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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.

2
votes
1answer
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
1answer
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/...
-1
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0answers
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 $\Re(\rho) \in (1/2, 1/2 + \beta]$,...
3
votes
1answer
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 ...
1
vote
0answers
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 ...
8
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0answers
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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0answers
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
1answer
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
1answer
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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0answers
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 $s=1$ ? ...
0
votes
1answer
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 ...
2
votes
1answer
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 $$\...
7
votes
4answers
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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0answers
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 ...
3
votes
1answer
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 ...
6
votes
2answers
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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0answers
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}...
0
votes
1answer
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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0answers
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 ...
18
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2answers
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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0answers
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 ...
9
votes
0answers
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 ...
4
votes
1answer
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
1answer
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 ...
4
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0answers
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}...
5
votes
1answer
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>...
3
votes
2answers
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 ...
1
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0answers
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 ...
2
votes
1answer
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}}(\...
4
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0answers
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 ...
4
votes
0answers
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+...
7
votes
1answer
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 ...
10
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4answers
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^...
1
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0answers
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\...
2
votes
0answers
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 ...
2
votes
2answers
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 $...
1
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0answers
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 ...
5
votes
1answer
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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0answers
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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0answers
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\}...
4
votes
2answers
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 $\ge n$, ...
6
votes
1answer
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 $q$, ...
2
votes
1answer
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
0answers
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\...
8
votes
1answer
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
0answers
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_{...
3
votes
0answers
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 ...
5
votes
2answers
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
0answers
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
1answer
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_{\...