Questions tagged [analytic-number-theory]

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

761 questions with no upvoted or accepted answers
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28
votes
0answers
476 views

$\forall n\in\mathbb N:n^x\in\mathbb Q$ implies $x\in\mathbb Z$ - elementary proof?

Consider the following two problems: Show that if for some $x\in\mathbb R$ and for each $n\in\mathbb N$ we have $n^x\in\mathbb N$, then $x\in\mathbb N$. Show that if for some $x\in\mathbb R$ ...
28
votes
0answers
726 views

Using the Brun Sieve to show very weak approximation to twin prime conjecture

I recently stumbled across MIT OCW for analytic number theory. As a budding number theorist, my ears perked up and I looked through some of the material haphazardly. I don't really know much about ...
21
votes
0answers
458 views

Why is Fourier Analysis effective for studying uniform distributions

On his great expository article about the naturality of the Zeta function in number theory, Tim Gowers makes the following claim: When it comes to the primes, we find that we do not have a good ...
14
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0answers
989 views

Values of hypergeometric functions

Let $_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;c)$ denote the generalized hypergeometric function. Let $A \subset \mathbb R$ be the set of all values of $\ _pF_q(\cdot)$ at rational points $a_i,b_j,c\in \...
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
538 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
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
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
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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
votes
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
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
670 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
votes
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
votes
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
votes
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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0answers
183 views

How to prove that below quantity is purely imaginary?

How do I prove that the following quantity is purely imaginary: $$\sum_{0\leq l_1<l_2<l_3<l_4\leq q-1} e^{-2\pi i \frac{(l_1^2-l_2^2+l_3^2-l_4^2)}{q} } $$ where $q$ is an odd number?
8
votes
0answers
282 views

Fourier transform of the critical line of zeta?

Is there a known expression for the (distributional) Fourier transform of the Riemann zeta function, taken along the critical line? I'd love to say that it's a weighted sum of delta distributions, ...
8
votes
0answers
146 views

Weierstrass product expression for Klein's j-invariant

The first sentence of @ccorn's answer to a previous question of mine was: “Because of the modular symmetries of $j(\tau)$, the zeros of $j(\tau)$ are precisely the $\operatorname{SL}(2,\mathbb{...
7
votes
0answers
104 views

Chebyshev function: variational formulation

The Wikipedia article on the Chebyshev function $\psi(x)$ states that, for $x=e^t$, it minimizes the functional $$J[f] = \int_0^\infty \dfrac{f(s)\zeta'(s+c)}{\zeta(s+c)(s+c)}ds - \int_0^\infty \...
7
votes
0answers
173 views

Density of multiplication table

Is there any easy way to show $$\lim_{N \to \infty} \frac{1}{N^2}\#\{ab : 1 \le a,b \le N\} = 0$$ A quick calculation I did shows that the number of positive integers $\le N^2$ with a prime divisor $...
7
votes
0answers
231 views

Is the Euler function $\phi$ constant in arbitrarily large intervals?

Is it true that for every $k \in \mathbb{N}$ there exists a natural number $x$ such that $\phi(x)=\phi(x+1)=\cdots=\phi(x+k)$, where $\phi$ is the Euler's totient function? I thought about this ...
7
votes
0answers
152 views

Can we use $n\log n$ instead of $n$-th prime?

Denote $\pi(x)$ be the number of primes $\leq x,$ $p(n)$ be the $n$-th prime number. We have $\pi(p(n))=n.$ It's well known that $$\pi(x)\sim \frac{x}{\log x} \\p(n)\sim n\log n.$$ Is it always ...
7
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0answers
709 views

Sum of reciprocal of primes in arithmetic progression

In http://www.math.dartmouth.edu/~carlp/Lehmer0.5.pdf on page 6 (top) the author states that: $$ \sum_{p \le x, \ p \equiv 1 \bmod l} \frac{1}{p} = \frac{\log \log x}{\phi(l)} + O \left ( \frac{\log ...
7
votes
0answers
405 views

Partial summation of a harmonic prime square series (Prime zeta functions)

I am trying to find the following series: $S=\displaystyle\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}\dfrac{1}{p_ip_j},A\leq p_1 < p_2 < \dots < p_n \leq B, \lbrace A,B\rbrace \in \mathbb{N}$ ...
7
votes
0answers
202 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 ...
6
votes
0answers
144 views

Almost a prime number recurrence relation

For the number of partitions of n into prime parts $a(n)$ it holds $$a(n)=\frac{1}{n}\sum_{k=1}^n q(k)a(n-k)\tag 1$$ where $q(n)$ the sum of all different prime factors of $n$. Due to https://oeis....
6
votes
0answers
128 views

Estimation of the number of solutions for the equation $\sigma(\varphi(n))=\sigma(\operatorname{rad}(n))$

For integers $n\geq 1$ in this post we denote the square-free kernel as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p,$$ that is the product of distinct primes dividing an ...
6
votes
0answers
180 views

Prove that every integer greater than 1 is sum of square and squarefree.

I tried to put some sieve method on this. Here, we denote the classic squarefree sieve function $\sum_{d^2|a}\mu(d)$ (notice that if $a$ is squarefree, the value is 1, and if not squarefree, the value ...
6
votes
0answers
335 views

Useful reformulation of Goldbach's conjecture?

Let us assume there exists some infinite order differential equation whose solution is: $$ y= \sum_{n=1}^\infty A_n \exp(p_n^sx) $$ Where $p_n$ is the $n$'th prime. Substituting $ y=\exp(\lambda x)$...
6
votes
0answers
70 views

Generalisation of the transcendence of $e$

Let $(e_n)$ be a sequence of numbers from the set $\{-1,0,1\}$. Is it true that $$\sum\frac{e_n}{n!}$$ is always transcendental except trivial cases (when the series is finite, i.e. $e_n=0$ for ...
6
votes
0answers
161 views

On the change $u=x^{1+\frac{1}{p_n}}$ in $\log \zeta(s)=s\int_0^\infty\frac{\pi(x)}{x(x^s-1)}dx$, where $p_n$ is the nth prime number

In [1] Wikipedia say that for $\Re s>1$ the Riemann zeta function satisfies $$\log \zeta(s)=s\int_0^\infty\frac{\pi(x)}{x(x^s-1)}dx,$$ where $\pi(x)$ is the prime counting function, and say too ...
6
votes
0answers
120 views

For a given integer $n$, how many primes $p_1,p_2 \leq n$ such that $\tau(p_1-1)=\tau(p_2-1)$

This is a curiosity question. Let $N$ be positive integer, I just want to know how many (an approximation) pair of primes $(p_1,p_2)$ that are less than $n$ and verify the following identity: $$\tau(...
6
votes
0answers
47 views

Equidistribution estimates after Zhang

I was curious what were the best equidistribution estimates toward the Elliott-Halberstam conjecture $$\sum_{q\le Q}\max\left|\Theta(x;q,a)-\frac{x}{\phi(q)}\right|<<\frac{x}{(log x)^A}$$ with $...
6
votes
0answers
483 views

Is there an equivalent statement of Riemann Hypothesis in term of Random Matrix or physics theory?

We all know that Riemann Hypothesis has many equivalent statements. After Montgomery’s works on pair-relationship, we now know that ZEROs of Riemann Zeta function has similar properties as ...
6
votes
0answers
233 views

Prime number theorem for Dedekind domains

Let $\mathscr P\subseteq \mathbb N$ be the set of prime numbers. The prime number theorem tells us that if $\pi(x)=|\{p\in\mathscr P\colon p\leq x\}|$ then $\pi(x)\sim \frac{x}{\log x}$. Now one could ...
5
votes
0answers
115 views

For what $k\in \mathbb{C}$ do we have $\lim_{x\to\infty}\frac{1}{x^{k+1}}\sum_{n=1}^x \sigma_k(n)=\frac{\zeta(k+1)}{k+1}$?

Can the following claim be extended for some complex $k$. Perhaps: all $k$ with real part greater than or equal to $1$? Do the arguments below fall apart for complex $k$ for some reason? Claim. ...
5
votes
0answers
120 views

The ordinary generating function for the square-free kernel: reference request about singularities and its phase plot

Let $n\geq 1$ an integer, in this post I denote the product of distinct prime numbers dividing dividing $n$ as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p,$$ is the famous ...
5
votes
1answer
185 views

An approximation for $1\leq n\leq N$ of the number of solutions of $2^{\pi(n)}\equiv 1\text{ mod }n$, where $\pi(x)$ is the prime-counting function

We denote the prime-counting function with $\pi(x)$ and we consider integer solutions $n\geq 1$ of the congruence $$2^{\pi(n)}\equiv 1\text{ mod }n.\tag{1}$$ Then the sequence of solutions starts as $$...
5
votes
0answers
79 views

Erdős-Kac and moments

One way to prove Erdős-Kac is by proving that for each non-negative integer $k$, the $k$-th moment $$M_k(x):= \frac{1}{x}\sum_{1 \leq n \leq x} \left(\frac{\omega(n)-\log \log x}{\sqrt{\log \log x}}\...
5
votes
0answers
334 views

Functional equation of the complete $L$-function of the twisted $L$-function of a cuspidal modular form

Let $f(z)=\sum a(n)n^{(k-1)/2}q^n\in S_k(\Gamma_0(N),\chi)$ a cuspidal modular form of integral weight with nebentypus $\chi.$ I am looking for the expression of $\Lambda(\psi\otimes f,s)$ the ...
5
votes
1answer
251 views

Large gap between two consecutive square-free numbers

Let $q_n$ denote the $n$-th square-free number. By Chinese remainder theorem (see this post), it is not difficult to show that there is arbitrarily large gap between two consecutive square-free ...
5
votes
0answers
225 views

Equidistribution theorem of Weyl

Have you examples of applications of Equidistribution theorem of Weyl in proofs of irrationality of numbers? I don't know if "if and only if" is true for this theorem.
5
votes
0answers
96 views

Relation between Bombieri theorem and p-adic squares

Koblitz states in his book on p-adic numbers on page 84: Suppose that $\alpha \in \mathbb Q$ is such that $1 + \alpha$ is the square of a nonzero rational number $a/b$. Let $S$ be the set of all ...
5
votes
1answer
88 views

Number of solutions to $x_1x_2+x_3x_4 = 1$ (mod $n$)

Show that the number of solutions $N$ to $x_1x_2+x_3x_4 = 1 \pmod n$ is $$ N=n^2\phi(n)\prod_{p|n} \left( 1+\frac{1}{p} \right). $$ Only thing I know how to start the problem is to consider: $$ N = ...
5
votes
0answers
89 views

Kloosterman sum and multiples of 16

A Kloosterman sum is defined as $$K(a,b;m)=\sum_{\substack{ 0\leq x \leq m-1\\\gcd(x,m)=1}} e^{2\pi \mathcal{i} (ax+bx^*)/m}$$ where $a,b,m \in \mathbb{N}$ and $x^*$ is the inverse of $x$ modulo $m$...
5
votes
0answers
376 views

Where's my mistake applying Perron's Formula?

I applied Perron's Formula to Riemann Zeta Function and got a weird result. First, I started with a simple definition of Riemann Zeta Function, $$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}$$ where $\...
5
votes
0answers
625 views

Maximum length of sequence of non-coprimes of $N$ - least upper bound for Jacobsthal's function

I am looking at the length of the longest sequences of adjacent integers that are not coprime to $N$ for very large $N$. Let $F_N$ be the set of integers less than $N$ which are not coprime with $N$: ...