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

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3
votes
1answer
176 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 ...
16
votes
1answer
236 views

Why are $L$-functions a big deal?

I've been studying modular forms this semester and we did a lot of calculations of $L$-functions, e.g. $L$-functions of Dirichlet-characters and $L$-functions of cusp-forms. But I somehow don't see, ...
2
votes
1answer
35 views

Small primes congruent to $a$ mod $p$.

Let $p$ be a prime and $a$ be an integer such that $0 \lt a \lt p$. Is there a prime number, $q$, congruent to $a$ mod $p$ such that $q\lt p^2$? I have checked that this is true for the first $3000$...
1
vote
1answer
40 views

Examples of Weil's explicit formula

In Bombieri, PROBLEMS OF THE MILLENNIUM: THE RIEMANN HYPOTHESIS, Clay Mathematics Institute (2000), from page 8, V. Further evidence: the explicit formula the author tell us that there is a ...
0
votes
0answers
13 views

On $\sum_{\substack{\zeta(\frac{1}{2}+i\gamma)=0\\0<\gamma<T}}\prod_{n=1}^\infty \left| 1-\frac{(\gamma\log x)^2}{n^2\pi^2}\right|$ as $O(\log x)$

On assumption that the identity (2) for a representation of $\pi(x)$ holds, see here Two Representations of the Prime Counting Function in this site Mathematics Stack Exchange, and since using the ...
1
vote
0answers
11 views

What's about of an analogous Riemann's function $R(X)$ for twin primes?

It is well know the so-called Riemann's explicit formula for the prime counting function $\pi(x)$ involving the density $J(x)$ for prime powers and how by Möbius inversion one recovers $\pi(x)$ and ...
0
votes
1answer
35 views

Bounds for the Fourier transform of characteristic functions on $\mathbb{Z}/N\mathbb{Z}$ supported on large sets

Suppose $A \subseteq \mathbb{Z}_N := \mathbb{Z}/N\mathbb{Z}$ with $|A| \geq N/2$. Let $$ \hat{A}(h) := \sum_{a \in A} e_N(ha), $$ where $e_N(x) := e^{2\pi i x/N}$. Clearly $|\hat{A}(h)| \leq |A|$ for ...
1
vote
0answers
43 views

Limit at infinity of a function series

In my researches I got stuck on two similar calculations, and I'd like to deal with them in one fell swoop. 1. I want to say that $$ \lim_{x \to \infty} \sum_{n > 1} z_n \!\!\!\sum_{\substack{d \...
0
votes
0answers
33 views

Can do you repeat these calculations combining the explicit formula and Nicolas criterion, on assumption of the Riemann Hypothesis?

I did easy calculations to get for $x=N_k=\prod_{n=1}^k p_k$ the kth primorial, combining the so-called explicit formula$\dagger$ for the second Chebyshev function and Nicolas criterion for the ...
2
votes
2answers
75 views

number of primitive Pythagorean triangles whose hypotenuses do not exceed n?

i just read "mathematical constants" book; it said that Lehmer proved the following theorem in 1900 where P_h(n) , P_p(n) is number of primitive Pythagorean triangles whose hypotenuses and ...
8
votes
0answers
69 views

How many elliptic curves have complex multiplication?

Let $K$ be a number field. Suppose we order elliptic curves over $K$ by naive height. What is the natural density of elliptic curves without complex multiplication? More generally, suppose we order $...
3
votes
2answers
38 views

Prove or refute that $\{p^{1/p}\}_{p\text{ prime}}$ to be equidistributed in $\mathbb{R}/\mathbb{Z}$

I've tried follow the Example 3 (see minute 30'40" of the reference), where is required the related Theorem (stated at minute 21') combined with Serre's formalism for $\mathbb{R}/\mathbb{Z}$ (also ...
1
vote
1answer
26 views

Can you provide us an asymptotic for this series involving Mertens functions?

Let for integers $k\geq 1$, the Möbius function denoted by $\mu(k)$, and $M(n)=\sum_{k\leq n}\mu(k)$ the Mertens function, then one can prove easily that $$\sum_{k=1}^n\mu(k)\frac{e^{\mu(k)}+1}{e^{\...
7
votes
1answer
196 views

An Inequality Involving The Riemann Zeta Function

I'm having trouble proving the following inequality for $2<r<3$: $$(1+2^{-r})\frac{(3^r+1)^2}{3^{2r}+1}>\frac{\zeta(r)}{\zeta(2r)}.$$ I can easily plot the graph, and the inequality clearly ...
5
votes
1answer
295 views

Question about primes in square-free numbers

For any prime, what percentage of the square-free numbers has that prime as a prime factor?
11
votes
1answer
317 views

Maximum integer not in $\{ ax+by : \gcd(a,b) = 1 \land a,b \ge 0 \}$

Ryan asked about a variation of the coin problem, which was whether for any coprime natural numbers $x,y$ every sufficiently large natural number is $ax+by$ for some coprime natural numbers $a,b$. ...
4
votes
1answer
73 views

(Non-)Canonicity of using zeta function to assign values to divergent series

This article http://blogs.scientificamerican.com/roots-of-unity/does-123-really-equal-112/ got me thinking about the "identity" $$1 + 2 + 3 + \cdots = -1/12,$$ and I wanted to convince myself there ...
1
vote
1answer
87 views

a formula involving order of Dirichlet characters, $\mu(n)$ and $\varphi(n)$

Let $p$ a prime number, ${q_{_1}}$,..., ${q_{_r}}$ are the distinct primes dividing $p-1$, ${\mu}$ is the Möbius function, ${\varphi}$ is Euler's phi function, ${\chi}$ is Dirichlet character $\bmod{...
1
vote
0answers
59 views

Is this stronger Kintchine theorem true?

Let $\phi(n)$ be an increasing real valued function on the positive integers. Suppose that almost every $x \in (0,1)$ has $a_n \geq \phi(n)$ for infinitely many $n$, where $a_n$ is the n'th integer ...
3
votes
1answer
34 views

Chowla's Construction of prime having least quadratic non-residue $\gg \log p$

This paper by NC Ankeny mentions that " S. Chowla has proved that there exist infinitely many primes $k$ where the first $c_1\log k$ residues $(\bmod k)$ are all quadratic residues". I recently ...
1
vote
1answer
48 views

Help on an application of Dirichlet's theorem for primes in progression

Suppose that I have an infinite sequence of positive integers $$a_1,\ldots,a_m,\ldots$$ with the following recursion $$a_{m+1} -a_m =b(m+1)$$ So that $$a_{m+1} =b(m+1) +a_m$$ Suppose ...
-1
votes
2answers
37 views

On a superior limit involving the multiplication formula for the Gamma function and the divisors $d\mid n$ of a positive integer

I did the specialization for the $m's$ in the multiplication formula for the Gamma function, see the identity (4) in page 250 of Apostol, Introduction to Analytic Number Theory Springer (1976) as the ...
0
votes
0answers
28 views

On Dirichlet series and Firoozbakht's conjecture

On assumption of the Firoozbakht's conjecture (this is the Wikipedia, but the reference is for Carlos Rivera's Page) one has that can writes informally the Dirichlet series in LHS of this inequality $$...
3
votes
1answer
38 views

Generalizing Dirichlet characters

Suppose I want to consider Dirichlet characters $$\chi: \mathbb{F}_p(\zeta_r)^{*} \longrightarrow \mathbb{C}$$ Can I prove something similar to the Polya Vinogradov inequality for these characters? ...
1
vote
1answer
69 views

Primitive, quadratic Dirichlet character to odd prime power modulus

Let $p$ be an odd prime number and let $\alpha \geq 1$ be an integer. Let $\chi$ be a real, non-principal, primitive Dirichlet character mod $p^{\alpha}$. How does one show that $\alpha = 1$? If we ...
1
vote
0answers
18 views

Can I presume that this inequality is a good aproximation for a divisor function?

I've used the Lemma 7.9 from page 73 from Krizek, Luca and Somer, 17 Lectures on Fermat Numbers From Number Theory to Geometry Springer CMS (2001) (you can see this page as a Google Book, type here ...
2
votes
0answers
20 views

can we have probabilistic interpretation of $L(1,( \frac{\cdot }{ p}))^{-1}$

the zeta function has a probabilistic interpretation: $$ \zeta(2)^{-1}= \prod_p \left( 1- \frac{1 }{p^2 } \right) $$ can we have probabilistic interpretation of $L(1, \frac{\cdot }{ p})$ which ...
2
votes
2answers
102 views

Asymptotic expression for sum of first n prime numbers?

Is one known? If not, what are the best known bounds? Is there reason to think that an asymptotic expression is beyond current methods if none exists?
1
vote
0answers
38 views

On relationships between the general terms of sequences from different equivalences to the Riemann Hypothesis

The following are simple deductions using easy calculations for inequalities and limits. I define the following sequences, whose shape is inspired in Nicolas, Robin and Lagarias, respectively, ...
0
votes
1answer
64 views

sequence of diophantine approximants of $\pi$

I define the sequence of optimal diophantine approximants of $\pi$ to be the sequence $u_m = \frac{n}{m}$ where $n$ is given by $\min_{\forall n \in \mathbb{N}} |\frac{n}{m}-\pi|$ and we define $\...
0
votes
1answer
45 views

What is the equivalent statement of GRH in term of Redheffer Matrix or Farey Sequences?

We all know that Riemann Hypothesis (RH) has many equivalent statements. There is one statement which expresses RH in term of Redheffer matrix, there is another equivalent statement of RH which ...
0
votes
1answer
44 views

If $(a,b)=1$ then there exist positive integers $x$ and $y$ s.t $ax-by=1$. [duplicate]

How can i prove that if $\gcd(a,b)=1$ there exist $x>0$ and $y>0$ such that $ax-by=1$?
1
vote
1answer
58 views

Proof of Landsberg-Schaar relation

From the Wikipedia page, Landsberg-Schaar relation is the following equation: $$\frac{1}{\sqrt{p}}\sum_{n=0}^{p-1}\exp(\frac{2\pi i n^2 q}{p})=\frac{e^{\pi i/4}}{\sqrt{2q}}\sum_{n=0}^{2q-1}\exp (-\...
4
votes
1answer
43 views

Showing $L(1,\chi)$ is positive given that it's nonzero

Let me first provide context for this question. There is a series of four exercises in Ireland & Rosen's book (in second edition it's exercises 14-17 in chaprer 16), aim of which is (although ...
1
vote
1answer
55 views

What is the exact procedure to represent any positive integer '$n$' in the $m-adic$ form?

I've just started graduate number theory.This seems to be an elementary question,but i'm not getting exact procedure to represent any positive integer '$n$' in the $m-adic$ form. In particular,what ...
0
votes
2answers
66 views

What is the relative density of the abundant numbers in the positive integers?

The Art and Craft of Problem Solving by Paul Zeitz has the following problem. Now, I have been able to solve parts (a) and (b), part (a) by showing that it can get arbitrarily large, and part (b) by ...
12
votes
1answer
195 views

How to prove that $n\sum_{d\mid n}\frac{|\mu(d)|}{d}=\sum_{d^2\mid n}\mu(d)\sigma\left(\frac{n}{d^2}\right)$?

This is problem 11 part b in chapter 3 of Tom M. Apostol's "Introduction to Analytic Number Theory". A variation on Euler's totient function is defined as $$\varphi_1(n) = n \sum_{d \mid n} \frac{|\mu(...
1
vote
3answers
71 views

What are the applications of Sigma Function?

I read about the Sigma Function today.It tells that- The $\sigma(n)$ is the sum of all the positive divisors of $n$. But I had no idea how they can be useful.What are the practical applications ...
0
votes
0answers
16 views

Jacobi sum identity

Let $\chi,\chi'$ be Dirichlet characters modulo $q$, such that $\chi,\chi',\chi\chi'$ are all non-principal characters. By computing the sum $\sum_{n,n'\in \mathbf{Z}/q\mathbf{Z}}\chi(n)\chi'(n')e((n+...
0
votes
0answers
38 views

Additive combinatorics modulo $N$: Reference request

For integers $N, t \geq 1$, would you know of any special sets $A$ of integers in literature for which either an explicit formula (hopefully nice enough) or good estimate is known for the number $$ \#\...
1
vote
1answer
45 views

Asymptotics of $\sum\limits_{n/2 < p \leq n} \frac{1}{p}$

I'm reading a paper which asserts the following: $$\sum_{n/2 < p \leq n} \frac{1}{p} \sim \frac{\log 2}{\log n}$$ follows from prime number theorem, where the sum is taken over $p$ prime. What is ...
1
vote
0answers
22 views

Justify $\lim_{n\to\infty}n^p\int_0^1\sum_{k=n}^\infty\frac{\sigma(k)e^{x/k}}{k^{p+2}\log\log k} dx=\frac{e^\gamma\int_0^1f(x)dx}{p}$

Inspired in PROBLEM 207, La Gaceta de la Real Sociedad Matemática Española, Vol. 16, N0. 3 (page 507 in spanish, proposed and solved by Furdui), I've tried write examples of this new statement ...
2
votes
1answer
32 views

Explaining an integral involving the divisor function

In a 1973 paper by Martinet, Deshouilliers and Cohen, $A(x)$ is defined as $$A(x)=\lim_{N\to\infty}\frac{\#\{n\leq N\mid \frac{\sigma(n)}{n}≥x \}}{N}$$ where $\sigma(n)$ is the "sum-of-divisors" ...
13
votes
6answers
315 views

Is $ \sin: \mathbb{N} \to \mathbb{R}$ injective?

I was trying to show that $\sin(x)$ is non-zero for integers $x$ other than zero and I thought that this result might emerge as a corollary if I managed to show that the result in question is true. ...
0
votes
1answer
39 views

How Changing the order of integration(Elementary proof of the prime number theorem)?

I'm studying the exchange of integration order, I need help, any hint? For every real number $\rho \geq 0$, write $V(\rho)=e^{-\rho}R(e^{\rho})=e^{-\rho}\psi(e^{\rho})-1$ where $\psi(x)$ is the ...
1
vote
2answers
78 views

Integral solution of separable differential equation

On page 524 of Tenenbaum's Introduction to Analytic and Probabilistic Number Theory (3rd edition) it is essentially stated that the solution to the first-order differential equation $$y' = e^{-x}y/x \...
1
vote
0answers
67 views

Are complex numbers complete in every way?

I was told many times a story. Indeed a fascinating one to me as a student learning mathematics. First there were natural numbers. People started adding things and finding solutions to finding the ...
0
votes
1answer
58 views

Contour Integration, Riemann Zeta (-n)

I was reading Riemann's Zeta Function by H. Edwards, and could not understand the equation on the page 12. \begin{align*} \zeta(-n) &= \frac{\prod(n)}{2\pi i}\int_{+\infty}^{+\infty} \frac{(-x)^{-...
1
vote
0answers
30 views

The asymptotic behaviour of $\sum_{1\leq k\leq N-1}\int_{p_k}^{p_{k+1}}\log x d[x]$, where $p_n$ is the nth prime number

Let $p_k$ is the kth prime number and consider for $N\geq 2$ the arithmetic function $$f(N)=\sum_{k=1}^{N-1}\int_{p_k}^{p_{k+1}}\log(x) d[x]$$ where $[x]$ is the integer part function (provide us in ...
1
vote
1answer
40 views

Alternative proofs that Dirichlet products are associative?

Is there alternative proof of the following fact: Dirichlet product on arithmetic function is associative. I'm looking for something different than that given in Dirichlet's product with ...