Questions tagged [analytic-number-theory]
On the 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.
3,066 questions
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Explicit formula for Artin L-functions
The classical explicit formula for the Riemann Zeta function states that
$$
\psi(x)=x-\sum_{\rho} \frac{x^{\rho}}{\rho}+O(1),
$$
where $\psi(x)=\sum_{n \leq x} \Lambda(n)$ and the sum is over all non-...
- 63
4
votes
1
answer
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The function $f(g):=\sum_{\gamma \in \text{SL}(2,\mathbb Z)}\frac{1}{\|g\gamma \|^4}$ for $g\in \text{SL}(2,\mathbb R)$
For $g\in \text{SL}(2,\mathbb R)$ and the Hilbert-Schmidt norm $\|\cdot\|$ (square root of sum of squares), define
$$f(g):=\sum_{\gamma \in \text{SL}(2,\mathbb Z)}\frac{1}{\|g\gamma \|^4}.$$
It is ...
- 457
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vote
1
answer
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Bound for some trigonometric polynomials
Let $e(x)=e^{2\pi i x}$ and consider the following functions defined for $x\in [0,1]$:
$$
f_1(x)=\frac{e(10x)-e(x)}{e(x)-1}, \quad f_2(x)=\frac{e(110x)-e(11x)}{e(11x)-1},
$$
and
$$
f_3(x)=\frac{e(...
- 178
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answers
63
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Calculate the number theory function $\sum_{n \equiv a(\bmod q)} \frac{\phi(n)-\mu(n)}{n^{s+1}}$
I am trying to simplify the function:
$$\sum_{n \equiv a(\bmod q)} \frac{\phi(n)-\mu(n)}{n^{s+1}}=\frac{1}{\phi\left(q^{\prime}\right)} \sum_\chi \bar{\chi}\left(a^{\prime}\right) \sum_{n^{\prime}=1}^{...
- 101
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votes
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answer
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What are the best known upper bounds for $\frac{1}{L(s, \chi)}$?
Let $\chi$ be a Dirichlet character and $L(s, \chi)$ be the corresponding Dirichlet L-function. What are the best known bounds for $\frac{1}{L(s, \chi)}$ in the half-plane of convergence?
I'm aware of ...
- 103
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what is the relationship betwen $L(s,sym^mf\times sym^mg)$ symmetric L function of $f$ and $g$ and $\lambda_{f}(n^m)$, $\lambda_{g}(n^m)$?
what is the relationship betwen $L(s,sym^mf\times sym^mg)$ symmetric L function of $f$ and $g$ and $\lambda_{f}(n^m)$, $\lambda_{g}(n^m)$ ?
- 11
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votes
2
answers
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Proof of an asymptotic formula by Tricomi
Firs of all I ask my question, then I explain how this question arises in my mind and lastly what I tried to solve it.
QUESTION:
Let $P_{n,N}(k)$ be the number of composition of an integer $k$ in $n$ ...
- 361
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votes
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answer
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Does the sum of reciprocal of integers with average power at least two converge?
$\DeclareMathOperator{\ap}{ap}$ $\DeclareMathOperator{\rad}{rad}$ The average power of an integer $m$ is given by
$$
\ap(m):=\log_{\rad(m)}(m)=\frac{\log(m)}{\log(\rad(m))},
$$ where $\rad(m)=\prod_{p|...
- 1,362
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answers
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Reference for explicit formula used by Ramanujan
In his work on highly composite numbers http://math.univ-lyon1.fr/~nicolas/ramanujanNR.pdf , Ramanujan used a version of an explicit formula (equation (329) on page 133) relating primes and zeros of ...
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Large values of $\zeta(1/2+it)$ from sums of short moments
In a now classical paper, Iwaniec proved the following theorem.
Theorem. Let $T \geq 2$, $T^{1/2} < T_0 \leq T$ and $T \leq t_1 < t_2 < \cdots < t_R \leq 2T$, $t_{r+1} - t_r \geq T_0$. ...
- 1,990
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A generalisation of theorem of Landau on sum of two squares?
Let
$r(B)$ be the number of integers $1 \leq n \leq B$ such that $n = x^2 + y^2$ for some $x, y \in \mathbb{Z}.$
Then it is a known theorem of Landau that
$$
r(B) \sim C \frac{B}{\sqrt{\log B}}
$$
...
- 3,625
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votes
1
answer
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views
Estimating a sum involving the von Mangoldt function
I'd like to know the estimate of the following sum
$$\sum_{n\leq x}\sum_{d|n}\Lambda(d)\frac{\phi(d)}{d} $$
where $\Lambda(d)$ is the von mangoldt function and $\phi(d)$ is the Euler totient function. ...
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Optimal exponents in upper bound for 4-dimensional exponential sum
A classical result of Fouvry and Iwaniec states that if $\alpha_1,\ldots, \alpha_4$ are nonzero, $M_1,\ldots, M_4 \geq 1$, $X > 0$, and $|\varphi_{m_1,m_2}|,|\psi_{m_3,m_4}|\leq 1$ are complex ...
- 1,990
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Why are Deligne-type exponential sum estimates so hard to use?
Let $F$ by a finite field, and $R(x_1,x_2,\ldots,x_n) := r_1(x_1,x_2,\ldots,x_n)/r_2(x_1,x_2,\ldots,x_n)$ a rational function in $n$ variables, frequently in analytic number theory or harmonic ...
- 13k
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What is the density of numbers which have at least two divisors whose sum is a perfect square?
Note: This question was posted in MSE about two years ago but it not receive an answer. Hence posting in MO.
A positive integer is said to have square-sum divisors if it has at least two divisors ...
- 5,470
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views
Artin's conjecture for polynomials and rational functions over finite fields
Artin's conjecture on primitive roots over the integers states that a given integer
$0\ne h\in \mathbb{Z}$ that is neither a square number nor $-1$ is a primitive root modulo infinitely many primes $p$...
- 11
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Equidistribution of $(2^nx)$, $x$ irrational
I've run into a scenario where it would be extremely useful to know if, for $x$ irrational, the sequence $\{(2^nx)\}$ is equidistributed on in $[0,1]$, where $(\cdot)$ denotes the ``fractional part&...
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Which sets of natural numbers are "lambda-analytic"?
Begin with a bit of notation. Let $t = t_0, \ldots, t_d$ be a finite sequence of real numbers. Define
$$\lambda^t(x) = x^{t_0} \log(x)^{t_1} \log(\log(x))^{t_2} \cdots.$$
for all real numbers $x \in ...
- 13.3k
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$\psi(x)-x$ on average
This is a reference question:
Let $\psi(x)$ be the psi-Chebyshev function. Is there any unconditional result in the literature that proves that there exists $0<a<2$ such that
$$
\int_2^x (\psi(y)...
- 3,062
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vote
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answers
290
views
Solutions of a linear diophantine equation
Let $N(h)$ be the number of solutions of the following linear diophantine equation:
\begin{equation}
x_1 + 2x_2 + 3x_3 + \dots + (h-1)x_{h-1} = 6h-6;
\end{equation}
where $h\geq 2$ and solution means ...
- 8,998
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votes
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answers
246
views
Function on $\mathbb{Z}/p^k \mathbb{Z}$ with small Fourier transform?
For $f:\mathbb{Z}/p^k \mathbb{Z}\to \mathbb{C}$, define the Fourier transform $\widehat{f}:\mathbb{Z}/p^k \mathbb{Z}\to \mathbb{C}$ in the usual way, viz., $\widehat{f}(\xi) = \sum_x f(x) e(-\xi x/p^k)...
- 20.2k
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views
Are there infinitely many twin primes p and p+2 with p+1 divisible by 4?
Every pair of twin primes $p$, $p+2$ will have either $p \equiv 1 \pmod{4}$ or $p \equiv 3 \pmod{4}$. Is there any work towards whether one of these is "more common", the way that Dirichlet'...
- 949
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Replacing a sharp cut-off by a smooth one
This question is more of a check/validation of a concept.
Suppose I want to study $$\sum _{n\leq X}a_n$$ (e.g. $a_n=d(n)$, the divisor function). As is well-known it's standard practice to replace ...
- 1,381
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votes
1
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165
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What does it mean to have a number of size $B$?
I have a really stupid question that I don't seem to know the answer to and have been too embarassed to ask. In some number theory papers, I encounter sums of the form $$\sum_{\substack{{x \asymp B}\\...
user495559
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views
When does $\lim_{s\to 1_-} (1-s)\sum_{n=0}^\infty a_ns^n$ exist?
Let $a=\{a_n\}_{n\geq 0}$ be a sequence of positive real numbers with $a_n\leq 1$, for all $n$, and observe that, for
any real number $s\in [0,1)$, one has that
$$
\sum_{n=0}^\infty a_ns^n \leq \...
- 2,263
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votes
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answer
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Spectral decomposition of $\Gamma\backslash X$
Let $X$ be a reasonable manifold of non-positive curvature (could be $\mathbb{H}^n$, symmetric or locally symmetric space, homogeneous Hadamard manifold etc.), and let $\Gamma$ be a reasonable group ...
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Asymptotic equivalence of two infinite products of prime numbers in residue classes
I am trying to figure out if the infinite product $$\omega=\frac{5\sqrt{3}}{12}\prod\limits_{\substack{p\equiv 1\pmod3 \\
p\ge 13}}\left(\frac{p-2}{p-1}\right)\prod\limits_{\substack{p\equiv 2\pmod3 \\...
- 309
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views
Cardinality of a subset of smooth numbers
Recall: An $n$-smooth number is an integer whose prime factors are all less than or equal to $n$.
Let $k$ and $N$ two integers with $0\leq k\leq N$. Let's put $H= \{n\in[0,N]\ s.t.\ n\ is\ k-smooth\} $...
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Can we arrange {1,...,9} in 3×3 grid so the set of products of rows equals the set of products of columns? [closed]
I find a interesting question of Prmo mock and Promys 2020
For which $n\in\mathbb{N}$ is it possible to arrange $\{1,…,n^2\}$ in an $n\times n$ grid so that the set of products of columns equals the ...
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Recent works on the Hardy-Littlewood conjecture on primes represented by quadratic polynomials
I have been working on my master's thesis which is about the equivalence of the Hardy-Littlewood conjecture on primes represented by quadratic polynomials and the Lang-Trotter conjecture for CM ...
- 309
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vote
1
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views
Best possible unconditional partial sum estimate of $\sum_{p\leq x}\frac{\ln(p)}{({p})^{n/2}}$:
Consider the following partial sum:
$$S(x,n)=\sum_{p\leq x}\frac{\ln(p)}{({p})^{n/2}}$$
Here p runs through primes and $n$ is constant
What is the best possible unconditional( using best known version ...
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votes
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answers
79
views
Reference request for literature on the following function--power counting zeta function
I'll start by writing the character of interest, and describing some properties of it, before I get to the meat of the question. Any help is greatly appreciated, even an offhand suggestion/comment/...
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Some conjectures about prime gaps
I checked some relations between primes, here $1<n<10^5$ and $p_n$ is the $n$th prime.
$a) p_n^{1/3} - p_{n-1}^{1/3}<1/2$
$b) p_n^{1/n} - p_{n-1}^{1/n}<1/n $
$c) (\log p_n)^{1/2} - (\...
- 521
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views
What is the winding behavior of the Riemann zeta function around zero along the line $s=1+it$?
Let $\phi: \mathbb R \setminus \{0\} \to S^1 \subset \mathbb C$ be defined by
$$\phi(t)= \zeta(1+it)/|\zeta(1+it)|$$
(the nonvavishing of the denominator being a bit weaker than the prime number ...
- 64k
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votes
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views
Elliptic analogue of primes of the form $x^2 + 1$
I have a project in mind for an undergraduate to investigate next quarter -- a curiosity really, but I'm surprised I can't find it in the literature. I do not want a detailed analysis here... but ...
- 13.3k
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votes
2
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649
views
How can one deduce an approximation for the density function of prime numbers from this Euler's theorem?
The author of Riemann's Zeta Function, H.M.Edwards, says:
According to Euler, $\sum_{p<x}\frac{1}{p}\sim \log(\log(x))$ when $x\longrightarrow\infty$.
$\log(\log(x))=\int_{1}^{\log(x)} \frac{du}{...
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63
views
Set from a diophantine equation with similar statistics to primes
While doing some computational calculations with some diophantine equations, I came across with some sequences from solutions of quartic and quintic equations with slowly decreasing frequency, similar ...
4
votes
1
answer
423
views
Is there a "convolution" of asymptotic growth?
Suppose that I have two asymptotic counts given by
$$
\#\{x \in [0,H] \cap \mathbb Z: f(x) \leq H\} \sim F(H)
$$
and also
$$
\#\{x \in [0,H] \cap \mathbb Z: g(x) \leq H\} \sim G(H).
$$
From these two ...
user494572
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On Carmichael function and aliquot parts of odd perfect numbers
I've asked nine months ago this question on Mathematics Stack Exchange with identifier 4430381 and same title. There is not answer for this question on Mathematics Stack Exchange, I wondered if this ...
1
vote
0
answers
142
views
Partial exponential sums over lattice points of lattice cones
Consider the usual lattice $M:=\mathbb{Z}^2\subseteq\mathbb{R}^2$, and
let $v_1,v_2\in\mathbb{Z}^2\subseteq\mathbb{R}^2$ be two non-zero lattice points which are $\mathbb{Z}$-linearly independent. ...
- 7,609
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votes
1
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474
views
Curious infinite product, convergence, connection to prime numbers
I have been playing with the following function:
$$
f(x)=\frac{\pi x (1-x^2)}{\sin\pi x}\prod_{k=2}^\infty \frac{\sin(\pi x/k)}{\pi x/k}
$$
It is hard to get correct numerical values. I'll start with ...
- 3,098
3
votes
0
answers
97
views
Research of average number of equivalence classes of solutions to generalised Pell's equation
Statement of the problem
Firstly, consider the infamous Pell's equation:
$x^{2}-dy^{2}=1$. Here $x$ and $y$ are integers and $d$ is a nonsquare integer. It is known ([3]) that all solutions of this ...
- 31
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votes
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101
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Prime races in two competing arithmetic progressions - error bound
I read an article by Andrew Granville on the subject, there's actually quite a bit of recent literature on the topic. My problem is as follows. I have two sequences of primes: $(p_{1,n})$ and $(p_{3,n}...
- 3,098
4
votes
1
answer
364
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Zeros of Dirichlet function $L(s,\chi_4)$
I am wondering if there are some know results for the non-trivial roots at ${\rm Re}(s) = \frac{1}{2}$, even maybe a table of the first few roots with $t>0$. This sister function
$$
L_4^* (s,\chi_4)...
- 3,098
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votes
1
answer
285
views
Positivity of partial Dirichlet series for a quadratic character?
Let $\chi\colon(\mathbb{Z}/N\mathbb{Z})^\times\rightarrow\{\pm1\}$ be a primitive quadratic Dirichlet character of conductor $N$. For any integer $m=1,2,\cdots,\infty$, consider the partial Dirichlet ...
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votes
1
answer
96
views
A sum related to the first moment of quadratic $L$-functions at $s=1$
Let $(\frac{m}{n})$ be the Jacobi quadratic symbol defined for positive squarefree odd integers $n,m$. Does the following sum go to infinity?
$$
\sum_{1\leq n \leq (\log x)^{100} } \mu^2(2n) \sum_{(\...
- 3,062
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votes
2
answers
260
views
Inequalities for two functions related to the primorial function
Added: As remarked in the answers below, my question has a negative (and well-known) answer.
We denote by $\mathcal P=\lbrace 2,3,5,7,\ldots\rbrace$ the set of prime-numbers and by
$\mathcal P^*=\...
- 17.9k
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votes
1
answer
140
views
Weak Siegel–Walfisz property
Let $f:\mathbb N \to \mathbb C$ be an arithmetic function. There are various ways to define what the Siegel–Walfisz (S–W) property is for $f(n)$. One simple way is that
there exists some function $g(...
- 3,062
14
votes
0
answers
297
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An 'onion-structure' for roots of a series associated to prime numbers?
The series $$\sum_{n=1}^\infty\frac{z^{p_n-n}}{n!}$$ associated to the
sequence $p_1=2,p_2=3,p_3=5,p_4=7,p_5=11,\ldots$ of prime numbers
defines a holomorphic function in the open disc of radius $e$.
...
- 17.9k
3
votes
0
answers
76
views
Divisor of given order in short intervals
Is the following Open question or Conjecture already known, or eventually settled ?
Open question : For sufficiently large $x$ there is at least a positive integer in the interval $[x,x+\log^2(x)]$ ...
- 433