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

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11 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 ...
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10 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 ...
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1answer
33 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 ...
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1answer
34 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 ...
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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 \...
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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 ...
8
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62 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
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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 ...
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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^{\...
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2answers
74 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 ...
4
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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 ...
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1answer
85 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
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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 ...
0
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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 $$...
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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
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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 ...
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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 ...
13
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1answer
205 views
+100

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, ...
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0answers
105 views

can you proof this equation? [closed]

let ${q_{_1}}$,..., ${q_{_r}}$ be the distinct primes dividing $p-1$, that $p$ is a prime number,${\mu}$ is mobius function, ${\varphi}$ is Euler's phi function, ${\chi}$ is Dirichlet characteristic ...
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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
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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 ...
0
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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 $\...
3
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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? ...
0
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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
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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 $ax-by=1$?
4
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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
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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 ...
1
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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 (-\...
0
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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+...
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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, ...
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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
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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 ...
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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
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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" ...
1
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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 ...
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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
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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 ...
0
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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)^{-...
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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 ...
2
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0answers
75 views

When does $\sum_{p\in\mathbb{P}} \frac{1}{|p|^2}$ diverges?

We know $\sum_{p\in\mathbb{P}} \frac{1}{|p|^2}$ diverges where $\mathbb{P}$ denotes set of all primes in $\mathbb{Z}[i]$ (because that sum is greater that $\sum_{p \equiv 3 \mod 4} \frac{1}{p}$, which ...
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0answers
59 views

Next book in in learning Analytic Number Theory

I have just finished the book "Tom M. Apostol - Introduction to Analytic Number Theory". My aim is to reach to graduate level to do research, especially on Rationality/Irrationality and Algebraic/...
3
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2answers
52 views

Estimate for $\sum_{q=1}^{M}\frac{\varphi(q)}{q^{2}}$ Related to Bourgain Paper [duplicate]

Let $N\gg 1$ be a large parameter, which I ultimately want to let tend to infinity. I am reading an old paper of Bourgain, where he claims the lower bound (Equation 2.50, pg. 118) $$\sum_{q=1}^{N^{1/...
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1answer
70 views

about integral logarithm

I would to ask for a logarithm integral, used for Gauss. I read that he uses it to calculate the number of primes, less than a given natural number. It is like: $Li= \int_{0}^x(dt/lnt)$ I read that he ...
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2answers
67 views

Theorem 3.16. in Analytic Number Theory by Apostol

The below texts are from the book Introduction to Analytic Number Theory by Apostol: I have two questions which I couldn't find solutions for them: $1-$ According to Thm 3.16., $\sum_{n\le x} \...
2
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1answer
95 views

Prove $\sum_{d\le x} \sum_{q\le x/d} \dfrac{1}{q^{\beta}} = \sum_{d\le x} \dfrac{1}{d^{\beta}} \sum_{q\le x/d} 1$?

The below texts are from the book Introduction to Analytic Number Theory by Apostol: Trying to calculate $\sum_{n\le x} \sigma_{\alpha} (n)$ for negative $\alpha$ I followed the advice of the book, ...
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30 views

Probability distribution of $\omega'(n)$. [duplicate]

$\omega(n)$ is the number of distinct prime factors of $n$ and $\omega'(n)$ is the number of distinct prime factors of $n$ with multiplicity. For example if $p,q$ are prime numbers then $\omega(p^2q)=...
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22 views

Get an upper bound of $\left| F(1+it) \right|$ in an example of Perron type formula

From Proposition 3 of Tao, A cheap version of Halasz’s inequality, I know how get for example upper bounds for $x,T\geq 1$ $$\frac{1}{x}\sum_{n\leq x}\frac{\mu(n)\log n}{n}\ll\int_{-T}^{T} \left| \...
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76 views

Theorem 3.3 Apostol's Analytic Number Theory

The below texts are from the book Introduction to Analytic Number Theory by Apostol: Note. Part (d) Thm 3.2 [green-underlined] is $$\sum_{n\le x}n^a=\dfrac{x^{a+1}}{a+1} +O(x^a) \ \ \text{if} \ a\...
0
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1answer
20 views

An upper bound of $ \left| \frac{1}{s}\log\zeta(s) \right| $ for $\Re s=\sigma>1$, from this integral formula and a related comparison

For $\Re s=\sigma>1$ one has the following known formula $$\frac{1}{s}\log\zeta(s)=\int_1^\infty \Pi(x)x^{-s-1}dx,$$ then if we take the derivative we can write $$\frac{1}{s}\log\zeta(s)=s(s+1)\...
-1
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1answer
59 views

Given $N$ find the number of natural numbers less than $N$ that may be written in the form $\frac{(k)(k+1)}{2}$

Given $N$, find the number of natural numbers less than $N$ that may be written in the form $$\frac{k(k+1)}{2},$$ where $k\in \Bbb N$. I know that the answer to this problem is approximately $\sqrt {...