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

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15
votes
6answers
3k views

A good reference to begin analytic number theory

I know a little bit about basic number theory, much about algebra/analysis, I've read most of Niven & Zuckerman's "Introduction to the theory of numbers" (first 5 chapters), but nothing about ...
5
votes
0answers
125 views

Dedekind sums + strange integral

I need to write $\displaystyle \int_0^1 f(x)\,dx$, where $f(x) = \# \Sigma_{pq} \cap (x, x+1)$ for $x \in [0, 1]$, where $$\Sigma_{pq} = \left \{ \frac{k}{p} + \frac{l}{q}:\ 1 \leq k \leq p-1, 1 \leq ...
6
votes
1answer
124 views

Limit of ratios of numbers with $m$ factors and primes

This is my first question. Let $a_1, a_2,\ldots, a_k$ be natural numbers $\leq n$ with $m$ prime factors. Let $p_1, p_2, \ldots, p_r$ be the prime numbers $\leq n$. Let $$C_{m,n} = ...
2
votes
1answer
129 views

Some asymptotics for zeta function.

I need to use the functional equation for $\zeta(s)$ and Stirling's formula, to show that for $s=\sigma +it$ , with $\sigma <0$: $$ |\zeta(s)| << \left(\frac{t}{2\pi}\right)^{1/2-\sigma}$$ ...
3
votes
3answers
354 views

Other functional equations for $\zeta(s)$?

For the Riemann zeta function, we know of the standard functional equation that relates $\zeta(s)$ and $\zeta(1-s)$. I wanted to know whether there are functional equations that relates $\zeta(s)$ ...
8
votes
1answer
143 views

Comparing average values of an arithmetic function

Suppose $f(n)$ is a positive real-valued arithmetic function such that $$ \frac1n\sum_{k=1}^nf(k)\sim g(n) $$ for $g(x)$ a monotonic increasing function. What can be said about the asymptotic behavior ...
4
votes
1answer
502 views

Is there a simple way to prove Bertrand's postulate from the prime number theorem?

Is there a simple way to prove Bertrand's postulate from the prime number theorem? The prime number theorem immediately implies Bertrand's postulate for sufficiently large $n$, but it fails to ...
17
votes
1answer
447 views

What is the binomial sum $\sum_{n=1}^\infty \frac{1}{n^5\,\binom {2n}n}$ in terms of zeta functions?

We have the following evaluations: $$\begin{aligned} &\sum_{n=1}^\infty \frac{1}{n\,\binom {2n}n} = \frac{\pi}{3\sqrt{3}}\\ &\sum_{n=1}^\infty \frac{1}{n^2\,\binom {2n}n} = ...
6
votes
1answer
297 views

Limit inferior of the quotient of two consecutive primes

I have recently read an article about the prime number theorem, in which Mathematicians Erdos and Selberg had claimed that proving $\lim \frac{p_n}{p_{n+1}}=1$, where $p_k$ is the $k$th prime, is a ...
5
votes
1answer
228 views

Always a prime between $x$ and $x+cf(x)$

What is the asymptotically slowest growing function $f(x)$, such that there exists constants $a$ and $b$, such that for all $x>a$, there is always a prime between $x$ and $x+bf(x)$? $f(x)=x$ ...
7
votes
2answers
360 views

Question regarding Von-Mangoldt function.

Let $\psi(x) := \sum_{n\leq x} \Lambda(n)$ where $\Lambda(n)$ is the Von-Mangoldt function. I want to show that if $$ \lim_{x \rightarrow \infty} \frac{\psi(x)}{x} =1 $$ then also $$\lim_{x\rightarrow ...
1
vote
0answers
58 views

Asymptotic bounds: $\ll$ vs. $\ll_{\epsilon}$?

I am feeling a bit slow today. In Analytic Number Theory it is usual to express asymptotic bounds by specifying the relation of the constant to a specific variable, i.e. $\log n \ll_\epsilon ...
3
votes
1answer
269 views

Integers with odd number of prime factors

Let d(n) be the number of integers less then n which has an odd number of prime factors ( 2,3,5,7,8,11,12,13,17,18...). How to prove d(n)/n have a limit 1/2? Is there for all m an n such that ...
5
votes
1answer
205 views

A combinatorial number theory proof

How can I prove the following identity: $$\sum_{k=1}^{n}{\sigma_{\ 0} (k^2)} = \sum_{k=1}^{n}{\left\lfloor \frac{n}{k}\right\rfloor \ 2^{\omega(k)}}$$ where $\omega(k)$ is the number of distinct ...
6
votes
1answer
216 views

Exponentiation of a Dirichlet series

I'm trying to understand a proof in Chandrasekharan's Introduction to Analytic Number Theory. Specifically, the proof of the lemma on p.118 before Dirichlet's theorem on primes in arithmetic ...
42
votes
5answers
4k views

What is so interesting about the zeroes of the Riemann $\zeta$ function?

The Riemann $\zeta$ function plays a significant role in number theory and is defined by $$\zeta(s) = \sum_{n=0}^\infty \frac{1}{n^s} \qquad \text{ for } s > 1 \text{ and } s= \sigma + it$$ The ...
1
vote
3answers
558 views

Asymptotic formula for the logarithm of the hyperfactorial

Background: I was trying to derive an asymptotic formula for the following: $$\sum_{m\leqslant n}\sum_{k\leqslant m}(m\ \mathrm{mod}\ k),$$ which I think I succeeded in doing (I will skip some steps ...
8
votes
3answers
4k views

Calculating the Zeroes of the Riemann-Zeta function

Wikipedia states that The Riemann zeta function $\zeta(s)$ is defined for all complex numbers $s \neq 1$. It has zeros at the negative even integers (i.e. at $s = −2, −4, −6, ...)$. These are ...
8
votes
1answer
1k views

Divisor summatory function for squares

The Divisor summatory function is a function that is a sum over the divisor function. $$D(x)=\sum_{n\le x} d(n) = 2 \sum_{k=1}^u \lfloor\frac{x}{k}\rfloor - u^2, \;\;\text{with}\; u = \lfloor ...
1
vote
1answer
61 views

To show $\lim_{\xi \to 1} (\xi -1) L(\xi,\chi_{0}) = \frac{\Phi(N)}{N}$

How to show if $\chi_{0}$ is the trivial $\text{Dirichlet Character}$ then $$\lim_{\xi \to 1} (\xi -1) L(\xi,\chi_{0}) = \frac{\Phi(N)}{N}$$ where $\Phi$ is the $\text{Euler's Totient}$.
3
votes
0answers
95 views

Products of primes of the form $an + b$

What is the asymptotic order of numbers divisible by no primes except those of the form $an+b$ ($a$, $b$ fixed)? Surely (except for the trivial cases) they are of order strictly between that of he ...
9
votes
0answers
244 views

Equidistribution of roots of prime cyclotomic polynomials to prime moduli

Here is a relevant - and longstanding, I'm told - conjecture. Let $f \in \mathbb{Z}[x]$ be irreducible and of degree > 1. Set $E_p = \{x/p \: | \: 0 \leq x < p, f(x) \equiv 0 \: (p) \}$ = { ...
3
votes
1answer
133 views

Dirichlet series 'shifted' by a polynomial

Let $F(x) \in \mathbb{Z}[x]$ and $$ \xi(s) = \sum^\infty_{n=1}g(n)n^{-s} $$ be the Dirichlet series associated an arithmetic function $g(n)$. Define a new Dirichlet series $$ \xi_F(s) = ...
13
votes
2answers
1k views

How to show that the Laurent series of the Riemann Zeta function has $\gamma$ as its constant term?

I mean the Laurent series at $s=1$. I want to do it by proving $\displaystyle \int_0^\infty \frac{2t}{(t^2+1)(e^{\pi t}+1)} dt = \ln 2 - \gamma$, based on the integral formula given in Wikipedia. ...
4
votes
1answer
97 views

Asymptotics for almost all $x$

Theorem 2.2 in Shparlinski 2006 says: For all positive integers $n\le x$ except possibly $o(x)$ of them, the bound $$M(x)\ll\frac{x}{\log x}\exp\left((C+o(1))(\log\log\log x)^2\right)$$ holds. ...
6
votes
0answers
150 views

Density of products of a certain set of primes

I have a set S of prime numbers and I would like to find the size (in some sense, ideally some nice asymptotic expression) of the set of positive integers which are the product of with all prime ...
5
votes
2answers
268 views

Is $\eta^{24}(\tau)\,j(\tau) = {E_4}^3(q)$?

Given the j-function $j(\tau)$, $j(\tau) = 1728J(\tau)$, where $J(\tau)$ is Klein’s absolute invariant, the Dedekind eta function $\eta(\tau)$, and the following Eisenstein series, $\begin{align} ...
3
votes
1answer
104 views

$kN^{2}$ relatively prime pairs in $\mathbf{Z}^{2}$

If $k= 1- \sum _{p\in \mathbf{P}} \frac{1}{p^{2}}$, then there are at least $kN^{2}$ pairs $(n_{1},n_{2})$ in $\mathbf{Z}^{2}$, with $1\le n_{1}, n_{2} \le N$ and $gcd(n_{1},n_{2})=1$. Also since ...
28
votes
2answers
664 views

Rounding is asymptotically useless?

Recently I came across the nice result that $$\left\lfloor n \right\rfloor - \left\lfloor\frac{n}{2}\right\rfloor + \left\lfloor\frac{n}{3}\right\rfloor - \left\lfloor \frac{n}{4}\right\rfloor + ...
8
votes
2answers
196 views

Complex integral with zeta

this is a homework problem I am stuck on: Compute the following integral for $\sigma > 1$ $$\displaystyle \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T}\left|\zeta{(\sigma + it)}\right|^2dt .$$ I ...
2
votes
1answer
102 views

The bound behavior for an integral important in studying Chebyshev's prime counting function

I am working through a problem set in an analytic number theory course, and the following problem was included: Let $$I(y,T) = \int_{c-iT}^{c+iT} \frac{y^s}{s} ds.$$ (1) If $y = 1$, $c>0$ and ...
5
votes
1answer
623 views

The value of a Dirichlet $L$-function (and its derivative) at $s=0$

I am working through a problem set in an analytic number theory course, and the following problem was included: Show that if $\chi$ is a non-principal Dirichlet character $\pmod{m}$, and where $L(s, ...
1
vote
1answer
433 views

about the riemann zeta function and the prime counting function

i have posted this question on MO, and they referred me to post here . one starts with the formal definition of zeta : $$\displaystyle \zeta (s)=\prod_{p}\frac{1}{1-p^{-s}} $$ then : $ \ln(\zeta ...
5
votes
0answers
184 views

$2, 5, 13, 17, 29, 421, 401, 53, 281,…,\rightarrow \infty$? $a_{n+1}=\operatorname{ GPF}(qa_n+p)$

I denote by $\operatorname{ GPF}(n)$ the greatest prime factor of $n$, eg. $\operatorname{ GPF}(17)=17$, $\operatorname{ GPF}(18)=3$. Is there a way to prove that the sequence $a_{n+1}=\operatorname{ ...
33
votes
1answer
2k views

Are these zeros equal to the imaginary parts of the Riemann zeta zeros?

Edit 8.8.2013: See this question also. The Fourier cosine transform of an exponential sawtooth wave times $e^{-x/2}$: $$\operatorname{FourierCosineTransform}(\operatorname{SawtoothWave}(e^x)\cdot ...
4
votes
1answer
238 views

Zeta function identity

How does one prove the zeta function identity $$\sum_{s=2}^{\infty}\left(1-\sum_{n=1}^{\infty}\frac{1}{n^s}\right)=-1 \;?$$
1
vote
1answer
99 views

Analytical number theory replacing integer with variable in the summation

I know that $\sum_{p \leq N} \frac{1}{p} \geq \log\log N -1 $ However, want to show that $\sum_{p \leq x} \frac{1}{p} \geq \log\log x -1 $. If let $N=[x]$, then we get a bound for x, i.e. $N \leq x ...
5
votes
4answers
587 views

Evaluating $\sum\limits_{n=2}^{\infty} \frac{1}{ GPF(n) GPF(n+1)}\,$, where $\operatorname{ GPF}(n)$ is the greatest prime factor

$\operatorname{ GPF}(n)=$Greatest prime factor of $n$, eg. $\operatorname{ GPF}(17)=17$, $\operatorname{ GPF}(18)=3$. $\operatorname{ LPF}(n)=$Least prime factor of $n$, eg. $\operatorname{ ...
9
votes
1answer
348 views

Interpolating the primorial $p_{n}\#$

The primorial $p_{n}\#$ is given by the product $p_n\# = \prod_{k=1}^n p_k$ (where $p_{k}$ is the $k$th prime) -- is there a natural (a la the gamma function $\Gamma(z)$) way of interpolating it for ...
22
votes
4answers
661 views

Evaluating $\sum\limits_{n=1}^{\infty} \frac{1}{n\operatorname{ GPF}(n)}$, where $\operatorname{ GPF}(n)$ is the greatest prime factor

$\operatorname{ GPF}(n)=$Greatest prime factor of $n$, eg. $\operatorname{ GPF}(17)=17$, $\operatorname{ GPF}(18)=3$. How to evaluate convergence/divergence/value of the sum $$\sum_{n=1}^{\infty} ...
8
votes
1answer
419 views

Showing that $\log(\log(N+1)) \leq 1+\sum\limits_{p \leq N} \frac{1}{p}$

I can't see how you get this. I want to show that $$\log(\log(N+1)) \leq \sum_{p \leq N} \frac{1}{p}+1$$ Can't see how it follows from this. So you show that $$0 \lt -\log(1-x)-x \lt ...
11
votes
3answers
1k views

Lower bound for $\phi(n)$: Is $n/5 < \phi (n) < n$ for all $n > 1$?

Is it true that : $\frac {n}{5} < \phi (n) < n$ for all $n > 1$ where $\phi (n)$ is Euler's totient function . Since $\phi(n)$ has maximum value when $n$ is a prime it follows that ...
9
votes
2answers
279 views

Relationship between different L-functions

What's the relationship between between Artin $L$-functions and Dirichlet or Hecke $L$-functions if $L/K$ is an abelian extension? I've been told that one can interpret the Artin $L$-functions as ...
2
votes
1answer
328 views

Deriving (a different) expression for the sum of divisors function.

I always really loved the derivation of the exact expression for the partition function, and I happened to recently stumble across this generating function for the sum of divisors function ...
10
votes
1answer
724 views

Rate of convergence of series of squared prime reciprocals

It is well known that $\sum_{p \text{ prime}} \frac{1}{p}$ diverges, and in fact - it behaves like log of the harmonic series: $$ \sum_{p \le x} \frac{1}{p} = \log \log x + O(1). $$ It is also well ...
4
votes
1answer
202 views

Is there a $p$-adic version of Liouville theorem?

That is, if a function $f$ is analytic and bounded in all $K$, a $p$-adic field (or more generally a complete non-archimedean field), has to be constant? And does the theorem work for functions on ...
7
votes
3answers
439 views

Reference requests: Jitsuro Nagura

I spent some time today looking for any biographical information on Jitsuro Nagura and came up empty-handed. Any suggestions welcome. Also, the Wiki note on the Chebyshev $\psi$ function says that ...
6
votes
1answer
245 views

Inequality appearing in proof of Mills' Theorem

I'm reading this (very short, 1 page long) paper by W.H. Mills where he determines that there exists a real number $A$ such that $f(n) = \lfloor {A^3}^n \rfloor$ is a prime number for all positive ...
5
votes
2answers
1k views

Erdős and the limiting ratio of consecutive prime numbers

The following is a piece of math lore from the late forties, which was described in an Intelligencer article entitled "The Elementary Proof of the Prime Number Theorem". It reads: Turán, who was ...
13
votes
1answer
310 views

Chebyshev: Proof $\prod \limits_{p \leq 2k}{\;} p > 2^k$

How do I prove the following: $$\prod_{p \leq 2k} \; p > 2^k \text{ with } p \in \mathbb{P}$$ I tried induction, but I didn't know how to go on because I don't have a look at all numbers. ...