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

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19
votes
1answer
458 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
576 views

Trig identity to zeta function identity?

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 ...
22
votes
3answers
462 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 ...
4
votes
1answer
597 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
240 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.
15
votes
2answers
839 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 ...
2
votes
2answers
243 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 ...
6
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 ...
5
votes
2answers
479 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} ...
6
votes
4answers
1k 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 ...
26
votes
3answers
1k 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 ...
8
votes
1answer
214 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}. ...
5
votes
2answers
554 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
495 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 ...
21
votes
3answers
1k 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
124 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, ...
40
votes
6answers
4k 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 ...
22
votes
4answers
3k 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 ...
8
votes
3answers
904 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} = ...
11
votes
3answers
547 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 ...
6
votes
1answer
138 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 ...
0
votes
1answer
193 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
189 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 ...
4
votes
1answer
638 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 ...
6
votes
1answer
1k 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) = ...
10
votes
1answer
605 views

Continued Fraction expansion of $tan(1)$

Prove that 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
142 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 ...
7
votes
2answers
330 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 ...
10
votes
5answers
748 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
2answers
599 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
147 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
27
votes
2answers
979 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 ...
5
votes
3answers
784 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 ...
6
votes
2answers
978 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 ...