0
votes
0answers
29 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
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
vote
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
votes
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
votes
1answer
24 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
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} + ...
2
votes
2answers
57 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 ...
1
vote
0answers
29 views

Mobius inversion formula

Let $e$ be a positive natural number, there is the following equality of formal power series ...
2
votes
1answer
35 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 $$\lim_{n\to\infty}\frac{\pi(n)m_n}{\phi(m_n)n}$$ ...
-1
votes
0answers
33 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 ...
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 ...
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
vote
0answers
72 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 ...
4
votes
1answer
56 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 ...
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) &= ...
0
votes
0answers
93 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
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 ...
0
votes
0answers
33 views

Mersenne numbers with two distinct prime factors

For an integer $k$, denote with $p_k$ the $k$-th prime factor. Let $q$ be an odd prime such that $M_q = 2^q-1$ has exactly two distinct prime factors, say $p_s, p_{s+i}$. What is the largest possible ...
5
votes
1answer
80 views

On the square coeffecients of a modular form

Let $k\in \mathbb{N}$. Let $f\in M_k(\Gamma_0(N),\chi)$ be a modular form of weight $k$ on $\Gamma_0(N)$ with a Dirichlet character $\chi$. If $f$ has a Fourier expansion of the form $$ ...
0
votes
1answer
71 views

Can anyone recommend an easy to read algebraic number theory book?

Can anyone recommend an easy to read algebraic number theory book ? I prefer a book with good examples. (hints or answers to selected questions if possible. Not sure if it is possible for a book of ...
3
votes
3answers
118 views

Upper bound for Euler's totient function on composite numbers

I've seen before the general bound $\phi(n) \leq n - n^{1/2}$ for composite $n$. Can this bound be improved at least for those $n$ when we don't have equality above? Say could we possibly have at ...
0
votes
0answers
47 views

Twisting modular forms by Dirichlet characters

Let $\chi,\chi_1$ be Dirichlet characters modulo $M$ and $N$. In Koblitz's book "Introduction to Elliptic Curves and Modular Forms", Proposition III.3.17, it is proved that if ...
0
votes
0answers
45 views

Number of prime factors of Mersenne numbers

Let $p$ be a prime and let $M_p = 2^p-1$. Is it known whether the number of prime factors of $M_p$ is unbounded above as $p \to \infty$? Also do the probabilities estimating the chance that $M_p$ is ...
0
votes
0answers
26 views

Upper bounds for Euler's totient function on a set of composite numbers with bounded number of prime divisors

For an integer $c$, denote with $\omega(c)$ the number of distinct prime divisors of $c$. Now fix an integer $k \geq 2$ and let $W_k$ be some set, with $\#W_k = \infty$, of positive composite integers ...
3
votes
1answer
83 views

Divisor function asymptotics

Define $\tau_{r}(n) = \sum_{d_1...d_r = n}1$. One exercise in a book on sieve theory asked for an elementary proof by induction of the fact that $$\sum_{n\le x}\tau_r(n) = \frac{1}{(r - 1)!}x(\ln ...
0
votes
1answer
95 views

Computing infinite product over primes

How can I compute $$ \prod_p \left(1+\frac{k}{p}\right)\exp(-k/p) $$ where $0<k<e$ and the product is over all primes $p$? Background L. G. Sathe proved [1] that there are $$ ...
2
votes
3answers
56 views

Smallest prime factor of a Mersenne number

The Mersenne numbers $M_n$ are integers of the form $2^n-1$, where $n$ is a positive integer. In the case when $n$ is a prime, are there any results known on the smallest prime factor, $p_n$, of ...
3
votes
0answers
25 views

how that if $P(x_1,…,x_n) \in C[x_1,…,x_n]$ takes only prime values at all non-negative integer values $x_i$, then $P$ is constant.

Show that if $P(x_1,...,x_n) \in C[x_1,...,x_n]$ takes only prime values at all non-negative integer values $x_i$, then $P$ is constant. To start, how would you express $P(x_1,...,x_n)$? I really ...
2
votes
1answer
41 views

Prove that for $P(X) \in \mathbb{Z}[X]$ the set $S = \left\{p : \text{prime and }p \mid P(n) \text{ for } n \in \mathbb{Z}^+\right\}$ is infinite

Prove that for $P(X) \in \mathbb{Z}[X]$, $P(x)$ non-constant, the set $S = \left\{p : \text{prime and }p \mid P(n) \text{ for some } n \in \mathbb{Z}^+\right\}$ is infinite. could someone please give ...
0
votes
1answer
80 views

Size of N in primes in arithemtic progression algorithm

I've been implementing the search for Primes in Arithmetic Progression (PAP) as explained by Weintraub (1976), and in his paper he refers to a number N which he sets to what seems to be an arbitrary ...
1
vote
0answers
42 views

Montgomery&Vaughan's Multiplicative number theory theorem 13.3

I can't understand well the proof of theorem 13.3 There exist a constant $C>0$ s.t. if RH is true, then for every $x\ge 2$ the interval $(x,x+Cx^{1/2}\log x)$ contains at least $x^{1/2}$ prime ...
1
vote
1answer
63 views

the average order of divisor function

In Analytic number theory by Apostol there's a theorem: $$\sum_{n\le x} \sigma(n)= \frac{1}{2} \zeta(2)x^2 + O(x\log x)$$ and then it claims that because we know that $\zeta (2)= \frac{\pi^2}{6} $ ...
2
votes
1answer
55 views

Identities for L-series and Euler product

It is a mabe a stupid question for many experts here. There is something wrong in the following reasoning, and now I could not find it. Could someone help me out? Any advice will be highly ...
0
votes
0answers
102 views

Arithmetic progression and average of two prime numbers

Let $A=(a_n : n \in \mathbb{N})$ be the sequence given by: $$ \ a_n = a_1 + (n - 1)d,\quad a_1,\ d,\ n \in \mathbb N,\quad d\gt a_1,\quad \gcd(a_1,\ d)=1. $$ For all terms of $A$ greater than $\ ...
0
votes
1answer
16 views

Given $d$, for how many $m$'s is $d$ a quadratic residue mod $m$?

Let $d$ be a fixed, square-free integer, and let $M$ be some very large number. I would like to count the numbers $m \leq M$ such that $m \perp d$ and $d$ is a quadratic residue modulo $m$. Call this ...
1
vote
1answer
64 views

Congruences of weights of modular forms modulo primes

I'm trying to prove that for two modular forms $f$ and $g$ of weight $k$ and $k'$ respectively, that are congruent modulo a prime $\ell\ge 5$, their weights are congruent modulo $\ell-1$. This is what ...
1
vote
1answer
42 views

Transformation property for classical Siegel modular forms of weight 2

Let $\mathbb{H}_g = \{ \tau \in GL_g(\mathbb{C}) | \; {^t\tau} = \tau, Im(\tau) >0\}$ be the Siegel upper half space. There are Eisenstein series $$ E_{2k}(\tau) := \sum_{\gamma\in (P_0\cap ...
0
votes
0answers
37 views

Riemann Zeta Function at Real Values of the variable s

My question is: Is the Riemann Zeta function for real values of $s$ $( s = \sigma + 0\,i)$ a monotone function of $\sigma \,$?
2
votes
0answers
46 views

Tau Summatory Function

It is well known that the divisor summatory function can be calculated in $O(x^{1/2})$ via $$D(x)=\sum_{n\le x} d(n) = 2 \sum_{k=1}^{\lfloor \sqrt{x}\rfloor} \lfloor\frac{x}{k}\rfloor - \lfloor ...
1
vote
0answers
52 views

Wolstenholme Number

Does Wolstenholme Numbers have perfect squares other than 1 and 49? The first few are 1, 5, 49, 205, 5269, 5369, 266681, 1077749 seems to be a complicated problem
1
vote
1answer
37 views

Twin primes : prove the convergence of $ \lim_{N\sim\infty} \frac{1}{N} \sum^{N}_{p\in T} (\log(p)+\frac{1}{p})²$

let $T$ be the twin primes set : $p \in T $ if and only if $ p$ and $p+2$ are primes. Can you help me establish the convergence of : $$ \lim_{N\sim\infty} \frac{1}{N} \sum^{N}_{p\in T} ...
5
votes
2answers
194 views

least common multiple $\lim\sqrt[n]{[1,2,\dotsc,n]}=e$

The least common multiple of $1,2,\dotsc,n$ is $[1,2,\dotsc,n]$, then $$\lim_{n\to\infty}\sqrt[n]{[1,2,\dotsc,n]}=e$$ we can show this by prime number theorem, but I don't know how to start I ...
2
votes
0answers
34 views

Growth rate of arithmetical function

I'm interested in how one would estimate the growth rate of $$f(n)=\sum_{k\le n}\mu^2(k)\log(k)$$ I.e. sum of logarithms of square free integers. I can think of some trivial methods in my head ...