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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12
votes
1answer
312 views

Show that $\sum\limits_pa_p$ converges iff $\sum\limits_{n}\frac{a_n}{\log n}$ converges

I am going through A. J. Hildebrand's lecture notes on Introduction to Analytic Number Theory. I'm currently stuck at the exercises at the end of Chapter 3 (Distribution of Primes I - Elementary ...
12
votes
1answer
367 views

Sum of Reciprocals of Primes in Imaginary Quadratic Field Diverges (2014 Miklós Schweitzer)

Problem 5 of the 2014 Miklós Schweitzer states: Let $\alpha$ be a non-real algebraic integer of degree two, and let $P$ be the set of irreducible elements of the ring $\mathbb{Z}[\alpha]$. Prove that $...
12
votes
0answers
155 views

Is $\sqrt p - \lfloor\sqrt p\rfloor$, $p$ running over primes $1 \pmod 4$ , dense in $[0,1]$?

A result I would like to know is if there are infinitely primes congruent to $1 \pmod 4$, with fractional part in an interval strictly contained in $\left(0, \dfrac 1 4 \right)$. The title question ...
12
votes
0answers
537 views

More elegant $\zeta(s)$ zeros counting function than $N(T)$

The explicit formula expresses the deep connection between the primes $p$ and the non-trivial zeros $\rho$ of $\zeta(s)$. The prime-counting function is given by the following formula giving primes in ...
12
votes
1answer
429 views

Finding the integer $\le n$ with largest number of divisors

As mentioned in an answer to this question an integer less than $n$ with largest number of divisors can be found exploring the numbers $x$ of the form $$ x = 2^{a_1} 3^{a_2} \dots p_k^{a_k} \dots $$ ...
11
votes
2answers
1k views

A problem about the largest prime factor of $n^2+1$

Let $f(n)$ be the largest prime factor of $n$. The image of function $g(n)=\sqrt{f(n^2+1)}$ is like this: Question: If we want to draw a horizontal line which bisects the points from $n=1$ to $n=x,...
11
votes
2answers
959 views

Importance of the zero free region of Riemann zeta function

I have heard that for improving the error term in the Prime Number Theorem, we need better and better estimates on the zero free region. I have also heard that the best possible error term comes from ...
11
votes
3answers
998 views

Equivalence to the prime number theorem

I was just reading this question and answer: How will this equation imply PNT and it raised a whole new question: Given that $\sum_{n\le x} \Lambda(n)=x+o(x)$, prove that $$\sum_{n\le x} \frac{\...
11
votes
5answers
1k views

Proving $\sqrt{2}\in\mathbb{Q_7}$?

Why does Hensel's lemma imply that $\sqrt{2}\in\mathbb{Q_7}$? I understand Hensel's lemma, namely: Let $f(x)$ be a polynomial with integer coefficients, and let $m$, $k$ be positive integers such ...
11
votes
1answer
955 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 ...
11
votes
1answer
143 views

For primes sufficiently large, must digit products be zero?

Let $\{P_n\}, n\in \mathbb{N}$ be the sequence prime numbers such that $P_1=2, P_2=3\dots$. Define a new sequence $\{M_n\}$, $n\in \mathbb{N}$, such that $M_n=$Product of the digits of the $nth$ ...
11
votes
1answer
270 views

Is there a special value for $\frac{\zeta'(2)}{\zeta(2)} $?

The answer to an integral involved $\frac{\zeta'(2)}{\zeta(2)}$, but I am stuck trying to find this number - either to a couple decimal places or exact value. In general the logarithmic deriative of ...
11
votes
2answers
754 views

Prime Numbers and a Two-Player Game

In this question, $\mathbb{N}_0$ is the set of all nonnegative integers. The notation $\mathbb{N}$ is reserved for the set of all positive integers. Alex and Beth are playing the following game. ...
11
votes
1answer
292 views

The asymptotics of the products over primes $\prod\limits_{2<p\le n}\left(1 - \frac1{p-1}\right)$

Short version If we define $$ f(n) = \prod_{2 < p \le n} \left( 1 - \frac{1}{p-1}\right) $$ where the product is over prime numbers $p$, then is it true that asymptotically $$ f(n) \sim \frac{c}{\...
11
votes
2answers
372 views

Numbers divisible by the square of their largest prime factor

Let $p(n)$ be greatest prime factor of $n$, denote $A=\{n\mid p^2(n)\mid n,n\in \mathbb N\}.$ $A=\{4,8,9,16,18,25,27,32,36,49,50,\cdots\},$ see also A070003. Define $f(x)=\sum_{\substack{n\leq x\\n\...
11
votes
1answer
354 views

Maximum integer not in $\{ ax+by : \gcd(a,b) = 1 \land a,b \ge 0 \}$

Ryan asked about a variation of the coin problem, which was whether for any coprime natural numbers $x,y$ every sufficiently large natural number is $ax+by$ for some coprime natural numbers $a,b$. ...
11
votes
1answer
863 views

Question about a proof in Iwaniec-Kowalski's Analytic Number Theory book

My question is about the end of the proof of theorem 1.1, in page 27. Namely, it is stated that whenever we have a multiplicative function $f:\mathbb{N} \to \mathbb{C},$ let the sequence $\Lambda_{f}...
11
votes
1answer
216 views

Combinatorial number theory, what is $\lim_{n \to \infty} {\ln f(n)\over \ln n}$?

Let $\textbf{Z}_n$ be the set $\{0, 1, \ldots, n - 1\}$ with addition mod $n$. Consider subsets $S_n$ of $\textbf{Z}_n$ such that $(S_n + k) \cap S_n$ is nonempty for every $k$ in $\textbf{Z}_n$. Let $...
11
votes
1answer
2k views

how to understand $\log\zeta(s)$ (Riemann zeta function)?

I know that if a function $f$ is analytic and has no zeros in a simple connected region, then we can define $\log{f}$ making it analytic in that region. Let's assume $Re(s)>1$. Is $\zeta(s)$ ...
11
votes
1answer
439 views

Is there any relationship between the Riemann z function and strange attractors?

I have this question in mind since the first time I saw a graphical representation of the zeta function (like in the sample below). Just by looking to them I wondered if there is any relationship ...
11
votes
0answers
306 views

Weak version of Fortune's conjecture

Let $p\#=2\cdot3\cdot5\cdots p$ denote the primorial and $N(x)$ the smallest prime greater than or equal to $x$. Then Fortune's conjecture is that $N(p\#+2)-p\#$ is prime for all $p$. (Heuristic: to ...
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 $$\int_{2}^{x} \frac{\mathrm dt}{\log^{n}{t}} = O\Bigl(\frac{x}{\log^{n}{x}}\Bigr)?$$
10
votes
3answers
2k views

A cohomological statement equivalent to the Riemann Hypothesis

Is there a possibility for looking for a theory of cohomology and an equivalent cohomological statement for Riemann hypothesis over $\mathbb{Z}$?
10
votes
3answers
456 views

Where is the fallacy in the argument using Prime Number Theorem

I am reading about Prime Number Theorem from book by Ingham. As as application of PNT I found the following theorem: Now my doubt is at the step $\frac{\log(y)}{\log(x)}\rightarrow 1$, we can say $\...
10
votes
2answers
531 views

Bounds on a sum of gcd's

Does there exist a positive real number $C$ and a positive integer $M$ such that for all $n > M$ we have: $$\sum_{i=1}^n\sum_{j=1}^n\gcd (i, j)\ge Cn^2 \log n$$ This originally appeared as an ...
10
votes
1answer
2k views

Why the Riemann hypothesis doesn't imply Goldbach?

I'm interested in number theory, and everyone seems to be saying that "It's all about the Riemann hypothesis (RH)". I started to agree with this, but my question is: Why then doesn't RH imply the (...
10
votes
2answers
700 views

Two Dirichlet's series related to the Divisor Summatory Function and to the Riemann's zeta-function, $\zeta(s)$

Considering the $\textit{Divisor Summatory Function}$, $D(n)$, defined as $$ D(n) = \sum_{k=1}^{n}d(k) , $$ where $$ d(n) = \sum_{k|n}^{n}1. $$ One can observe the following pattern in the values of $...
10
votes
3answers
579 views

For what $t$ does $\lim\limits_{n \to \infty} \frac{1}{n^t} \sum\limits_{k=1}^n \text{prime}(k)$ converge?

The average of all primes is $$\lim\limits_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} \text{prime}(k) ,$$ which diverges. What is the smallest $r$ such that for $t>r$, $$\lim_{n \to \infty} \frac{1}...
10
votes
1answer
583 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?
10
votes
3answers
351 views

M and n are positive integers such that $2^n - 3^m > 0$. Prove (or disprove) that $2^n - 3^m \geqslant 2^{n-m}-1$.

Given that $2^n - 3^m > 0$, I know that $n > m\log_{2}3$ (*). If $2^n - 3^m \geqslant 2^{n-m}-1$, $n>= m + \log_{2}\frac{3^m-1}{2^m-1}$ (**). This is the result when I graph it out ($m$ -> $...
10
votes
2answers
2k views

how to prove this extended prime number theorem?

A Generalized Prime Number Theorem? Conjecture Let $n$ and $k$ be positive integers with $n - 50 > k^2 > 0$ and $n$ sufficiently large. Then for the odd primes we have, when $p$ is the biggest ...
10
votes
1answer
225 views

Fourier Series involving the Jacobi Symbol

We know that the Fourier Series $$s(x)=\sum_{k\neq0}\frac{1}{k}\exp\left(2\pi ik x\right)$$ corresponds to the sawtooth function, $s(x)=\left\{x\right\} -\frac{1}{2}$. Suppose that $\left(\frac{k}{d}\...
10
votes
2answers
362 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 ...
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 + ...
10
votes
1answer
338 views

Converting an infinite product to sum; Ramanujan $\tau$ function

I've gotten what seems most of the way, but I'm quite stuck at this point. Define $\tau(n)$ by \begin{align*} q\prod_{n=1}^\infty (1-q^n)^{24} = \sum_{n=1}^\infty\tau(n)q^n. \end{align*} ...
10
votes
2answers
339 views

Probability of determinants being coprime

I have a question that is not of particular significance, but I would love to understand the underlying principles. Suppose we have two square $3 \times 3$ matrices, $M_1$ and $M_2$ with $$M_1 = \...
10
votes
2answers
526 views

How often is a sum of $k$ consecutive primes also prime?

Let's define a $k$-sum as a sum of $k$ consecutive primes. For example, $15=3+5+7$ is a $3$-sum. How many $k$-sums are themselves prime? Here's one way to formulate the question more precisely: What ...
10
votes
2answers
253 views

What is your idea about this conjecture?

I conjecture that in a consecutive sequence of $n$ natural numbers all greater than $n$, there exists at least one number which is not divisible by any prime number less than or equal to $n/2$. Can ...
10
votes
1answer
333 views

some standard estimates in Yitang Zhang's paper

I'm trying to understand Zhang's paper on prime gaps, but I can't figure out some "standard" estimates for which Zhang omitted details. As a layman in analytic number theory, I really need some hints (...
10
votes
1answer
299 views

Analytic number theory primer — sequences and series

For a book like Titchmarsh, or Iwaniec and Kowalski, it seems a thorough knowledge of asymptotics is a prerequisite. What are good books for training oneself in such manipulation of asymptotics, ...
10
votes
1answer
806 views

Polar Density of a Set of Primes

In Chapter 7 of Marcus' Number Fields, he defines the polar density of a set $A$ of primes of a number field $K$ as follows: Definition: If some $n$th power of the function $$\zeta_{K,A}(s) = \...
10
votes
1answer
337 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,...
10
votes
0answers
348 views

Number of ways to express a binary number in a certain way

So I'm working on a problem where I get to a point where I have to count the number of solutions to an equation or at least find a decent upper bound to be used in an estimate I need later. The ...
10
votes
0answers
912 views

Proof of Hardy-Ramanujan inequality in number theory.

On page 3 of http://www.math.dartmouth.edu/~carlp/Lehmer0.5.pdf the author write that the following inequalities follow from "the Hardy-Ramanujan inequality", but he doesn't point to a proof. The ...
9
votes
2answers
197 views

Evaluating $\sum_{\gcd\left(m,n\right)=1}\frac{1}{m^2n^2}$

I was wondering how one would evaluate the sum $$\sum_{\gcd\left(m,n\right)=1}\frac{1}{m^2n^2}.$$ The first thought that came to mind to to try something like this: $$\sum_{\gcd\left(m,n\right)=1}\...
9
votes
4answers
822 views

The asymptotic expansion for the weighted sum of divisors $\sum_{n\leq x} \frac{d(n)}{n}$

I am trying to solve a problem about the divisor function. Let us call $d(n)$ the classical divisor function, i.e. $d(n)=\sum_{d|n}$ is the number of divisors of the integer $n$. It is well known that ...
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 ...
9
votes
3answers
185 views

Is there a monotonic $f$ such that $\sum f(n)$ diverges but $\sum f(p)$ converges?

(where the former summation is over natural numbers $n$ and the latter is over prime numbers $p$, and $f: \mathbb{N} \to \mathbb{R}$ is a monotonic function.) For the class of functions $f_s(n) = \...
9
votes
2answers
1k views

The Möbius function is the sum of the primitive $n$th roots of unity. [duplicate]

Did you know that the Möbius function $\mu$ is the sum of the primitive nth roots of unity? I want to know about meaning of this. This statement is expressed as, $$\mu(n) = \sum_{\substack{k=1 \\ (...
9
votes
3answers
316 views

What is the set $\{x\in\Bbb R\mid \forall q\in\Bbb Q: q^x\in\Bbb Q\}$?

What is the set $\{x\in\Bbb R\mid \forall q\in\Bbb Q: q^x\in\Bbb Q\}$? Of course $\Bbb Z$ is a subset of this set. Are there any other? if not what is the proof? is there a good reference for it?