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

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0
votes
2answers
24 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
25 views

Mobius inversion formula

Let $e$ be a positive natural number, there is the following equality of formal power series ...
0
votes
0answers
14 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
13 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
17 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} ...
25
votes
1answer
2k views

Books about the Riemann Hypothesis

I hope this question is appropriate for this forum. I am compiling a list of all books about the Riemann Hypothesis and Riemann's Zeta Function. Here is my list: The Riemann Hypothesis: A Resource ...
2
votes
2answers
22 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 ...
2
votes
1answer
33 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
33 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
28 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
28 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
1answer
79 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 ...
1
vote
3answers
34 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 ...
0
votes
1answer
20 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 ...
3
votes
1answer
49 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 ...
1
vote
0answers
81 views

How to evaluate this finite sum? [closed]

$$\sum_{i=1}^{N}i^4\lfloor N/i\rfloor$$ Where brackets represent floor function, others have usual meaning.
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.
3
votes
0answers
71 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
vote
0answers
63 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 ...
5
votes
2answers
199 views

Is $\eta^{24}(\tau)\,j(\tau) = {E_4}^3(q)$?

Given the j-function $j(\tau)$, $j(\tau) = 1728J(\tau)$, where $J(\tau)$ is Klein’s absolute invariant, the Dedekind eta function $\eta(\tau)$, and the following Eisenstein series, $\begin{align} ...
0
votes
0answers
35 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
votes
0answers
23 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 ...
16
votes
1answer
262 views

Are there infinite many $n\in\mathbb N$ such that $\pi(n)=\sum_{p\leq\sqrt n}p$?

Are there infinite many $n\in\mathbb N$ such that $$\pi(n)=\sum_{p\leq\sqrt n}p,\tag{1}$$ where $\pi(n)$ is the Prime-counting_function? For example, ...
1
vote
0answers
24 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 ...
4
votes
1answer
53 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 ...
2
votes
1answer
106 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
34 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 ...
0
votes
1answer
33 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 ...
5
votes
3answers
77 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) &= ...
1
vote
1answer
90 views

Bessel function and upper bound

I'm stuck on this following problem: Let $G$ a function such that $0\leq G(t)\leq 1$, and $G(t)=1$ if $B^2\leq t\leq 4B^2$, with $\operatorname{supp}G\subset [\frac{1}{4}B^2, 9B^2]$ and $G^{(j)}\ll ...
2
votes
0answers
27 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
76 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 ...
1
vote
0answers
22 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, ...
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 ...
6
votes
1answer
372 views

Evaluating a series with the Möbius function and greatest common divisor.

Problem: Let $\gcd(a,b,c,d)$ refer to the largest integer $r$ such that $r$ divides each of $a,b,c,d$. Evaluate the series ...
19
votes
6answers
914 views

A closed form for the sum $\sum_{n=1}^{\infty}\left(\frac{H_n}{n}\right)^2$

How can I find a closed form for the following sum? $$\sum_{n=1}^{\infty}\left(\frac{H_n}{n}\right)^2$$ ($H_n=\sum_{k=1}^n\frac{1}{k}$).
4
votes
2answers
97 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) ...
5
votes
1answer
372 views

Residue of Rankin-Selberg L-function for non-trivial nebentypus

Let $f\in S_k(\Gamma_0(N),\chi)$ be a normalized holomorphic newform (i.e. weight $k$, level $N$, nebentypus $\chi$) and write its Fourier expansion as $$ f(z)=\sum_{n\ge 1} ...
35
votes
3answers
702 views

How to prove that $\sum_{n=1}^{\infty}\frac{(H_n)^2}{n^3}=\frac{7}{2}\zeta(5)-\zeta(2)\zeta(3)$

Prove that $$\sum_{n=1}^{\infty}\frac{(H_n)^2}{n^3}=\frac{7}{2}\zeta(5)-\zeta(2)\zeta(3)$$ $H_n$ denotes the harmonic numbers.
12
votes
2answers
666 views

How to show that the Laurent series of the Riemann Zeta function has $\gamma$ as its constant term?

I mean the Laurent series at $s=1$. I want to do it by proving $\displaystyle \int_0^\infty \frac{2t}{(t^2+1)(e^{\pi t}+1)} dt = \ln 2 - \gamma$, based on the integral formula given in Wikipedia. ...
8
votes
4answers
465 views

The asymptotic expansion for the weighted sum of divisors $\sum_{n\leq x} \frac{d(n)}{n}$

I am trying to solve a problem about the divisor function. Let us call $d(n)$ the classical divisor function, i.e. $d(n)=\sum_{d|n}$ is the number of divisors of the integer $n$. It is well known that ...
3
votes
1answer
63 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
92 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
54 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
35 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
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 ...
0
votes
0answers
29 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
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
52 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
32 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 ...