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

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0
votes
1answer
49 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
71 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
1answer
21 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
16 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
29 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
80 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
56 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
1answer
43 views

Is $\sum_{p\text{ prime, } p \geq 2}\frac{(-1)^{\frac{p^2-1}{8}}}{p}$ 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} ...
0
votes
0answers
50 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
31 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
88 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
55 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
36 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
40 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} - ...
2
votes
1answer
95 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
44 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
101 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
88 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 ...
1
vote
2answers
64 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
71 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
51 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
64 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
155 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
43 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
1answer
46 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 ...
1
vote
0answers
42 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
50 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
92 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
25 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
19 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
81 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
1answer
243 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
12 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
132 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
55 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 ...
0
votes
1answer
39 views

Question about the Chebyshev Inequality.

Let $p_1 < p_2 <\dots < p_n$ be the $n$ first primes listed in crescent order. Using the Chebyshev Inequality (for $x$ sufficiently large) $$0.92\leq \frac{\pi(x)\log x}{x}\leq 1.11,$$ How ...
0
votes
0answers
17 views

Bertrand's Postulate and and Chebyshev Inequality

Let $\theta(x) = \sum_{p\leq x}\log p$ and $\pi(x) = |\{p\leq x:p\text{ is prime}\}|$. Using Abel's formula, one can prof the following $$\pi(x) = \frac{\theta(x)}{\log x} + ...
1
vote
1answer
52 views

Partial summation formula and integral

I have to prove that $\forall k \geq 1$ $$ \sum_{n\leq x} \frac{f(n)}{n} = \frac{1}{(k+1)!} \log^{k+1} x + O(\log^k x), $$ where $$ \sum_{n\leq x} f (n) = \frac{x}{k!} \log^k x + O(x\, \log^{k-1}x). ...
0
votes
0answers
23 views

Sum of convolution of divisor function [duplicate]

For every integer $k$ let $d_k: \mathbb{N} \rightarrow \mathbb{C}$ be defined recursively as $d_0 = \mathbf{1}$, $d_k = d_{k-1} * \mathbf{1}$. So for example $d_1 (n) = d (n) = \sum_{d \vert n} 1$ is ...
2
votes
2answers
77 views

Is there a way to show that $d(n)$, which counts the number of divisors of $n$ is non-increasing? [closed]

Is there a way to show that $d(n)$, which counts the number of divisors of $n$ is non-increasing? I'm trying to use the Cauchy condensation test to show that $\sum_{n\ge{2}}\frac{d(n)}{n\log^2n}$ is ...
2
votes
0answers
33 views

Why do so many identities for the Logarithmic Integral begin with the terms $\log \log n + \gamma +…$?

Several identities for the log integral lead with the terms $\log \log n + \gamma$, where $\gamma$ is the Euler–Mascheroni constant. So, for example, there's the well-known $$\text{li}(n) = \log ...
0
votes
2answers
54 views

Why is $\mu \star E =e $ , where $\star$ denotes the Dirichlet Convolution operator?

Let $$ E(n) = 1 \qquad \forall n \in \mathbb{Z} $$ be the constant function, and let $\mu$ be the Möbius function. Based on the following definition of the latter function, where $\mu(n) = 1$ for ...
1
vote
0answers
75 views

Apply Möbius inversion to formal power series

Let $e$ be a positive natural number, there is the following equality of formal power series ...
0
votes
0answers
17 views

How to show that $\sum_p \int_{p^m}^\infty f(x) dx = \int_0^\infty \pi(x^{1/m}) f(x) dx$

How do you show that for some function $f(x)$, $$\sum_p \int_{p^m}^\infty f(x) dx = \int_0^\infty \pi(x^{1/m}) f(x) dx$$ where the sum on left is taken over the set of all prime numbers $p$ and ...
0
votes
0answers
34 views

Upper bound for number of primes in an interval

Let $S(x,y)$ be the number of primes $p$ in $(x, x + y]$ such that also $p + 6$ and $p + 12$ are primes. I know that $$ T(x, y) \leq 48 c \frac{y}{\log^3 y} \left( 1 + O \left ( \frac{\log \log ...
0
votes
0answers
39 views

Summation of Legendre symbol

Let $\chi_{2,q}$ be the real Dirichlet character modulo a prime $q>2$, which is not the principal one (the so-called Legendre symbol). Is it true that $$ \sum_{n=0}^{+\infty} ...
2
votes
1answer
40 views

Show that the first derivative of the Riemann Zeta function $\zeta'(s) < 0$ if $s \in (1-\epsilon,1)$ and $\epsilon > 0$ is sufficiently small.

Show that $\zeta'(s) < 0$ if $s \in (1-\epsilon,1)$ and $\epsilon > 0$ is sufficiently small. Using the fact that \begin{align} \zeta(s) = \frac{s}{s-1}-s\int_1^\infty\frac{\{t\}}{t^{s+1}}dt ...
3
votes
1answer
54 views

A question about the convergence of partial products of zeta of one.

Recently I've been toying around with the Totient function and the Prime Number Theorem and came up with the odd result that the following limit $$\lim_{n\to\infty}\frac{\pi(n)m_n}{\phi(m_n)n}$$ ...
0
votes
0answers
55 views

A short question on the estimation of $\sum_{1\leq n\leq x} \mu(n)n^{-1}$.

$ \ \ $ I want to ask an estimation of $\sum_{1 \leq n\leq x} \mu(n)n^{-1}$. According to a paper: http://arxiv.org/pdf/0908.4323v5.pdf of Terry Tao (See the theorem 1.3 on page 4 if you want), for an ...
1
vote
1answer
43 views

Is there an expression for $\mu(n)^2$ where $\mu$ is the mobius function?

Is there an expression for $\mu(n)^2$ where $\mu$ is the mobius function? I know that \begin{align} \sum_{d|n} \mu(d)=\left\{ \begin{array}{cc} 1 & \text{if }n=1\\ 0 & \text{if }n>1 ...