# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### Mobius inversion formula

Let $e$ be a positive natural number, there is the following equality of formal power series ...
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### 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 ...
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### 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 $$\lim_{n\to\infty}\frac{\pi(n)m_n}{\phi(m_n)n}$$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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) &= ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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} ...
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### 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 ...
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### 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 ...
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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 $$... 3answers 59 views ### Smallest prime factor of a Mersenne number The Mersenne numbers M_n are integers of the form 2^n-1, where n is a positive integer. In the case when n is a prime, are there any results known on the smallest prime factor, p_n, of ... 0answers 26 views ### how that if P(x_1,…,x_n) \in C[x_1,…,x_n] takes only prime values at all non-negative integer values x_i, then P is constant. Show that if P(x_1,...,x_n) \in C[x_1,...,x_n] takes only prime values at all non-negative integer values x_i, then P is constant. To start, how would you express P(x_1,...,x_n)? I really ... 1answer 41 views ### Prove that for P(X) \in \mathbb{Z}[X] the set S = \left\{p : \text{prime and }p \mid P(n) \text{ for } n \in \mathbb{Z}^+\right\} is infinite Prove that for P(X) \in \mathbb{Z}[X], P(x) non-constant, the set S = \left\{p : \text{prime and }p \mid P(n) \text{ for some } n \in \mathbb{Z}^+\right\} is infinite. could someone please give ... 1answer 80 views ### Size of N in primes in arithemtic progression algorithm I've been implementing the search for Primes in Arithmetic Progression (PAP) as explained by Weintraub (1976), and in his paper he refers to a number N which he sets to what seems to be an arbitrary ... 0answers 45 views ### Montgomery&Vaughan's Multiplicative number theory theorem 13.3 I can't understand well the proof of theorem 13.3 There exist a constant C>0 s.t. if RH is true, then for every x\ge 2 the interval (x,x+Cx^{1/2}\log x) contains at least x^{1/2} prime ... 1answer 65 views ### the average order of divisor function In Analytic number theory by Apostol there's a theorem:$$\sum_{n\le x} \sigma(n)= \frac{1}{2} \zeta(2)x^2 + O(x\log x)$$and then it claims that because we know that \zeta (2)= \frac{\pi^2}{6}  ... 1answer 56 views ### Identities for L-series and Euler product It is a mabe a stupid question for many experts here. There is something wrong in the following reasoning, and now I could not find it. Could someone help me out? Any advice will be highly ... 0answers 106 views ### Arithmetic progression and average of two prime numbers Let A=(a_n : n \in \mathbb{N}) be the sequence given by:$$ \ 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. $$For all terms of A greater than \ ... 1answer 16 views ### Given d, for how many m's is d a quadratic residue mod m? Let d be a fixed, square-free integer, and let M be some very large number. I would like to count the numbers m \leq M such that m \perp d and d is a quadratic residue modulo m. Call this ... 1answer 64 views ### Congruences of weights of modular forms modulo primes I'm trying to prove that for two modular forms f and g of weight k and k' respectively, that are congruent modulo a prime \ell\ge 5, their weights are congruent modulo \ell-1. This is what ... 1answer 43 views ### Transformation property for classical Siegel modular forms of weight 2 Let \mathbb{H}_g = \{ \tau \in GL_g(\mathbb{C}) | \; {^t\tau} = \tau, Im(\tau) >0\} be the Siegel upper half space. There are Eisenstein series$$ E_{2k}(\tau) := \sum_{\gamma\in (P_0\cap ...
My question is: Is the Riemann Zeta function for real values of $s$ $( s = \sigma + 0\,i)$ a monotone function of $\sigma \,$?