Questions tagged [analytic-number-theory]

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

6
votes
1answer
380 views

Cusp forms' Fourier coefficients sign changes

I need some clarification on the following, if possible: I have seen in that for every $ f \in S_k$ which Fourier transform is $\sum_{n=1}^\infty a(n)q^n$ there is an upper bound $\sum_{n=1}^N \|a(n)\...
20
votes
3answers
3k views

How to prove Chebyshev's result: $\sum_{p\leq n} \frac{\log p}{p} \sim\log n $ as $n\to\infty$?

I saw reference to this result of Chebyshev's: $$\sum_{p\leq n} \frac{\log p}{p} \sim \log n \text{ as }n \to \infty,$$ and its relation to the Prime Number Theorem. I'm looking into an information-...
9
votes
0answers
352 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 ...
6
votes
1answer
232 views

Abelian theorem regarding Riesz summability

This is my first time to post something here. If there is anything wrong, please inform me... Anyway, here is my question: Let $k$ be a nonnegative integer. We say a sequence $(a_n)$ is $(R, k)$-...
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 ...
7
votes
1answer
395 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
299 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 ...
21
votes
1answer
526 views

On existence of an integer between $\sqrt{n}$ and $\sqrt{2n}$ coprime to $n$

I have one proof of the following statement, and I would like to know if there is a simpler proof. I am not sure if “simpler” is the right word or not but, for the purpose of this question, I prefer ...
8
votes
2answers
645 views

How is $3 + 4\cos \theta + \cos 2\theta \geq 0$ related to $\zeta(s)^3|\zeta(s + it)^4\zeta(s + 2it)| \geq 1$?

The inequality $$\zeta(s)^3 | \zeta(s + it)^4 \zeta(s + 2it)| \ge 1$$ follows from $$3 + 4 \cos(\theta) + \cos(2 \theta) \ge 0.$$ How is that done? What is the relationship between zeta and the ...
28
votes
3answers
761 views

Computing the product of p/(p - 2) over the odd primes

I'd like to calculate, or find a reasonable estimate for, the Mertens-like product $$\prod_{2<p\le n}\frac{p}{p-2}=\left(\prod_{2<p\le n}1-\frac{2}{p}\right)^{-1}$$ Also, how does this behave ...
3
votes
1answer
738 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.
18
votes
2answers
1k views

A Conjecture of Schinzel and Sierpinski

Melvyn Nathanson, in his book Methods in Number Theory (Chapter 8: Prime Number's) states the following: A conjecture of Schinzel and Sierpinski asserts that every positive rational number $x$ can ...
3
votes
2answers
261 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 $x \in \mathbb{N}$...
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 ...
7
votes
2answers
666 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}^{...
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 ...
28
votes
3answers
2k views

Ramanujan's First Letter to Hardy and the Number of $3$-Smooth Integers

A positive integer is $B$-smooth if and only if all of its prime divisors are less than or equal to a positive real $B$. For example, the $3$-smooth integers are of the form $2^{a} 3^{b}$ with non-...
9
votes
1answer
230 views

When does the “Zetor function” converge?

Let $p_n$ be the n'th non-trivial zero of the Riemann zeta function. We define the Zetor function (acronym of 'zeta' and 'zero') as follows: $$\zeta \rho (s) = \sum_{n=1}^{\infty} \frac{1}{(p_n)^s}. $$...
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 ...
2
votes
1answer
1k views

Is Riemann Zeta Function symmetrical about the real axis?

From wikipedia, http://en.wikipedia.org/wiki/Riemann_zeta_function "Furthermore, the fact that $\zeta(s) = \zeta(s^*)^*$ for all complex s ≠ 1 ($s^*$ indicating complex conjugation) implies that the ...
23
votes
3answers
2k views

On Dirichlet series and critical strips

(I'll keep this one short) Given a Dirichlet series $$g(s)=\sum_{k=1}^\infty\frac{c_k}{k^s}$$ where $c_k\in\mathbb R$ and $c_k \neq 0$ (i.e., the coefficients are a sequence of arbitrary nonzero ...
7
votes
1answer
158 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(...
53
votes
6answers
10k views

How hard is the proof of $\pi$ or $e$ being transcendental?

I understand that $\pi$ and $e$ are transcendental and that these are not simple facts. I mean, I have been told that these results are deep and difficult, and I am happy to believe them. I am curious ...
37
votes
5answers
7k views

Riemann zeta function at odd positive integers

Starting with the famous Basel problem, Euler evaluated the Riemann zeta function for all even positive integers and the result is a compact expression involving Bernoulli numbers. However, the ...
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{...
14
votes
3answers
1k views

Motivation for Hecke characters

The context is the definition of Hecke Größencharakter: http://en.wikipedia.org/wiki/Hecke_character This is supposed to generalize the Dirichlet $L$-series for number fields. Dirichlet characters ...
7
votes
1answer
211 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 ...
1
vote
1answer
216 views

Showing $e^{\psi(x)}= \text{lcm}[ 1,2,\cdots , \lfloor{x\rfloor}]$

Let $$ \theta(x) = \sum\limits_{p \leq x} \log{p} \quad \ ; \ \psi(x)=\sum\limits_{n=1}^{\infty} \theta(x^{1/n})$$ then how does one prove $$e^{\psi(x)}= \text{lcm}[ 1,2,\cdots , \lfloor{x\rfloor}]$$
0
votes
2answers
204 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 ...
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 ...
8
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) = \...
11
votes
1answer
951 views

Continued Fraction expansion of $\tan(1)$

Prove that the continued fraction of $\tan(1)=[1;1,1,3,1,5,1,7,1,9,1,11,...]$. I tried using the same sort of trick used for finding continued fractions of quadratic irrationals and trying to find a ...
2
votes
3answers
311 views

Markov-Hurwitz equation

Prove that the Markov-Hurwitz equation $x^2+y^2+z^2=dxyz$ is solvable in positive integers iff d= 1 or 3. Of course the reverse direction is easy, just set x=y=z=1, d=3. But I really have no idea ...
8
votes
2answers
460 views

A Question on RH relating to Prime Number theorem

Well, in a previous post regarding the explanation of Riemann Hypothesis Matt answered that: The prime number theorem states that the number of primes less than or equal to $x$ is approximately equal ...
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 $$\int_{2}^{x} \frac{\mathrm dt}{\log^{n}{t}} = O\Bigl(\frac{x}{\log^{n}{x}}\Bigr)?$$
8
votes
2answers
802 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 ...
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
29
votes
3answers
1k views

Are there infinitely many $x$ for which $\pi(x) \mid x$?

Let $\pi(x)$ denote the Prime Counting Function. One observes that, $\pi(6) \mid 6$, $\pi(8) \mid 8$. Does $\pi(x) \mid x$ for only finitely many $x$, or is this fact true for infinitely many $x$...
6
votes
3answers
1k views

On Zeta function zeros in the critical strip

I have been reading about Riemann Zeta function and have been thinking about it for some time. Has anything been published regarding upper bound for the real part of zeta function zeros as the ...
8
votes
2answers
2k views

Dirichlet's Divisor Problem

We know that if $ \displaystyle d(n)= \sum\limits_{d \mid n} 1$, then we have $$ \sum\limits_{n \leq x} d(n)= x\log{x} + (2C-1)x + \mathcal{O}(\sqrt{x})$$ I have referred Apostol's "Analytic Number ...