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

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1
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0answers
14 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
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0answers
10 views

How can I compare two bivariate functions [on hold]

If we assume two functions $f_1(x,y)$ and $f_2(x,y)$ is there any way to compare them, e.g. find how similar they are? Both functions do not have any analytical solution and everything need to be ...
0
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0answers
20 views

Partial Summation [duplicate]

In many places, it is stated that $$\sum_{p\le x}\frac{1}{p} = \log\log x + O(1)$$ easily follows from $$\sum_{p\le x}\frac{\log p}{p} = \log x + O(1)$$ by partial summation. However, I don't see how ...
0
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0answers
56 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
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0answers
33 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
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0answers
5 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
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1answer
30 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
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1answer
24 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
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1answer
25 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
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0answers
9 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
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1answer
37 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
16 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
58 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
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0answers
25 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
41 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
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0answers
30 views

Mobius inversion formula

Let $e$ be a positive natural number, there is the following equality of formal power series ...
0
votes
0answers
15 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
15 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
18 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
36 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
36 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
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0answers
34 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
30 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 ...
2
votes
2answers
25 views

Summation of non-principal real Dirichlet character

Let $q > 3$ be a prime and $$ S_q := \sum_{k=1}^{q-1} \chi_{2,q} (k) \, k, $$ where $\chi_{2,q}$ is the real Dirichlet character modulo $q$ which is not the principal one. I have to prove that ...
1
vote
3answers
39 views

Divisor function convolution

I need some help to prove that $$ (d*d)(p^k) = \frac{(k+3)(k+2)(k+1)}{6} \qquad \forall p \in \mathcal{P},\quad \forall k \in \mathbb{N}, $$ where $d$ is the divisor function and $\mathcal{P}$ the set ...
2
votes
1answer
81 views

Useful device in complex analysis (Perron's formula)

I've come across the following useful device from complex analysis: $$\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}{\frac{y^z}{z}}{dz} = \left\{\begin{array}{lll} 0 & \text{if} & 0<y<1 ...
0
votes
1answer
21 views

Question about the covergence of a Dirichlet series

Suppose that F($s$) = $\sum \frac{a(n)}{n^s}$ is a Dirichlet series, where the sum is taken for all intergers $n\geq 1$. It's also known that the series converges for all complex numbers $s$ with ...
4
votes
1answer
53 views

Divergence of sum containing number of divisors function.

Show that $\sum_{n\ge{2}}\frac{d(n)}{n\log^2n}$ is divergent. I've tried to do this using the comparison test, i.e looking for a divergent series with a summand smaller than the summand in the ...
0
votes
0answers
27 views

How many solution are possible for this multivariable equation? [duplicate]

$$2(a+b+c+d+e+f)+g=N$$ where $$a,b,c, \cdots ,N \in \mathbb{N}$$ Any lead will be appreciated.
1
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0answers
75 views

Any Computational Number Theory Book, include software programs for key steps of the proofs of major theorem?

All: Can anyone recommend some Computational Number Theory Books, which include software programs for key steps of the proofs of major theorem ? Some computational number theory books only include ...
0
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0answers
37 views

Analytic Continuation of the zeta function

Is the analytic continuation of the Riemann zeta function to the upper half plane unique? I don't know much complex analysis, so I can't see why that is the case.
0
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0answers
24 views

How to differentiate an expression involving big-o notation?

From Apostol - Introduction to analytic number theory (Theorem 3.3) we have $$ x\geq1, \sum_{n\leq x}d(n)=x\log x+(2\gamma-1)x+O(\sqrt{x}):=E(x), $$ I want to differentiate $E$ -- to get a rough ...
3
votes
0answers
72 views

Brun's sieve bounds

Working from Halberstam-Richert they state the following bounds \begin{align} S(\mathcal{A}; \mathfrak{P}, z) \leq XW(z)\left(1 + 2 \frac{\lambda^{2b + 1}e^{2\lambda}}{1 - \lambda^2 e^{2 + ...
1
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0answers
25 views

Zeta zero sum & reciprocals of prime powers

Below is a plot of $$\dfrac{1}{x}\sum_{n=1}^{75}2\Re\left(\operatorname{Ei}\left(\rho_n\log\left(x\right)\right)\right)$$ where $\rho_n$ is the $n$th zeta zero, with grid lines at primes and prime ...
2
votes
1answer
109 views

Between $n$ and $2n$ there is always a prime number. [duplicate]

Between $n$ and $2n$ there is always a prime number. I was thinking of this and looked it up on the google to find that this is true. Now, I am wondering what is the proof for it? Does any ...
2
votes
0answers
35 views

Showing $n! = \sqrt{2\pi n} (\frac{n}{e})^n \big(1 + \mathcal{O}(\frac{1}{n})\big)$ from $\log(n!) =n\log(n) - n + \mathcal{O}\big(\log(n)\big)$

I wish to prove Stirling's Formula in this way, in particular showing the first term in the series is $\mathcal{O}\big(\frac{1}{n}\big)$, and I've come across a "proof" that simply states there is ...
2
votes
0answers
28 views

How to find the analytic continuation of this series?

I have the following series: $$ \sum_{n = 0}^{+\infty} \frac{n^2}{(n^2 + a^2)^{\epsilon}} $$ with $a\in \mathbb{R}$. How can I find its analytic continuation for $\epsilon \in \mathbb{C}$? In ...
4
votes
1answer
58 views

Asymptotics on the largest prime for which $x^n+1\equiv y^n$ has no nonzero solution

It $\let\epsilon\varepsilon\let\leq\leqslant\let\geq\geqslant$is a well known result that for every $n\in\mathbb N$, $x^n+1\equiv y^n\pmod p$ is non-trivially solvable for sufficiently large primes ...
1
vote
0answers
24 views

L-function Like Convergence

Question: Let $p_1<p_2<p_3<\cdots$ be all the odd primes. (1) Show that $$S=\sum_{k=1}^{\infty}\frac{(-1)^{(p_k+1)/2}}{p_k}$$ diverges. (2) Show that $S$ converges to a real in $(0, ...
4
votes
2answers
98 views

Why is there a 'missing' $1$ in the Euler–Mascheroni constant?

It is easy to show that: $$ \sum_{k=1}^n \frac{1}{k} > \ln(n+1), $$ but the Euler–Mascheroni constant is defined as: $$ \gamma = \lim_{n \to \infty} \left( \sum_{k=1}^n \frac{1}{k} - \ln(n) ...
4
votes
1answer
79 views

What are major algebraic number theory attempts, results and progressions toward Goldbach's Conjecture?

To my understanding, most progress toward Goldbach's Conjecture has been made in analytic number theory. Progress has often based on sieve, asymptotic estimation or other analytic methods. What are ...
5
votes
3answers
78 views

How would you show that the Riemann Zeta function, $\zeta(s) < 0$ for $s \in (0,1)$?

How would you show that the Riemann Zeta function, $\zeta(s) < 0$ for $s \in (0,1)$? So far I have that along the critical strip \begin{align} \zeta(s) &= ...
3
votes
1answer
66 views

A double sum involving the Riemann zeta function

Evaluate the sum $S=\sum_{k=2}^{\infty} \frac{\zeta (k)-1}{k+1}$, where $\zeta (s)$ denotes the Riemann zeta function. The sum is equal to $\sum_{k=2}^{\infty} \sum_{n=2}^{\infty} ...
0
votes
0answers
96 views

Proof that $G(3)\le 7$

Let $G(k)$ be the minimal $n$ s.t. every sufficiently large integer is the sum of $n$ nonnegative $k$th powers. Does anybody know where I can find Vaughan's proof that $G(3)\le 7$? I can't find a ...
2
votes
1answer
40 views

Integers Free of Small Prime Factors

I am trying to understand (a version of) the elementary proof of the Prime Number Theorem. I've been following Tenenbaum and Mendès France's book The Prime Numbers and Their Distributions. My goal is ...
2
votes
0answers
55 views

Are there any fundamental new discoveries in Algebraic Number Theory than in ‘traditional’ number theory ? [closed]

I am new to Algebraic Number Theory. I wonder if there is any fundamental new discoveries in Algebraic Number Theory than in ‘traditional’ number theory ? I want to know, beside ‘generalizing’ or ...
2
votes
0answers
36 views

Binomial Congruence Mod primes

So while I was messing around with binomial coefficients I noticed that $$ \binom{3p-1}{p}\equiv 2 \pmod{p^3} $$ For all the primes I tested above 2. I looked around and found similar congruences ...
0
votes
0answers
33 views

Another question/observation about Mersenne numbers and Euler's totient function

This is a follow up to this question Upper bound for Euler's totient function on composite Mersenne numbers and an ongoing project with lots of questions related to Mersenne numbers. I'm sorry if ...
0
votes
0answers
56 views

Using The Abel Summation formula to calculate $\prod\limits_{p \leq x}(1-\frac{1}{p})$

Using The Abel Summation formula to calculate $\prod\limits_{p \leq x}(1-\frac{1}{p})$ Can anyone give me some hints on how to solve this? I've tried using logs and get \begin{align} ...
0
votes
0answers
51 views

Which is the best book on Goldbach conjecture research

Is there a book which summarizes the major research results in the past, and current research trends, for the Goldbach conjecture? I know, much progress has been made in Analytic Number theory in ...