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

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5
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3answers
88 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
81 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} ...
2
votes
1answer
43 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
62 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
38 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
44 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
73 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
66 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
36 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 ...
0
votes
0answers
49 views

Lower bound for Euler's totient function

Is there a constant $k > 0$ such that $\phi(n) \geq n - k \sqrt{n}$ for infinitely many composites $n$ with more than one prime divisor? If not, can we replace $k$ with some function $E(n) \ll n$ ...
5
votes
1answer
95 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
172 views

Bounding error when iterating a function

If I am iterating some function $f$ that goes to infinity as x goes to infinity with error $o(g(x))$, for example, is there anyway to bound the error? To be more specific, if I have some sequence ...
3
votes
1answer
100 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 ...
2
votes
3answers
69 views

Euler's totient function and limits

Suppose we have an infinite set $S$ of positive composite numbers such that the prime factors $p$ of $n \in S$ have the property that $p \to \infty$ as $n \to \infty$. What is $$ \lim_{\substack{n ...
0
votes
1answer
52 views

Upper bound for Euler's totient function on composite Mersenne numbers

Are there any good upper bounds for Euler's phi function on composite Mersenne numbers. That is, any good $f(n)$ such that $\phi(2^n-1) \leq f(n)$? It might be useful to know that $n \mid \phi(2^n-1)$ ...
3
votes
3answers
166 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 ...
1
vote
2answers
43 views

Asymptotic behaviour of $\prod_{p \leq x} (1 + 4/(3p) + C p^{-3/2})$

I'm reading Montgomery and Vaughan and in it they state quite simply \begin{equation} \prod_{p \leq x} \left(1 + \frac{4}{3p} + \frac{C}{p^{3/2}} \right) \ll (\log x)^{4/3} \end{equation} as $x ...
0
votes
0answers
72 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
26 views

Upper bounds for Euler's totient function on an increasing sequence of numbers

Let $\omega(k)$ be the number of distinct prime divisors of an integer $k$. Let $\delta > 0$ and suppose we have an increasing sequence of positive numbers $S = \{a_n \}_{n=1}^{\infty}$ with the ...
2
votes
1answer
47 views

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

If we take an infinite set $S$ of positive numbers with the property that the number of prime divisors of the elements is unbounded above, then can we make $\phi(n)/n$ arbitrarily small for infinitely ...
0
votes
0answers
59 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 ...
5
votes
1answer
64 views

Is any elementary proof important (beside Selberg's work) ?

Is any elementary proof important (beside Selberg's work) ? Plus, why is the elementary proof of prime number theory by Selberg so important ? Selberg was awarded the Field medal is mainly because ...
0
votes
0answers
31 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 ...
4
votes
1answer
105 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
97 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
2answers
34 views

Inequality $k!\pi+\frac{\pi}{6}\le{m!}\le{k!}\pi+\frac{5\pi}{6}$

there Is the following statement is true? $\forall k \in \mathbf{N},\exists{m}\in\mathbf{N}, k!\pi+\frac{\pi}{6}\le{m!}\le{k!\pi}+\frac{5\pi}{6}$ I tried by descendant proof but was not satisfied ...
2
votes
3answers
80 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 ...
1
vote
1answer
35 views

Can we expect to find $r,$ large enough, so, $\sum_{n\in \mathbb Z} \frac{(1+n^{2})^{s}}{1+(n-y)^{r}}\leq C (1+y^{2})^{s} $ for all $y\in \mathbb R$?

Fix $y\in \mathbb R$ and $s>1.$ Consider the series: $$I(y)=\sum_{n\in \mathbb Z} \frac{(1+n^{2})^{s}}{1+(n-y)^{r}}.$$ My Question is: Can we expect to find $r$ large enough, so that ...
2
votes
1answer
44 views

A slightly various form of Dirichlet's theorem on arithmetic progressions

Are there infinitely many primes of the form $2n(n+1)+1$?
3
votes
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 ...
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
82 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 ...
2
votes
1answer
87 views

Dirichlet series and Riemann zeta function

Im trying to show, for $\Re(s)>1$, that $\displaystyle\sum_{n=0}^{\infty} \frac{d(n^2)}{n^s} = \frac{\zeta^3(s)}{\zeta(2s)}$, where $d(n)= |\{k \mid k|n \}|$, number of positive integers that ...
0
votes
1answer
42 views

Integral inequality with gamma function

I have some trouble with paper I'm reading. The goal is this: let $s=\frac{1}{2}+\frac{1}{\log n}+it$. $M$ is a function such that $M(s)=O(\log^{3}(N(|t|+2)))$. Define $$U(s)=\frac{1}{2\pi ...
0
votes
0answers
30 views

Distribution of $\lfloor n^{\log{n}} \rfloor$ modulo $q$.

Let $q$ be an arbitrary integer. I want to investigate the distribution of the set $\mathcal{S} = \{\lfloor n^{\log{n}} \rfloor : n \in \mathbb{N}\}$. After a few explicit computations with SAGE, it ...
1
vote
0answers
49 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
78 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} $ ...
1
vote
1answer
61 views

an exercise from apostol analytic number theory

this is the second exercise of chapter 3: if $x\ge 2$ prove that $$\sum_{n\le x} \frac{d(n)}{n}= \frac{1}{2} {\log^2 x} + 2C\log x +O(1)$$ where $C$ is Euler's constant. here's what i've done to ...
7
votes
1answer
277 views

Notation in Terry Tao's exposition on the PNT

The exposition I'm talking about can be found here (page 6): http://www.math.ucla.edu/~tao/preprints/Expository/prime.dvi Essentialy, Tao proves the prime number theorem in the elementary way, ...
2
votes
1answer
65 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
22 views

Gamma function whose argument is a reciprocal power with integer base and exponent

Consider the analytic continuation of the factorial function $n!$ given by $\Gamma(z)$ (note $n!=\Gamma(n+1)$), and suppose $z=a^{-n}$, where $a,n\in\mathbb{N}$ are positive integers. Are there any ...
0
votes
0answers
113 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 $\ ...
1
vote
0answers
43 views

Riemann functional equation question?

I was looking through the derivation of the Riemann functional equation, and I understand how to obtain the result $$ \pi^{-\frac s2} \Gamma (\frac s2) \zeta(s) = \pi^{-\frac{1-s}{2}} ...
0
votes
1answer
18 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
55 views

Rearrangements of Dirichlet Eta Function

I was wondering if explicit closed forms for rearrangements of $\eta(s)$, for $s$ such that the series is not absolutely convergent, are useful in studying the Dirichlet $\eta$ function. I am asking ...
4
votes
1answer
108 views

Analytic Continuation of Zeta Function using Bernoulli Numbers

In my complex analysis textbook by Stein and Shakarchi, as an exercise, I am supposed to extend $\zeta(s)$ to the entire complex plane using Bernoulli numbers, but I am stuck. I can prove that $$ ...
2
votes
1answer
68 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
65 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 ...
1
vote
1answer
56 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
56 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 ...