Questions tagged [analytic-number-theory]

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

Filter by
Sorted by
Tagged with
3
votes
1answer
137 views

The average number of large prime factors of $p-1$

How can one prove that $$\sum_{p \leq x} \mathop{\sum_{q | p-1}}_{q > x^{1/3}} 1 \leq 3\pi(x),$$ where both sums run over primes? The left-hand side is $\displaystyle{\sum_{x^{1/3} < q < x} \...
0
votes
2answers
310 views

A property of the Legendre symbol

There is a property of Legendre symbol: $$ \sum_{0\leq k \leq p-1} \left(\frac{k^2-d}{p}\right)=\sum_{0\leq j \leq p-1}\left(1+\left(\frac{j}{p}\right)\right)\left(\frac{j-d}{p}\right) $$ with $\...
7
votes
2answers
207 views

Asymptotic density of numbers of the form $p_{1}^{\alpha_{1}^{2}}\cdot p_{2}^{\alpha_{2}^{2}}\cdots p_{k}^{\alpha_{k}^{2}}$

If $n$ is a number of the form $p_{1}^{\alpha_{1}^{2}}\cdot p_{2}^{\alpha_{2}^{2}}\cdots p_{k}^{\alpha_{k}^{2}}$ (OEIS A197680) and $T(x)$ counts how many of these numbers are between $1$ and $x$, ...
6
votes
1answer
283 views

Probabilistic proof of existence of an integer

The prime number theorem (PNT) says that an integer $n$ is prime with probability $\frac{1}{\ln n}$. Using only PNT, it's conceivable that each integer upto $10^{10^{10}}$ is non-prime. However using ...
0
votes
1answer
237 views

A simple property of Kloosterman sum

Kloosterman sum is defined as $$K(a,b;m)=\sum_{0\leq x \leq m-1}_{\gcd(x,m)=1} e^{2\pi \mathcal{i} (ax+bx^*)/m}$$ where $a,b,m \in \mathbb{N}$ and $x^*$ is the inverse of $x$ modulo $m$. Now there ...
5
votes
1answer
214 views

Express Dirichlet series in terms of Dirichlet L-function

Assume $\gcd(a,b)=1$, how to express $$F(s)=\sum_{n \equiv a \pmod b} \frac{\mu(n)}{n^s}$$ in the half plane where $\Re(s)>1$ in terms of Dirichlet L-function?
7
votes
1answer
410 views

How will this equation imply PNT

So we have $$\sum_{n \leq x} \frac{\Lambda (n)}{n}=\log{x}+C+o(1)$$ where $C$ is a constant, its partial summation is $$\sum_{n \leq x} \frac{\Lambda (n)}{n}=\frac{\psi(x)}{x}+\int_1^x \frac{\psi (t)}{...
10
votes
4answers
1k views

Representing a number as a sum of at most $k$ squares

Fix an integer $k >0 $ and would like to know the maximum number of different ways that a number $n$ can be expressed as a sum of $k$ squares, i.e. the number of integer solutions to $$ n = x_1^2 + ...
14
votes
3answers
637 views

Rate of divergence for the series $\sum |\sin(n\theta) / n|$

In the following we consider the series $$ S(N;\theta)= \sum_{n = 1}^{N} \left| \frac{\sin n\theta}{n} \right| $$ parametrized by $\theta$. It is well known that this series (taking the limit $N\to\...
15
votes
2answers
2k views

Logarithmic derivative of Riemann Zeta function

Given the logarithmic derivative of the zeta function $\dfrac{\zeta^\prime (s)}{\zeta(s)}$ how does it behave near $s=1$? I mean if for $s=1$ the Laurent series for the logarithmic derivative becomes ...
4
votes
2answers
247 views

How to prove this inequality using prime number theorem

Define $s_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime number, now how to show that $$\lim_{n \rightarrow \infty} \inf \frac{s_n}{\log n} \leq 1$$ I used the result from the prime number theorem: $...
3
votes
2answers
229 views

Will partial summation work for this problem?

Define $$G(x)=\sum_{n \leq x} T\left(\frac{x}{n}\right)$$ and $G,T: [1,\infty) \to \mathbb R$ And function T satisfies the following conditions: 1) $T(x)=O(x)$ 2) $T(x) \sim cx (x \to \infty)$ How ...
9
votes
0answers
306 views

Quadratic characters and Liouville's function

I'm working through the problems in Montgomery & Vaughan's Multiplicative Number Theory. In Section 11.2 'Exceptional Zeros', Exercise 9a says that for a quadratic character $\chi$, show that for ...
2
votes
1answer
184 views

Prime asymptotics from Euler product

It is said that the Euler product $$\prod_p \frac{1}{1-p^{-s}}$$ diverges as $s \to 1^+$ proves we can't find constants $C$,$\theta$ with $\theta < 1$ such that $\pi(x) < C x^\theta$ because ...
34
votes
3answers
1k views

Sequence of numbers with prime factorization $pq^2$

I've been considering the sequence of natural numbers with prime factorization $pq^2$, $p\neq q$; it begins 12, 18, 20, 28, 44, 45, ... and is A054753 in OEIS. I have two questions: What is the ...
7
votes
2answers
304 views

Mean value of arithmetic function

Suppose we define a mean value of arithmetic function $G(f)$ as $$ G(f)=\lim_{x \rightarrow \infty} \frac{1}{x \log{x}} \sum_{n \leq x} f(n) \log{n},$$ and suppose now for an arithmetic function $f$, $...
10
votes
1answer
594 views

How do I prove $\sum_{n \leq x} \frac{\mu (n)}{n} \log^2{\frac{x}{n}}=2\log{x}+O(1)$? Can I use Abel summation?

I am wondering if it is possible to solve this problem using Abel summation: $$\sum_{n \leq x} \frac{\mu (n)}{n} \log^2{\frac{x}{n}}=2\log{x}+O(1)$$ Or maybe I am on the wrong track?
13
votes
3answers
2k views

Proving $\sum\limits_{p \leq x} \frac{1}{\sqrt{p}} \geq \frac{1}{2}\log{x} -\log{\log{x}}$

How to prove this: $$\sum\limits_{p \leq x} \frac{1}{\sqrt{p}} \geq \frac{1}{2}\log{x} -\log{\log{x}}$$ From Apostol's number theory text i know that $$\sum\limits_{p \leq x} \frac{1}{p} = \log{\log{...
2
votes
1answer
473 views

Exercise I.1.6 from Tenenbaum's “Introduction to analytic and probabilistic number theory”

So this question has been asked before, see here, but instead of how to go from part 4 to part 5, I am having a difficult time proving part 4: For each $\alpha > 0$ there exists a sequence of ...
10
votes
5answers
1k views

Bounding the integral $\int_{2}^{x} \frac{\mathrm dt}{\log^{n}{t}}$

If $x \geq 2$, then how do we prove that $$\int_{2}^{x} \frac{\mathrm dt}{\log^{n}{t}} = O\Bigl(\frac{x}{\log^{n}{x}}\Bigr)?$$
7
votes
1answer
330 views

Riemann's $\zeta$ function and the uniform distribution on $[-1,0]$

It seems that the $n$th cumulant of the uniform distribution on the interval $[-1,0]$ is $B_n/n$, where $B_n$ is the $n$th Bernoulli number. And also $-\zeta(1-n) = B_n/n$, where $\zeta$ is Riemann's ...
5
votes
1answer
1k views

Efficiently calculating the logarithmic integral with complex argument

My number theory library of choice doesn't implement the logarithmic integral for complex values. I thought that I might take a crack at coding it, but I thought I'd ask here first for algorithmic ...
4
votes
1answer
496 views

Counting fractions with $n$ digits in the numerator and denominator

Playing around with fractions, I eventually had to consider the following question: Is there a formula for counting how many proper fractions in lowest terms with $n$ base-$b$ digits in both the ...
3
votes
0answers
169 views

Subadditive in analytic number theory

I have just encountered the following question: Let $C_n$ be a sequence of real numbers with the following three properties: 1) $C_n$ is subadditive, such that $$C_{m+n} \leq C_m +C_n$$ 2) $C_n=O(\...
5
votes
1answer
847 views

Intuition and Stumbling blocks in proving the finiteness of WC group

After reading many articles about the Tate-Shafarevich Group ,i understood that "in naive perspective the group is nothing but the measure of the failure of Hasse principle, and coming to its ...
5
votes
1answer
1k views

One line Proof of the Prime Number Theorem

Whenever I am not doing anything, I generally happen to see pages of some good Mathematical Institutes in India, so as to know more about the faculty members and see what they are working on. While ...
8
votes
2answers
830 views

Accuracy of approximation to inclusion-exclusion formula in prime sieve

This thing came up in a combinatorics course I am taking. Choose a fixed set of primes $p_1,p_2,\dots,p_k$ and let $A_n$ be number of integers in $\{1,2,\dots,n\}$ which are not divisible by any of ...
5
votes
0answers
179 views

Euler summation and its transformation

The following results: For any function $f \in C^1[a,b]$ and any $q \in \mathbb{N}$, $$\sum_{a<k \leq b, (k,q)=1} f(k)=\frac{\varphi(q)}{q} \int_a^b f(x) dx + O(\tau(q) (\sup_{x \in [a,b]} |f(x)|+\...
10
votes
1answer
338 views

What might the (normalized) pair correlation function of prime numbers look like? [closed]

You might have read about the fortuitous meeting between Montgomery and Dyson. The background is that the nontrivial zeros of the Riemann zeta function, when normalized to have unit spacing on average,...
2
votes
3answers
3k views

non-trivial upper bound for the number of primes less or equal to n

Using a result of Erdos as in this question An upper bound for $\log \operatorname{rad}(n!)$ one can show that $\sum_{p\leq n} \log p \leq \log(4) n$ for any positive integer $n$. Trivially, $\...
3
votes
1answer
521 views

Periodic Zeta Function Functional Equation

Recall that the periodic zeta function has the Dirichlet series $$F(\lambda,s)= \sum_{n=1}^\infty \frac{e^{2\pi i n\lambda}}{n^s}.$$ This defines an analytic function for $\Re s>0$ and has a ...
1
vote
2answers
185 views

On functions similar to Hurwitz zeta function

Denoted as $\zeta(s,a)$ for a > 0 Where do I find topics on the Hurwitz zeta function for a < 0? Any links or resources would be appreciated. (Please dont mention wiki or mathworld) Thanks
5
votes
1answer
180 views

Equidistribution results vs transcendence degree

Consider $\alpha =(\alpha_1, \dots, \alpha_n) \in \mathbb{R}^n$ linearly independent over $\mathbb{Q}$, then the map $q \mapsto q \alpha:= ( q \alpha_1, \dots, q \alpha_n)$ gives a dense embedding ...
0
votes
1answer
204 views

Does $f \sim g$ imply $f \asymp g$ in certain conditions?

I got a good answer to this question over on MathOverflow a while ago. Harald Hanche-Olsen claimed that, if $f, g: D\to \mathbb{R}^+$, then $$ f(x) \sim g(x) \implies f(x) \asymp g(x) \qquad \qquad (*)...
9
votes
0answers
353 views

How can we prove a simple case of the High Indices Theorem?

Let $(a_n)$ be a sequence of real numbers such that $$f(x) = \sum_{n=1}^{\infty} a_n x^{2^n}$$ converges for $|x| < 1$ and $f(x)$ converges to $a$ as $x \to 1^{-}$. Then I have to prove that $\sum ...
9
votes
1answer
2k views

Why is the following evaluation of Apery's Constant wrong and do you have suggestions on how, if at all, this method could be improved?

Please let me summarize the method by which L. Euler solved the Basel Problem and how he found the exact value of $\zeta(2n)$ up to $n=13$. Euler used the infinite product $$ \displaystyle f(x) = \...
2
votes
1answer
162 views

Difference between zeta sum and Euler product?

The fact that $$\sum_{n=1}^\infty \frac{1}{n^s} = \prod_{p}\frac{1}{1-p^{-s}}$$ is a consequence of unique factorization of primes. We could form a similar sum and a similar product of irreducibles ...
6
votes
2answers
472 views

Asymptotic formula for d(n)/n summation

I was trying to show $$\sum_{n \le x} \frac{d(n)}{n} = \frac{1}{2}\log(x)^2 + 2\gamma \log(x) + O(1)$$ where $d(n)$ is the number of divisors of $n$ and $\gamma$ is the Euler constant using the ...
13
votes
2answers
1k views

Supplemental number theory text to Montgomery and Vaughan

We already have a large list of the Best book ever on Number Theory, but I'm looking for a more targeted response for analytic number theory. Specifically, I'm taking a trip on which I may or may ...
6
votes
1answer
405 views

Question on de la Vallee Poussin's simplified proof of Dirichlet's theorem on primes in arithmetic progressions

I've been trying to understand de la Vallee Poussin's "Demonstration Simplifiee du Theorem de Dirichlet sur la Progression Arithmetique" and I've got stuck at the following step on pg 18 where Poussin ...
5
votes
1answer
300 views

Eisenstein series

Show that all Eisenstein series $G_k$ can be expressed as polynomials in $G_4$ and $G_6$, e.g. express $G_8$ and $G_{10}$ in this way. Hint: Setting $a_n := (2n+1)G_{2n+2}$, show that $2n(2n-1)a_n ...
3
votes
1answer
743 views

On the convergence of $\sum \mu(n)/n^s$

I arrived at something during my maths ponderings which is really exciting for me. It is clearly stated in the book on Riemann Hypothesis by Borwein that the convergence of $\sum_{n=1}^{\infty} \...
1
vote
1answer
249 views

Proving $\pi(\sqrt{p_{1}p_{2}\cdots p_{n}})>2n$ for $n \geq 6$

I am having trouble in solving the following problem. Let $p_{n}$ denote the $n$-th prime. Then prove that $$\pi(\sqrt{p_{1}p_{2}\cdots p_{n}})>2n$$ for $n \geq 6$. No idea how to start.
7
votes
4answers
2k views

Why are complex numbers necessary to prove the Prime Number Theorem?

The standard proof of the Prime Number Theorem requires taking into consideration that there are no zeroes of the Riemann Zeta function in which the real part equals one. But consider the following ...
0
votes
2answers
205 views

Bounding the series $\sum_{m\leq x,m\neq n}\frac{1}{|\log(m/n)|}$

I am trying to reproduce the following bound: $\sum_{1\leq m\leq x, m\neq n}\frac{1}{|\log(m/n)|}=O(x\log(x))$, for $x\geq 2$ and some $n$, $1\leq n\leq x$ (the implicit constant shouldn't depend on ...
3
votes
2answers
262 views

Expressing $\pi(x) = \frac{P(x)}{Q(x)}$, where $P$ and $Q$ are polynomials

In Tom Apostol's Analytic Number Theory book there is a problem which states: That there do not exists polynomials $P(x)$ and $Q(x)$ such that $$\pi(x) = \frac{P(x)}{Q(x)}$$ for all $x \in \mathbb{N}$...
7
votes
2answers
670 views

Understanding an integral from page 15 of Titchmarsh's book “The theory of the Riemann Zeta function”

In Titchmarsh's book "The theory of the Riemann Zeta function" pg. 15 where the functional equation of the zeta function is being derived, I couldn't understand this part: $$\frac{s}{\pi} \sum_{n=1}^{...
9
votes
2answers
1k views

Infinite product representation of a function in terms of its non-trivial zeroes?

From Wikipedia's Weierstrass Factorization Theorem, I learned that every entire function can be represented as a product involving its zeroes. Examples are the sine and cosine function. The Riemann ...
7
votes
1answer
160 views

Nonnegativity of the quadratic Dirichlet L-function $L(\tfrac{1}{2},\chi)$ under GRH

I have been looking for a proof of the statement: "Assume the Generalized Riemann Hypothesis. Let $d$ be a fundamental discriminant and $\chi_d$ the associated primitive quadratic character. Then, $$L(...
7
votes
1answer
227 views

Why is width of critical strip what it is?

For Riemann zeta function and $L$-functions of number fields, the width of critical strip is $1$. For $L$-functions of modular forms of weight $k$, the width of the critical strip is $k$. Why is ...