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
votes
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
votes
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 ...
1
vote
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 ...
1
vote
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
$$\...
7
votes
4answers
1k views
Modern Algebraic Geometry and Analytic Number Theory
I am currently discovering the algebraic geometry of Grothendieck. I have the impression that this theory, which leads to categories, schemas, topos etc. alone can encompass all modern mathematics (...
1
vote
0answers
191 views
On Primes in Arithmetic Progressions
I was wondering if the following approach is being attempted to prove the twin-prime conjecture.
Tao and Green proved in their paper (2006), that there are arbitrarily long arithmetic progressions ...
3
votes
1answer
156 views
Local phase statistics of the nontrivial Riemann zeros
(The question is inspired by Owen Maresh's post)
The local phase of a nontrivial zero $s$ of the Riemann $\zeta$ is the argument of $\zeta'(s)$.
Numerical results on the first 10000 zeros suggest ...
6
votes
2answers
210 views
How to obtain an upper bound for $\prod_{p\mid N} (1 + 1/\sqrt{p})$ where $N$ is square free?
I am interested in obtaining an upper bound for $\prod_{p|N} (1 + 1/\sqrt{p})$ when $N$ is squarefree. It's not too hard to show that
$$
\prod_{p\mid N} (1 + 1/\sqrt{p}) \ll C^{\omega(N)} \ll N^{\...
0
votes
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
$$\sigma(n)\geq e^{\gamma}...
0
votes
1answer
69 views
meromorphic extension of dirichlet series
Suppose $\{a(n)\}_{n\ge 1}$ is a bounded complex sequence. Let $\phi(s)=\sum_{n\ge 1} \frac{a(n)}{n^s}$. Obviously, the Dirichlet series $\phi(s)$ is absolutely convergent for $\mathcal{R}(s)>1$. I ...
1
vote
0answers
134 views
Do we believe that the distribution of spacings of successive critical zeros of zeta is log-symmetric?
Let $\gamma^{+}(T)$ be the imaginary part of the critical zero of $\zeta$ closest to $1/2+iT$ with $\gamma^{+}(T)\ge T$ and define similarly $\gamma^{-}(T)$ with a reversed inequality. Let $g(T)$ be ...
18
votes
2answers
755 views
Motivation behind Analytic Number Theory
I am an undergraduate student of mathematics and recently took an introductory course in analytic number theory, where the instructor roughly followed Apostol's first text on the subject. I have now ...
0
votes
0answers
124 views
Approximation in the circle method
I am interested in the circle method and I am currently working on Vaughan's book. Let $f$ be the generating function $f$ of the squares, that is to say the power series sum of $z^{m^2}$.
One of the ...
9
votes
0answers
170 views
How small may the discriminant of an $S_d$-field be?
In every degree $d$, the Galois closure of the typical number field has the maximal possible Galois group $S_d$. Denote by $f(d)$ the least absolute value of a discriminant of an $S_d$-field of degree ...
4
votes
1answer
160 views
Zeros of derivatives of Dirichlet Eta function
Let
$$
\eta^{(d)}(z) =
\sum_{n=1}^\infty
\dfrac
{(-1)^d(-1)^{n-1}\ln(n)^d}
{n^z}
$$
be the derivative of Dirichlet Eta function of order $d$.
Does it exist any known or not known zero of $\eta^{(d)}...
3
votes
1answer
192 views
On the convergence of $\sum_{n=1}^{\infty} \frac{\lambda(n)}{n}$ and the Prime Number Theorem
Let $\lambda$ be the Lioville function of number theory.
I've heard several times that if $L=\sum_{n=1}^{\infty} \frac{\lambda(n)}{n} =O(1)$ then $L=0$ (the Prime Number Theorem). How can this be ...
4
votes
0answers
77 views
Smoothed Weyl sum inequality
One version of Weyl's inequality states that for any $\alpha\in\mathbb{R}$ and $(a, q) = 1$ such that $|\alpha - a/q|\le 1/q^2$, we have that
$$\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
vote
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:
$$D_k(x) = \sum_{n \leq x} \tau_k(n) = xP_k(\log x) + O(x ^{1 - \frac{1}{k-1}}(\...
4
votes
0answers
119 views
How many exceptional conductors are there?
We say that a conductor $q$ is exceptional if there is a primitive quadratic character $\chi$ modulo $q$ such that $L(s,\chi)$ has a real zero $\beta$ such that $\beta > 1-c/\log q$ (where $c$ is ...
4
votes
0answers
135 views
Chen's theorem in which constituent primes are close together
Chen's theorem states that every sufficiently large even integer can be written as $n=p+q$, where $p$ is a prime and $q$ is a product of at most two primes.
I would like a representation $n=p_1+...
7
votes
1answer
444 views
Smallest Mazur's good prime
Let $p$ and $\ell$ be primes $\geq 5$ such that $\ell$ divides $p-1$. Following Mazur, we say that a prime $q$ is a $\textit{good prime}$ if $\ell$ does not divide $q-1$ and $q$ is not a $\ell$th ...
10
votes
4answers
843 views
An interesting sum over lattice points in a large disk centered at the origin
A friend posed the following question to me:
Evaluate the the limit, as $r \rightarrow \infty $, of the sum $\displaystyle \sum \limits_{(m,n) \in D_r}$ $\displaystyle (-1)^{m+n} \over \displaystyle m^...
1
vote
0answers
104 views
Averaging the Mobius function on arithmetic progressions
The Mobius function $\mu\colon \mathbb{N}\to\{-1,0,1\}$ is given by $\mu(n)=(-1)^{k}$ if $n$ is the product of $k$ distinct prime numbers, and $\mu(n)=0$ otherwise. It is classical that for all $a,b\...
2
votes
0answers
179 views
Perron's formula
Is it possible to show (the trivial statement)
$\sum _{n\leq x}1=x+\mathcal O\left (1\right )$
using Perron's formula?
For $c$ a little bigger than $1$ and $1>c'>0$, a quantitative form of ...
2
votes
2answers
319 views
Questions concerning the Fourier analysis of $ nx\ \%\ m$
Let $x\ \%\ m$ be the residue of $x$ modulo $m$, i.e.
$$x \equiv x\ \%\ m\pmod{m}$$
The plots of the functions $f_{nm}(x) = nx\ \%\ m$ exhibit characteristic patterns, especially periods of length $...
1
vote
0answers
114 views
Probability of small solutions to an uniform random linear diophantine equation?
Take the set $$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\}$$ where $c_1,\dots,c_t\in(-q/2,q/2)\cap\mathbb Z$.
What is probability ...
5
votes
1answer
273 views
How many roots of polynomial in $\mathbb Z[x]$ and $\mathbb Q[x]$ are integers on average?
Given $d,B>0$ the number of polynomials in $\mathbb Z[x]$ of degree $d$ and coefficient size at most $B$ have at least one integer roots should be $B^{O(d)}f(d)$ at some function $f$ (from Random ...
0
votes
0answers
63 views
Independence of integer sequences arising from Dirichlet's pigeonhole
Dirichlet's pigeonhole says given sequence $a_1,\dots,a_n\in\mathbb Z$ and a prime $p$ there is an integer $t$ coprime to $p$ with $\|r_i\|\in[-p^{(n-1)/n},p^{(n-1)/n}]$ at every $i\in\{1,\dots,n\}$ ...
0
votes
0answers
41 views
Two sequence discrepancy and smallest boxes?
Take $p$ to be a prime and let $a_1,\dots,a_n\in\mathbb Z$ be some set of integers such that discrepancy of the set of fractional parts $$\{\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_{\...
