies log(f(x)) = log(g(x)) + O(1)$? Has it got something to do with the fact that \begin{align} f(x) \asymp g(x) \implies \exists c_1,c_2, \text{ such that}\\ ...
0
votes
1answer
12 views

Show that $\frac{1}{e^\gamma \text{log }x + O(1)} = \frac{1}{e^\gamma\text{log }x} + O\left(\frac{1}{(\text{log }x)^2}\right)$

Show that $\frac{1}{e^\gamma \text{log }x + O(1)} = \frac{1}{e^\gamma\text{log }x} + O\left(\frac{1}{(\text{log }x)^2}\right)$ I'm using one of Merten's estimates in a proof, the one that states ...
1
vote
1answer
27 views

Why is it impossible that $\frac{\phi(n^*)}{n^*} < \frac{\phi(n)}{n}$ when $n^* < n$

Why is it impossible that $\frac{\phi(n^*)}{n^*} < \frac{\phi(n)}{n}$ when $n^* < n$ and $n$ has $k$ prime factors, and $n^*$ is the product of the first $k$ prime factors? I understand that ...
1
vote
1answer
67 views

a problem about averages of fractional parts

I know a little analytic number theory , for example i know that: $\sum_{d\le x}(\frac{x}{d}-[\frac {x}{d}])=\frac{x}{a}(1-\gamma)+O(\sqrt x)$ when the $\sum$ is over $d$'s the integers less or equal ...
0
votes
1answer
46 views

Trying to show that $\phi(n) > c_1 \frac{n}{\text{log log }n}$ for some constant $c_1 > 0$

Trying to show that $\phi(n) > c_1 \frac{n}{\text{log log }n}$ for some constant $c_1 > 0$ where $\phi(n)$ is the euler phi function. I was wondering if I could use something like ...
0
votes
2answers
25 views

Is it true that $\sum\limits_{n=1}^N e^{in} = \frac{e^i(1-e^{iN})}{1-e^i}$ even if $| e^i | > 1$?

Is it true that $\sum\limits_{n=1}^N e^{in} = \frac{e^i(1-e^{iN})}{1-e^i}$ even if $| e^i | > 1$? I know this question is quite trivial and I will understand if it gets removed. I am trying to ...
0
votes
1answer
33 views

Is $\sum\limits_{\text{p prime}, p \geq 2}\frac{(-1)^{\frac{p^2-1}{8}}}{p}$ convergent or divergent??

Is $\sum\limits_{\text{p prime}, p \geq 2}\frac{(-1)^{\frac{p^2-1}{8}}}{p}$ convergent or divergent? So far I have that \begin{align} \sum\limits_{\text{p prime}, p \geq 2} ...
4
votes
2answers
47 views

$(n+1)^{\textrm{st}}$ prime less than $2^{2^n}$

Using elementary means, show that the $(n+1)^{\textrm{st}}$ prime is less than $2^{2^n}$ please do not use fancier stuff like the prime number theorem or beyond. using this how can you show that ...
0
votes
0answers
25 views

Singularities of zeta function

I have to prove (if $\gamma \ne 0$) that there is a analytic continuation for $\Re s >0$ of the function $$f(s)=\frac{\zeta (s)^2 \zeta(s-i\gamma )\zeta(s+i\gamma ) }{\zeta(2s)} $$ and that this ...
0
votes
0answers
17 views

Conductor of Dirichlet character divides every quasiperiod

Let $\chi$ be a Dirichlet character which has quasiperiods $d_1, d_2$. I.e., if $(n(n + kd_i), q) = 1$ then $\chi(n + d_i) = \chi(n)$ for any $k \in \mathbb{Z}$. Supposedly we can then show that ...
1
vote
2answers
68 views

Prove the Inequality on $\pi$-function

Prove that for each $y \geq 2$ , we have $\pi(x)+\pi(y)>\pi(x+y)$ for all sufficiently large $x$. I tried searching in the Internet for quite a while. The best result that I have found is L. ...
1
vote
1answer
51 views

Find all positive integer pairs $(x,y)$ and $(u,v)$ with certain relations.

Is there exists any positive integer pairs $(x,y)$ and $(u,v)$ for which, the relations, $x^2+y^2=u^2+v^2$ and $x^3+y^3=u^3+v^3$ are satisfied simultaneously?
3
votes
1answer
29 views

Can we have $\sum_{n\leq [x]}e^{-\sqrt{\frac{\log x}{r}}}\ll \frac{x}{e^{c \sqrt{\log x}}}$ for some constant $c>0$, where $x>1.$

Let positive interger $n$ is square-free, that is $n=p_1p_2\cdots p_r$ some $r$. Can we have $$\sum_{n\leq [x]}e^{-\sqrt{\frac{\log x}{r}}}\ll \frac{x}{e^{c \sqrt{\log x}}}$$ for some constant ...
2
votes
0answers
32 views

Sum of reciprocals of natural numbers with numerator being Legendre symbol mod 7 (L-series)

How do I show that $$\sum_{k=1}^\infty \left({k \over 7}\right)\Big/k = \sum_{k=0}^\infty \left(\frac{1}{7k+1} + \frac{1}{7k+2} - \frac{1}{7k+3} + \frac{1}{7k+4} - \frac{1}{7k+5} - ...
1
vote
1answer
61 views

Sum of inverse prime numbers

How can the following equation be proven? $\sum\limits_{p \le n; p \in P} \frac{\ln p}{p} \sim \ln(n) + O(1).$ I just wanna understand this sum Sum of reciprocal prime numbers
1
vote
1answer
39 views

Dirichlet characters - proof in a book

I found the following in a book and don't understand. Let $\chi$ denote a non-principal character modulo $q$ and $S(x)=\sum_{n\leq x}\chi (n)$. Then $\sum_{m>y} \frac{\chi(m)}{m} = \int_y^{\infty ...
1
vote
0answers
70 views

What are the “hidden” symmetries in Goldbach Conjecture?

What are the "hidden" symmetries in Goldbach Conjecture ? If Goldback conjecture is true, the basic instinct is that there must exist some "symmetries" which ensure (and lead) such properties. As we ...
2
votes
0answers
52 views

What are the missing gaps to prove Goldbach Conjecture?

When Andrew Wiles proved FLT, all he needed to do was to prove "semi-stable elliptic curve case" of Shimura-Taniyama conjecture. He did not need to start from scratch, he just needed to fill this ...
0
votes
2answers
50 views

Twin Prime Constant

How would one prove that the twin prime constant $$C_2 = \prod_{p > 2}1-\frac{1}{(p-1)^2} > 0$$ Simply computing the product for a large number of terms isn't rigorous, and simply establishes ...
1
vote
1answer
23 views

Prove the complex conjugate of an analytic function is analytic in the set of conjugates.

Given a function $f(z) \in C$ that is analytic, prove that $g(z) = \overline{f(\bar z)}$ is analytic in the set $\{\bar z : z \in C \}$. This is for homework: tips would be appreciated.
0
votes
0answers
39 views

Continuation of the Zeta Function

I already showed that für $\sigma >1$, $$\zeta (s) = \frac{1}{s-1} + \frac{1}{2} + \sum_{j=1; 2\mid j+1}^{k-1}\left( \prod_{i=0}^{j-1}(s+i) \right) b_{j+1}(0) - \left( \prod_{j=0}^{k-1}(s+j)\right) ...
3
votes
1answer
56 views

Application of the Green-Tao theorem

I am currently trying to find some good exercises in analytic number theory, suitable for undergraduates. I have mentioned the Green-Tao theorem for arithmetic progressions of primes but I am ...
7
votes
1answer
109 views

An Inequality Invollving The Riemann Zeta Function

I'm having trouble proving the following inequality for $2<r<3$: $$(1+2^{-r})\frac{(3^r+1)^2}{3^{2r}+1}>\frac{\zeta(r)}{\zeta(2r)}.$$ I can easily plot the graph, and the inequality clearly ...
1
vote
0answers
26 views

How do I determine asymptotic formulas for $\sum |\mu (n)|$ and $\sum \frac{\phi (n)}{n}$ using Perron summation?

Can someone please give a solution? I'm quite a beginner as far as techniques in analytic number theory are concerned and can't quite derive any formulas for these using Perron summation.
0
votes
0answers
36 views

How to use the fact that $\sum\frac{|\mu (n)|}{n^s}=\frac{\zeta (s)}{\zeta (2s)}$ to determine the number of square free integers below a number?

I want to use Perron summation to determine an asymptotic formula for $\sum |\mu (n)|$ with the sum being for all $n\leq x$ and which will give me the number of square free numbers below $x$. Please ...
0
votes
1answer
45 views

(absolute) Convergence of a series

I want to prove that the following series is convergent for $x>0$: $$ \sum_{n=1}^\infty \left( \prod_{p\mid n} \frac{1}{p-1}\right) n^{-x} $$ I tried to estimate the product but I didn't get so ...
0
votes
0answers
8 views

Represent the set of values for the mean of all pair combinations?

I apologize for not knowing how to ask this well. Its been a long time. If I have a set integers, how would you represent finding all the pair combinations and computing the average of each pair? ...
1
vote
0answers
30 views

Analytic continuation Dirichlet series

I have a Dirichlet series $A(s)$ with an absolutely convergent Euler product for $\sigma >0$. The zeros of the factors converge to $0+2\pi k$. I now have to proof that there can't be an analytic ...
0
votes
2answers
38 views

The average order of $\frac{\sigma_1(n)}{\sigma_0(n)}$

I want to calculate the average order of $\frac{\sigma_1(n)}{\sigma_0(n)}.$ I know that for every $e\gt0$,$$f(x):=\sum_{1\le n\le x}\frac{\sigma_1(n)}{\sigma_0(n)}=o(x^{2-e})$$ I wonder if it's true ...
3
votes
0answers
81 views

A question on the Prime number theorem

Let $N\geq1$. Could we infer $$\sum_{n\leq N}\mu(n)\ll N\exp(-c\sqrt{\log N})$$from $$\sum_{n\leq N}\Lambda(n)= N+O(N\exp( -c\sqrt{\log N})$$or $$\sum_{p \leq N}1=Li(x)$$ without resorting to the ...
1
vote
0answers
22 views

How to prove that $ \int_{2}^{x} \frac{dt}{(log(t))^{k}} = O \Big{(} \frac{x}{(log(x))^{k}} \Big{)} $ as $x \to \infty$? [duplicate]

For a homework exercise, we are asked to prove that $$ \int_{2}^{x} \frac{dt}{(\log(t))^{k}} = O \Big{(} \frac{x}{(\log(x))^{k}} \Big{)} \quad \text{, as } x \to \infty . $$ The following hint is ...
1
vote
0answers
17 views

limit of regular hyperbolic integrals is a unipotent integral (GL2 calculation)

In developing a simple trace formula for $G$=GL$_2$ over a number field $F$ one encounters the following identity of local integrals: $$\int_{G_v}f_v(g^{-1}\begin{pmatrix}1 & 1\\ 0 & ...
0
votes
0answers
73 views

Question about Riemann zeta function + my proof

First let me say that I am 16 years old so I am not very professional in math. English is also a second language so I apologize for any mistakes. Now i have been reading about the Riemann zeta ...
0
votes
0answers
60 views

Upper bound number of distinct prime factors

I want to prove that if $\omega (n)$ is the number of distinct prime factors of $n$ for $n>2$ we have $\omega (n) \leq \frac{\ln n}{\ln \ln n} + O(\frac{\ln n}{(\ln \ln n)^2})$. How can I do this? ...
0
votes
0answers
6 views

Points of a lattice inside square of side $N$

Let $\Lambda\subseteq \mathbb Z^m$ be a full-rank lattice of index $h$. I would like to know an upper bound for the quantity $H_N=|\Lambda\cap [-N,N[^m|$ where $[-N,N[^m=\{(a_1,\dots,a_m)\in \mathbb ...
1
vote
1answer
44 views

Best self study book with answers to selected questions for analytic number theory?

All: Can anyone recommend Best self study book with answers to selected questions for analytic number theory ? If a book have no answers to questions, but if you know if some professors choose the ...
0
votes
1answer
31 views

which algebraic number theory book with answers to selected questions for self-study?

All: Can anyone recommend some easy to follow algebraic number theory books with answers (hints) to selected questions for self-study ? If a have no answers to questions, but if you know if some ...