# Questions tagged [analytic-number-theory]

A beautiful blending of real/complex analysis with number theory. The study involves distribution of prime numbers and other problems and helps giving asymptotic estimates to these.

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172 views

### Frequency of digits in powers of $2, 3, 5$ and $7$

For a fixed integer $N\in\mathbb{N}$ consider the multi-set $A_2(N)$ of decimal digits of $2^n$, for $n=1,2,\dots,N$. For example,
$$A_2(8)=\{2,4,8,1,6,3,2,6,4,1,2,8,2,5,6\}.$$
Similarly, define the ...

**2**

votes

**1**answer

115 views

### On an oscillation Theorem involving the Chebyshev function and the zeros of the Riemann zeta function

Define $\theta(x)=\sum_{p\leq x} \log p $, where $p>1$ denotes a prime.
Nicolas proved that if the Riemann zeta function $\zeta(s)$ vanishes for some $s$ with $\Re(s)\leq 1/2 + b$, where $b\in(0, 1/...

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183 views

### On Robin's inequality and the zeros of the Riemann zeta function

Let $\zeta$ denote the Riemann zeta function. By an argument of Robin http://zakuski.utsa.edu/~jagy/Robin_1984.pdf, we know that $\zeta(\rho)=0$ for some $\rho$ with $\Re(\rho) \in (1/2, 1/2 + \beta]$,...

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votes

**1**answer

294 views

### Yet another question on sums of the reciprocals of the primes

I recall reading once that the sum $$\sum_{p \,\, \small{\mbox{is a known prime}}} \frac{1}{p}$$
is less than $4$.
Does anybody here know what the ultimate source of this claim is?
Please, let me ...

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vote

**0**answers

121 views

### A strengthening of Dirichlet prime number theorem

Dirichlet Theorem on arithmetic progression states that the sequence $\{a+kd\}_{k=1}^{\infty}$ contains infinitely many primes when $(a,d)=1$. In other words if we let $A=\{a+kd\}_{k=1}^{\infty}$ and ...

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168 views

### Prime character sums

Let $p$ be a (large) prime number, and let $\chi : (\mathbf{Z}/p\mathbf{Z})^{\times} \rightarrow \mathbf{C}^{\times}$ be a Dirichlet character of conductor $p$. We have good estimates on the character ...

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136 views

### Riemann Explicit Formula

I am writing my senior thesis on Montgomery's pair correlation conjecture, and in his first lemma, he uses the following explicit formula:
$$\sum_{n \leq x} \Lambda(n) n^{-s} = -\frac{\zeta'}{\zeta}(s)...

**6**

votes

**1**answer

314 views

### Sum over characters

Take $x>0$ large, $t\in \mathbb R$, $q\in \mathbb N$ and a non-principal character $\chi $ mod $q$. If you want, take $t\leq x$. How do I bound
\[ \sum _{n\leq x}\frac {\chi (n)}{n^{it}}?\]
My ...

**5**

votes

**1**answer

105 views

### Asymptotic's for Fourier coefficients of $GL(3)$ Maass forms

Let $f$ be a $GL(3)$ Hecke-Maass cusp form and $A(m,n)$ denote its Fourier coefficients.
Are there any lower bounds known for $\sum_{p\leq x}|A(1,p)|^2$ or $\sum_{n\leq x}|A(1,n)|^2$ ? (we know the ...

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66 views

### Enquiry on bounds for $\frac{1}{(n-1)!}\frac{d^n}{ds^n}((s-1)\zeta(s))$ at $s=1.$

Let $\zeta$ be the Riemann zeta function and $n$ be a positive integer. What are the known (conditional and unconditional) bounds for $f(n)=\frac{1}{(n-1)!}\frac{d^n}{ds^n}((s-1)\zeta(s))$ at $s=1$ ?
...

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votes

**1**answer

104 views

### Question on the Selberg-Delange Method

I am learning the Selberg-Delange method in order to read this article:
https://arxiv.org/abs/1712.09019
It says at the beginning of section 6.2 that they use a weakened version of the general ...

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votes

**1**answer

86 views

### How to get asymptotic expansion of the sum of modified Bessel function $\sum_{n=1}^\infty K_0(s\, n)$ as $s\to 0^+$?

I guess for the modified Bessel funcion $K_0(z)$,
$$\sum_{n=1}^\infty K_0(s\, n)
\sim
\frac{-2\log 2 - \log \pi + \gamma}{2} + \frac{\log s}{2} + \frac{\pi}{2\, s}, \quad s\to 0^+,$$
if taking
$$\...

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votes

**4**answers

1k views

### Modern Algebraic Geometry and Analytic Number Theory

I am currently discovering the algebraic geometry of Grothendieck. I have the impression that this theory, which leads to categories, schemas, topos etc. alone can encompass all modern mathematics (...

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vote

**0**answers

191 views

### On Primes in Arithmetic Progressions

I was wondering if the following approach is being attempted to prove the twin-prime conjecture.
Tao and Green proved in their paper (2006), that there are arbitrarily long arithmetic progressions ...

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votes

**1**answer

156 views

### Local phase statistics of the nontrivial Riemann zeros

(The question is inspired by Owen Maresh's post)
The local phase of a nontrivial zero $s$ of the Riemann $\zeta$ is the argument of $\zeta'(s)$.
Numerical results on the first 10000 zeros suggest ...

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votes

**2**answers

210 views

### How to obtain an upper bound for $\prod_{p\mid N} (1 + 1/\sqrt{p})$ where $N$ is square free?

I am interested in obtaining an upper bound for $\prod_{p|N} (1 + 1/\sqrt{p})$ when $N$ is squarefree. It's not too hard to show that
$$
\prod_{p\mid N} (1 + 1/\sqrt{p}) \ll C^{\omega(N)} \ll N^{\...

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votes

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121 views

### Request for an English proof of Robin's result on $\sigma(n)$

Define $\sigma(n)=\sum_{d\mid n} d$. It is a result of Robin that if the Riemann Hypothesis is false, then there exists constants $0<\beta<1/2$ and $C>0$ such that
$$\sigma(n)\geq e^{\gamma}...

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**1**answer

69 views

### meromorphic extension of dirichlet series

Suppose $\{a(n)\}_{n\ge 1}$ is a bounded complex sequence. Let $\phi(s)=\sum_{n\ge 1} \frac{a(n)}{n^s}$. Obviously, the Dirichlet series $\phi(s)$ is absolutely convergent for $\mathcal{R}(s)>1$. I ...

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134 views

### Do we believe that the distribution of spacings of successive critical zeros of zeta is log-symmetric?

Let $\gamma^{+}(T)$ be the imaginary part of the critical zero of $\zeta$ closest to $1/2+iT$ with $\gamma^{+}(T)\ge T$ and define similarly $\gamma^{-}(T)$ with a reversed inequality. Let $g(T)$ be ...

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755 views

### Motivation behind Analytic Number Theory

I am an undergraduate student of mathematics and recently took an introductory course in analytic number theory, where the instructor roughly followed Apostol's first text on the subject. I have now ...

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124 views

### Approximation in the circle method

I am interested in the circle method and I am currently working on Vaughan's book. Let $f$ be the generating function $f$ of the squares, that is to say the power series sum of $z^{m^2}$.
One of the ...

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**0**answers

170 views

### How small may the discriminant of an $S_d$-field be?

In every degree $d$, the Galois closure of the typical number field has the maximal possible Galois group $S_d$. Denote by $f(d)$ the least absolute value of a discriminant of an $S_d$-field of degree ...

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votes

**1**answer

160 views

### Zeros of derivatives of Dirichlet Eta function

Let
$$
\eta^{(d)}(z) =
\sum_{n=1}^\infty
\dfrac
{(-1)^d(-1)^{n-1}\ln(n)^d}
{n^z}
$$
be the derivative of Dirichlet Eta function of order $d$.
Does it exist any known or not known zero of $\eta^{(d)}...

**3**

votes

**1**answer

192 views

### On the convergence of $\sum_{n=1}^{\infty} \frac{\lambda(n)}{n}$ and the Prime Number Theorem

Let $\lambda$ be the Lioville function of number theory.
I've heard several times that if $L=\sum_{n=1}^{\infty} \frac{\lambda(n)}{n} =O(1)$ then $L=0$ (the Prime Number Theorem). How can this be ...

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77 views

### Smoothed Weyl sum inequality

One version of Weyl's inequality states that for any $\alpha\in\mathbb{R}$ and $(a, q) = 1$ such that $|\alpha - a/q|\le 1/q^2$, we have that
$$\sum_{n\le X} e(n^k\alpha)\ll X^{1 + \varepsilon}(q^{-1}...

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votes

**1**answer

124 views

### Where can I find this result of Ingham?

Sometime ago, I read somewhere (should be in Titschmarsh) that, if $N(\sigma, T)$ denotes the number of zeros of the Riemann zeta function $\zeta(s)$ with $\Re(s)\geq \sigma>1/2$ up to height $T>...

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votes

**2**answers

298 views

### Infinitely many primes in particular progressions

I'm faced with the following problem on primes. Does someone have any clue? Is it (a reformulation of) an open problem?
Let $d$ be a positive integer, $d\geq 2$. By Dirichlet's theorem, there is an ...

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vote

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67 views

### Is there an estimate available for a sum of the form $\sum_{\mathbf{x} \equiv \mathbf{a} (H) } \mu^2(x_1 x_2)$

I am interested in a sum of the shape
$$
\sum_{ \substack{ 1 \leq x_1, x_2 \leq B\\
\mathbf{x} \equiv \mathbf{a} (H) } } \mu^2(x_1 x_2).
$$
I figured it must have been considered before, but I have ...

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votes

**1**answer

190 views

### Piltz Divisor Problem

Let $\tau_k(n)$ count the number of ways of representing $n$ as the product of $k$ natural numbers. It is known that:
$$D_k(x) = \sum_{n \leq x} \tau_k(n) = xP_k(\log x) + O(x ^{1 - \frac{1}{k-1}}(\...

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votes

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119 views

### How many exceptional conductors are there?

We say that a conductor $q$ is exceptional if there is a primitive quadratic character $\chi$ modulo $q$ such that $L(s,\chi)$ has a real zero $\beta$ such that $\beta > 1-c/\log q$ (where $c$ is ...

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135 views

### Chen's theorem in which constituent primes are close together

Chen's theorem states that every sufficiently large even integer can be written as $n=p+q$, where $p$ is a prime and $q$ is a product of at most two primes.
I would like a representation $n=p_1+...

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votes

**1**answer

444 views

### Smallest Mazur's good prime

Let $p$ and $\ell$ be primes $\geq 5$ such that $\ell$ divides $p-1$. Following Mazur, we say that a prime $q$ is a $\textit{good prime}$ if $\ell$ does not divide $q-1$ and $q$ is not a $\ell$th ...

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843 views

### An interesting sum over lattice points in a large disk centered at the origin

A friend posed the following question to me:
Evaluate the the limit, as $r \rightarrow \infty $, of the sum $\displaystyle \sum \limits_{(m,n) \in D_r}$ $\displaystyle (-1)^{m+n} \over \displaystyle m^...

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vote

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104 views

### Averaging the Mobius function on arithmetic progressions

The Mobius function $\mu\colon \mathbb{N}\to\{-1,0,1\}$ is given by $\mu(n)=(-1)^{k}$ if $n$ is the product of $k$ distinct prime numbers, and $\mu(n)=0$ otherwise. It is classical that for all $a,b\...

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votes

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179 views

### Perron's formula

Is it possible to show (the trivial statement)
$\sum _{n\leq x}1=x+\mathcal O\left (1\right )$
using Perron's formula?
For $c$ a little bigger than $1$ and $1>c'>0$, a quantitative form of ...

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votes

**2**answers

319 views

### Questions concerning the Fourier analysis of $ nx\ \%\ m$

Let $x\ \%\ m$ be the residue of $x$ modulo $m$, i.e.
$$x \equiv x\ \%\ m\pmod{m}$$
The plots of the functions $f_{nm}(x) = nx\ \%\ m$ exhibit characteristic patterns, especially periods of length $...

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vote

**0**answers

114 views

### Probability of small solutions to an uniform random linear diophantine equation?

Take the set $$T(c_1,\dots,c_t)=\{(x_1,\dots,x_t)\in\mathbb Z^t\backslash\{(0,\dots,0)\}:\sum_{i=1}^tc_ix_i\equiv0\bmod q\}$$ where $c_1,\dots,c_t\in(-q/2,q/2)\cap\mathbb Z$.
What is probability ...

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votes

**1**answer

273 views

### How many roots of polynomial in $\mathbb Z[x]$ and $\mathbb Q[x]$ are integers on average?

Given $d,B>0$ the number of polynomials in $\mathbb Z[x]$ of degree $d$ and coefficient size at most $B$ have at least one integer roots should be $B^{O(d)}f(d)$ at some function $f$ (from Random ...

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votes

**0**answers

63 views

### Independence of integer sequences arising from Dirichlet's pigeonhole

Dirichlet's pigeonhole says given sequence $a_1,\dots,a_n\in\mathbb Z$ and a prime $p$ there is an integer $t$ coprime to $p$ with $\|r_i\|\in[-p^{(n-1)/n},p^{(n-1)/n}]$ at every $i\in\{1,\dots,n\}$ ...

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41 views

### Two sequence discrepancy and smallest boxes?

Take $p$ to be a prime and let $a_1,\dots,a_n\in\mathbb Z$ be some set of integers such that discrepancy of the set of fractional parts $$\{\frac{ma_1}p,\dots,\frac{ma_n}p\}$$ with $m\in\{1,\dots,p-1\}...

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votes

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243 views

### upper bound of consecutive integers which are not coprime with $n!$

Is there any research on getting upper bound of the maximal possible number of consecutive positive integers which are less than $n!$ and NOT coprime with $n!$?
Easy to see that lower bound $\ge n$, ...

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votes

**1**answer

232 views

### Functional equation for general number fields

When it comes to general number fields beyond $\mathbb{Q}$, the litterature is not so abundant in analytic number theory. For instance over $\mathbb{Q}$, for primitve Dirichlet characters modulo $q$, ...

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votes

**1**answer

186 views

### A truncated divisor sum

I am interested in an upper bound for
$$\sum_{\substack{d\mid N\\ d>A}}\frac{1}{d^3},$$
in particular, I can show that above is
$$\ll\frac{\text{exp}\left(C\frac{\log(N)}{\log\log(N)}\right)}{A^...

**0**

votes

**0**answers

88 views

### Number of $b$-separated Sidon sets with pairwise difference set intersection bounds

Given two integers integers $0<b<p$ and a real $\alpha\in(0,1)$ call a set of $m$ integers $a_1<\dots<a_m$ in the interval $(p^\alpha,p-p^\alpha)$ to be $b$-separated Sidon if:
$a_i-a_j\...

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votes

**1**answer

229 views

### Conjecture about an Exponential Sum

Let $X \subset \mathbb{N}$ and say that $X$ is super-equidistributed if
for all $\alpha \in \mathbb{R} \setminus \mathbb{Z}$ there exists $C(\alpha) > 0$ such that for all $N$
$$
\left| \sum_{x \in ...

**0**

votes

**0**answers

115 views

### Degree bounds and coefficient size in elimination theory?

Suppose we have polynomials is of form
$$h_1(x_1,\dots,x_{dn})-c_1\in\mathbb Z[x_1,\dots,x_{dn}]$$
$$\vdots$$
$$h_{nd}(x_1,\dots,x_{dn})-c_{nd}\in\mathbb Z[x_1,\dots,x_{dn}]$$
where $h_1(x_1,\dots,x_{...

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votes

**0**answers

81 views

### L-functions of tempered automorphic representations

Let $G$ be a reductive group over the adele ring $\mathbb A_F$ of a number field $F$. Let also $r$ be a complex finite dimensional representation of the $L$-group $r:{^LG}\to GL(V)$.
It is generally ...

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votes

**2**answers

363 views

### How often does the Mertens function vanish?

It is well known that the Mertens function
$$M(x)=\sum _{n\leq x}\mu(n)$$
has infinitely many zeros, and this seems to be a short proof.
Are there known results about how often the Mertens function ...

**3**

votes

**0**answers

128 views

### Maximum number of integral roots in degree $d$ polynomial?

Given $f(x_1,\dots,x_n)\in\mathbb Z[x_1,\dots,x_n]$ such that
Each coefficient is bound in absolute value by $B$
Degree of each variable in any monomial is bound by $d$
Total degree is $d'$
$f(x_1,\...

**8**

votes

**1**answer

237 views

### Question about Friedlander, Iwaniec: “The polynomial $X^2+Y^4$ captures its primes”

I have a question about the argumentation at the beginning of section 15 in this paper. The goal is to estimate the sum
$$V(\beta) = 2 \sum_{(z_1,z_2)=1} \beta_{z_1} \overline{\beta_{z_2}} \sum_{\...