Questions tagged [analytic-number-theory]

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

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9
votes
1answer
154 views

Is the intersection of $\sin(\mathbb{N})$ and $\cos(\mathbb{N})$ empty?

My guess is that the intersection is empty and this is as far as I got in an attempt to prove this by contradiction: $\exists n,m \in \mathbb{N}, \cos(n)=\sin(m) \land n \neq m \quad (1)$ $\cos^2(n)=...
9
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2answers
380 views

Generalized PNT in limit as numbers get large

If $\pi_k(n)$ is the cardinality of numbers with k prime factors (repetitions included) less than or equal n, the generalized Prime Number Theorem (GPNT) is: $$\pi_k(n)\sim \frac{n}{\ln n} \frac{(\ln ...
9
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2answers
629 views

Concrete Example of the Birch and Swinnerton-Dyer Conjecture

The Setup Consider an elliptic curve $E$ in Weierstrass form $y^2=x^3+ax+b$ with $a,b \in \mathbb{Z}$. As usual, we let $\Delta_E$ be the discriminant of the polynomial, and we set $N_p := $ #{...
9
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1answer
287 views

Is there a $k$ such that $a_n=\frac{n^k!}{(n^k!!)^2}$ converges?

Lately I have been playing around with the sequence $$a_n(k) := \frac{n^k!}{(n^k!!)^2}.$$ For $k=1$, it does not look much like it converges. I don't know $k=2$ it converges, but it doesn't really ...
9
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1answer
247 views

Putnam 2015 B6, sum involving number of odd divisors on an interval.

For each positive integer $k$, let $A(k)$ be the number of odd divisors of $k$ in the interval $[1, \sqrt{2k})$. What is$$\sum_{k=1}^\infty (-1)^{k-1} {{A(k)}\over{k}}?$$
9
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2answers
4k views

What does this $\asymp$ symbol mean? (subject: analytic number theory)

I'm reading a survey article by Andrew Granville on analytic number theory. On page 22 of the paper, there appears a strange looking symbol, undefined. I've circled it in red in the screenshot below....
9
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1answer
254 views

Relationship between GCD, LCM and the Riemann Zeta function

Let $\zeta(s)$ be the Riemann zeta function. I observed that as for large $n$, as $s$ increased, $$ \frac{1}{n}\sum_{k = 1}^n\sum_{i = 1}^{k} \bigg(\frac{\gcd(k,i)}{\text{lcm}(k,i)}\bigg)^s \approx \...
9
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2answers
1k views

Ramanujan's Tau function, an arithmetic property

The problem: Let $\tau(n)$ denote the Ramanujan $\tau$-function and $\sigma(n)$ be the sum of the positive divisors of $n$. Show that $$ (1-n)\tau(n) = 24\sum_{j=1}^{n-1} \sigma(j)\tau(n-j).$$ I'...
9
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1answer
584 views

How to derive an identity between summations of totient and Möbius functions

I have the following identities $$\sum_{n \le x} \varphi(n) = \frac{1}{2} \sum_{n \le x} \mu(n) \left[\frac{x}{n}\right]^2 + \frac{1}{2}$$ $$\sum_{n \le x} \frac{\varphi(n)}{n} = \sum_{n \le x} \frac{...
9
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4answers
793 views

Explicit formula for floor(x)?

In number theory we have so-called explicit formula's in terms of the Riemann zeta zero's. For instance to count the sum of the logarithms of the primes below some given integer. (second Chebyshev ...
9
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2answers
2k views

Derivative of the Riemann zeta function for $Re(s)>0$.

The Riemann zeta function can be analytically continued to $Re(s)>0$ by the infinite sum $$\zeta(s)=\frac{1}{1-2^{1-s}}\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^s}.$$ Can we differentiate this with ...
9
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2answers
792 views

Prove that $\sum_{n=1}^\infty \mu(n) \log n/n =1$.

One can show that the Prime Number Theorem is equivalent to the statement $$ A(x):= \sum_{n \leq x} \frac{\mu(n)}{n}=o(1),\qquad \qquad (1)$$ i.e. that $A(x) \to 0$ as $x \to \infty$. Given that the ...
9
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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) = \...
9
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1answer
942 views

Sum of square root of primes

I was playing around with prime numbers and a question came into my mind: Let $S(n)$ denote the sum of square roots of primes from $2$ to the $n$th prime number. Are there infinitely many numbers $n$ ...
9
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1answer
145 views

Looking for the closed form of $\sum_{n=1}^{\infty}{\zeta(2n+1)\over (2n+1)2^{4n}}$

We was able to determine $(1)$ to have this closed form $$\ln(2)-\gamma=\sum_{n=1}^{\infty}{\zeta(2n+1)\over (2n+1)2^{2n}}\tag1$$ then we when on and try to evaluate $(2)$ and we only half of the ...
9
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2answers
465 views

Intuitive explanation with rigorous details why zeta has infinitely many zeros?

I have seen a proof outline that $\zeta$ has infinitely many zeros on the critical line here but what I really want is: Simplest possible (least "magic") argument that explains why zeta has ...
9
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1answer
376 views

Squares in $(\operatorname{rad}(1)^2+1)\cdot(\operatorname{rad}(2)^2+1)\ldots(\operatorname{rad}(n)^2+1)$

For integers $n\geq 1$, $\operatorname{rad}(n)$ denotes the radical of an integer, see in Wikipedia this definition $$\operatorname{rad}(n)=\prod_{p\mid n}p,$$ if $n>1$ with factorization $n=\prod_{...
9
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1answer
252 views

the constant in the asymptotics of $\sum_{1\le k \le n} \frac{\varphi(k)}{k^2}$

The following thread at math.stackexchange.com proposes a constant term for the asymptotic expansion of $$\sum_{1\le k \le n} \frac{\varphi(k)}{k^2}.$$ I am getting a different term and I would like ...
9
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1answer
186 views

How to prove that the values of Eisenstein Series $E_2$ are algebraic integers?

I'm looking for a proof (see my older question here) that the following term is an algebraic integer whenever $\tau_N=\frac{N+\sqrt{-N}}{2}$ is a quadratic irrationality with class number $1$: $$A_N:=...
9
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1answer
215 views

Exercise 2 from Terry Tao's blog on Euler-Maclaurin, Bernouilli numbers, and the zeta function

In the blog post The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation, Terry Tao looks at the commonly-cranked 'absurd' formulae $$\begin{align} \...
9
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2answers
287 views

Convergence of an integral involving the radical of an integer versus the convergence of $\int_2^\infty\frac{dx}{x(\log x)^2}$

For positive integers $n\geq 2$, let $\operatorname{rad}(n)$ the radical of the integer $n$. For example $\operatorname{rad}(24)=6$. See this Wikipedia to know this definition. Question. I would ...
9
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1answer
319 views

Asymptotics of A283190

OEIS sequence A283190 gives the number of different values $n \mod k$ for $1 \le k \le \lfloor n/2 \rfloor$. Yes, I know this is taking $\mod k$ as a function rather than an equivalence relation: $...
9
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1answer
200 views

Prove $\sum\limits_{i=k}^{n-1}\{\frac{\binom{i}{k}}{n}\}=\frac{n-k^{w(n)}}{2}$

$k$ is an odd number, $(n,k!)=1$, prove that $$\sum_{i=k}^{n-1}\left\{\frac{\binom{i}{k}}{n}\right\}=\frac{n-k^{w(n)}}{2},$$ where $\{x\}=x-[x]$, $w(n)$ is the number of distinct prime factors of $n$. ...
9
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1answer
301 views

Riemann zeta function and Bernoulli function

I encountered the following problem: Show that $$\zeta(2n+1)=\frac{(-1)^{n+1}(2\pi)^{2n+1}}{2(2n+1)!}\int_0^{1}B_{2n+1}(x)\cot({\pi}x)dx$$ where $B_{2n+1}(x)$ is the Bernoulli polynomial. This ...
9
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1answer
546 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 ...
9
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0answers
218 views

An Engineer sets out to Prove Fermat's Last Theorem …

This started off as a joke post of mine on a Facebook Group called "Bad Maths that Gives the Right Answer", in which I pulled a Fermat and claimed that the last bit of the proof was too long to post. ...
9
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1answer
208 views

On integer values which are attained by $n/\pi(n)$ only once

Let $\pi (n)$ denote the prime counting function. I can prove that $\mathbb N \setminus \{1\} \subseteq \{n/ \pi(n) : n \in \mathbb N \}$ . Now for every integer $m>1$ , define $s(m) := \{ n \in \...
9
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0answers
270 views

A map from zeros of $\zeta(s)$ to zeros of $C(s)?$

Let $P(s),C(s),\zeta(s)$ be the prime zeta function, the analogous composite zeta function, and the classical zeta function. I do not know whether it is known that there are infinitely many zeros of ...
9
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0answers
1k views

Mathematics felt by Srinivasa Ramanujan [closed]

At the moment I am reading the book Ramanujan's Papers by B.J. Venkatachala, V. Vinay and C.S. Yogananda; when clarifying some doubt with a professor, he told me that Srinivasa Ramanujan used Galois ...
9
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0answers
538 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 $E_p = \{x/p \: | \: 0 \leq x < p, f(x) \equiv 0 \: (p) \}$ = { ...
9
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0answers
668 views

How to simplify $\newcommand{\bigk}{\mathop{\vcenter{\hbox{K}}}}\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_k(s)}{g_k(s)}\right)^{-1}$

I'd like to simplify $$\newcommand{\bigk}{\mathop{\huge\vcenter{\hbox{K}}}}B(s)=\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_{k}(s)}{f_{k}(s)}\right)^{-1}$$ to something of the form $$...
9
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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 ...
9
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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
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1answer
231 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: $$\zeta \rho (s) = \sum_{n=1}^{\infty} \frac{1}{(p_n)^s}. $$...
8
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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 ...
8
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2answers
1k views

Generating functions and the Riemann Zeta Function

The generating function for the terms of the harmonic series: $\frac{1}{n}$ is $-\ln(1 - x)$. Does an ordinary generating function exist for the terms of the zeta function $\zeta(s) = \sum_{n=1}^\...
8
votes
2answers
612 views

Bounds for $\zeta$ function on the $1$-line

I was going over my notes from a class on analytical number theory and we use a bound for the $\zeta$ function on the $1$ line as $\vert \zeta(1+it) \vert \leq \log(\vert t \vert) + \mathcal{O}(1)$ ...
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 ...
8
votes
1answer
2k 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 \sqrt{x}...
8
votes
3answers
271 views

Showing $\pi(ax)/\pi(bx) \sim a/b$ as $x \to \infty$

I'm having a bit of a problem with exercise 4.12 in Apostol's "Introduction to Analytic Number Theory". I don't think it's supposed to be a very hard exercise, it's the first one in its section (they'...
8
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3answers
231 views

Riemann zeta for real argument between 0 and 1 using Mellin, with short asymptotic expansion

The following would appear to be true. For real $0 < \sigma < 1,$ we seem to have a very satisfying sum minus integral limit, $$ \zeta(\sigma) \; \; = \; \; \lim_{n \rightarrow \infty} \; \;...
8
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2answers
152 views

Why is $G(k)$ “more fundamental” than the Hilbert-Waring function $g(k)$?

In the Wikipedia entry for Waring's problem, the section on $G(k)$ starts as: “From the work of Hardy and Littlewood, more fundamental than $g(k)$ turned out to be $G(k)$, which is defined...” There ...
8
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2answers
466 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 ...
8
votes
1answer
227 views

Are the unit partial quotients of $\pi, \log(2), \zeta(3) $ and other constants $all$ governed by $H=0.415\dots$?

Khinchin showed that given the simple continued fraction of a real number, $$r = a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1} {\ddots}}}$$ then it is almost always true that the partial quotients $a_i$ ...
8
votes
1answer
480 views

Proof involving the logarithmic integral

Another exercise from Apostol's book, this time we're supposed to prove $$\mathrm{Li}(x)=\frac{x}{\log x}+\int_2^x \frac{dt}{\log^2t}-\frac{2}{\log 2}.$$ which is easy to do via integration by ...
8
votes
2answers
2k views

Approximation of $\mathrm{Li}(x) = \int\limits_{0}^x \frac{dt}{\ln t}$ [duplicate]

I am reading about the Riemann hypothesis, and the article mentioned the Li function: $$\mathrm{Li}(x) = \int\limits_{0}^x \frac{dt}{\ln t}$$ They said that this function can be approximated: $$\...
8
votes
1answer
347 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 ...
8
votes
2answers
378 views

Does the sum of reciprocals of primes congruent to $1 \mod{4}$ diverge?

Let $P$ be the set of primes $p$ greater than $3$ such that $p\equiv1 \pmod{4}$. Does the following sum converge or diverge? $$ \sum_{p\in P}\frac{1}{p} $$
8
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2answers
578 views

Tying some pieces regarding the Zeta Function and the Prime Number Theorem together

I came across two remarks that I would appreciate help in making the connections. I) In Riemann's Explicit Formula: for $x > 1$ $\Pi = Li(x) - \sum_{\rho:\zeta(\rho)=0}Li (x^{\rho})- \log(2) +$ ...
8
votes
2answers
1k views

Why does the Riemann zeta function have zeros in the complex plane? How is it possible to find them?

I ask this because, according to Euler's product formula, Riemann's zeta function =(1/something), so how could that be zero? Also, how could one find zeros that are on the negative side and find a ...