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.
3,067 questions
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Inequalities involving binary representation of integers
Let $N\geq 1$ be a positive integer and assume that $N=2^{n_1}+2^{n_2}+\cdots+2^{n_{p}}$, $n_{1}>n_{2}>\cdots>n_{p}\geq 0$, is the binary representation of $N$. I believe that the following ...
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Extrema of real analytic Eisenstein series and more general modular functions
The real analytic Eisenstein series defined by the Poincare sum
$$E(s,z)=\sum_{(m,n)\neq (0,0)} {y^s\over |mz+n|^{2s}}$$
for $z\in{\mathbb H}$ and ${\rm Re}(s)>1$ is a manifestly $SL(2,{\mathbb Z})$...
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2
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Explicit bound on $\zeta(s)$ inside a zero-free region?
Does anybody know of a place in the literature where one can find an explicit result of the form $|\zeta(\sigma+it)|\leq C \log t$ for $t$ within a zero-free region (assuming $t$ is larger than an ...
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'Almost all' zeros of the Dirichlet L function lies 'near' the critical line?
Is there a well known result that states that as $t \to \infty$, 'almost all' zeros of any Dirichlet L function $L(s,\chi)$ lie in the region $R= \{\sigma+i t\mid |\sigma -\frac{1}{2}| \leq \Phi(t) \}$...
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Lower bounding the number of Galois radii of an integer
Recall that I call $r>0$ a Galois radius of an integer $n$ if $n-r=p^a$ and $n+r=q^b$ with $p$ and $q$ primes and positive $a$ and $b$ and a primality radius of $n$ if $a=b=1$.
Does it suffice to ...
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Entropy of a sequence
I am reading the paper Sign Changes in Hecke Eigenvalues by Matomaki and Radziwill, and in one place they mention the following,
It would be interesting to rule out the possibility of $\lambda_f(n)$ ...
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For what estimated value of $x$ Is $|M(x)/\sqrt{x}|$ expected to exceed $1.2$ and $1.8$?
Background:
Mertens conjecture stated that $-1 \lt \frac{M(x)}{\sqrt{x}} \lt 1$ when $x\gt 1$. see more information here
The conjecture was proved to be false in 1985 by showing $\liminf \frac{M(x)}{\...
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Generalization of Gauss's class number one problem
Gauss's class number one problem for imaginary quadratic fields, now a theorem due to Heegner, Stark, and Baker (independently), asserts that the set of imaginary quadratic fields having class number ...
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Weyl sums in the arithmetic progressions
For any $\alpha \in \mathbb{R}$ which has the Diophantine
Approximation that $$\alpha=\frac{l}{q}+\frac{\theta}{q^2},\quad (l,q)=1, \quad|\theta|\le 1.$$ It is known that
$$\sum_{m\le M} \min \left(N,...
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Does asymptotic Goldbach imply GRH?
It seems to me that a proof of $\alpha_{n}=o(n)$ where the quantity $\alpha_{n}$ is defined in About Goldbach's conjecture together with the main result of https://kyushu-u.pure.elsevier.com/en/...
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Density of numbers with a prime factor satisfying a congruence
Let $S$ be the set of integers with at least one prime factor in the arithmetic progression $km+d$, $(m, d)=1$. I am looking for results on the density of $S$. I found this post which talked about the ...
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Do consecutive integers have a big prime factor?
Let us say that three consecutive positive integers $(m-1,m,m+1)$ have a big prime factor if the largest prime factor $p$ of $N=(m-1)m(m+1)$ satisfies $e^p>N$.
I ckecked that it is true for all $m&...
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Upper bound of the analytic rank of the modular Jacobian varieties $J_1(N)$
Does there exist an upper bound of the analytic rank of the modular Jacobian varieties $J_1(N)$?
(Or more generally of $J_\Gamma$ for a congruence subgroup $\Gamma_0 \subseteq \Gamma \subseteq \...
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A similar inequality for the Dedekind psi function, than an inequality stated by Schinzel
I would like to ask about the next question that seems to me interesting. I know an article that was written by Andrzej Schinzel in which he stated Lemma 2. In this post we denote the Dedekind psi ...
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Quadratic non-residue problem
For a positive integer $n$, let $a(n)$ the smallest number $k>0$ such that $-n$ is not a quadratic residue modulo $k$.
Using CRT, we can prove that all values of $a(n)$ are prime powers, and every ...
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Normal numbers and law of the iterated logarithm
If I remember correctly, for the binary digits of a real number in $[0,1]$, I was told that satisfying the law of the iterated logarithm (LIL) is stronger than being normal. That is, supposedly, some ...
- 3,098
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Averages of Möbius function in arithmetic progressions
It is mentioned in multiple occasions here that the bound
$$
\mathop{\sum_{n=1}^{N}}_{n\equiv a\mod l} \mu(n) = o(N)
$$
is equivalent to the prime number theorem in arithmetic progressions. But I am ...
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The Langlands parameters of the symmetric cube lifts of cusp forms
I have a question concerning the Langlands parameters of the symmetric cube lifts. Lets $f$ be a $GL_2$-cusp form, and $\operatorname{sym}^2f$ the symmetric-square lift of it. Assume that $\alpha_1,\...
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Special values of partial Zeta functions
Let $\mathfrak m$ be an integral ideal of a number field $F$, let $C$ be a class in the ray class group $J^{\mathfrak m}/P^{\mathfrak m}$ modulo $\mathfrak m$ and define
$\zeta(C,s)=\sum_{\mathfrak a\...
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What are the current directions/topics of research taking place in Analytic Number Theory
I have studied analytic number theory from 2 volumes of Apostol , Lectures in Sieve Theory online and I am thinking of reading some research papers in Number Theory and then try to work on some ...
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Status of current research in Sieve Theory
I have done a course in Sieve Theory from the notes of Prof. Rudnick. Before this, I did 2 courses in Number Theory from the 2 volumes of Apostol.
I don't have any guidance by professor as I am living ...
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Significance of $N_0(T+1)-N_0(T)\sim \frac{1}{2\pi}\log \frac{T}{2\pi}$
Let $N(T)$ be the number of zeros of Riemann zeta function upto height $T$ in the critical strip and $N_0(T)$ be the number of zeros on the critical line.
What will be the significance of proving ...
user485483
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Ramanujan's conjecture on modular forms and Riemann hypothesis
I just watched Kannan Soundararajan's talk on the distributions of valus of zeta and $L$-functions at virtual ICM 2022. In his talk, he introduced a theorem on Ramanujan's ternary form $\phi_{1}: x^{2}...
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On the explicit upper bound of $|\log\zeta(s)|$ near $\Re(s)=1$
I learn from Montgomery & Vaughan's Multiplicative Number Theory I: Classical theory that there exists $c_1,c_2>0$ such that whenever $\sigma\ge1-c_1/\log|t|,|t|\ge4$ there is
$$
|\log\zeta(\...
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Global irreducible admissible representations analogue
Let ${A}_{\mathbb{Q}}$ denote the adele over the rational numbers. Then it is known that cuspidal modular forms of level $N$ correspond to some unitary automorphic representation of $\operatorname{GL}...
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Selberg's 1943 result on primes in short intervals and primality radius
This preprint: https://arxiv.org/abs/2207.05038 states in the last paragraph of the first page that a result of Selberg (1943) implies that under RH, almost all intervals of the form $(x,x+\left(\log ...
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The first case of the strong Littlewood conjecture
Let $A$ be a set of $n$ integers and consider the quantity:
$$\int_{0}^1 \left| \sum_{a \in A} e^{2\pi i a x} \right|dx. $$
The (now solved) Littlewood conjecture is the claim that this quantity is ...
- 13k
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Resources where one can find open problems in Analytic Number Theory
Edit: I have asked this question separately because I am in need of some resources which focus on analytic number theory only.So, this question is not a duplicate of:Resources where I can find open ...
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Getting prime by changing 2 digits
I just got a result, an exercise of Tenenbaum's book that one cannot get a prime from arbitrary natural number $n$ by changing only one digit of its decimal expansion. For example, you cannot get ...
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Why does this convolution of the prime counting function $\pi$ look like a parabola?
In this previous question it is shown that the convolution of the prime counting function $\pi$ with itself, is related to the Goldbach conjecture:
$$\pi^*(n):=\sum_{k=0}^n \pi(k) \pi(n-k)$$
The ...
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Generalization of the zeta values
Consider the constants $c(s_1,s_2,...)=\sum_{n=1}^\infty \frac{1}{n^{s_n}}$ where the $s_i$'s are fixed positive integers and the number of $i$ with $s_i=1$ is finite (so that the previous sum ...
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Estimates for certain double-Kloosterman sums
Sorry to disturb. I encounter a double-Kloosterman sum, which needs some help from the experts here.
For any $q\in \mathbb{N}^+$, how can we estimate the type of sum
$$ \sideset{_{}^{}}{^{\ast}_{}}\...
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Quadratic patterns in summands of Goldbach's conjecture
Let $n $ be even and define
$$ Q(n)=\sum_{\substack{ p,q \ \textrm{ primes} \\p+q=n }}\left(\frac{p}{q} \right),$$ where $\left(\frac{p}{q} \right)$ is the quadratic Legendre symbol.
Has this sum been ...
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Question about exponent pairs
In some of my recent research efforts, I've been applying a lot of estimates for exponential sums involving exponent pairs. Two seemingly simple questions have arisen from these calculations, and I ...
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If the Dirichlet series $L(z,\chi)$ diverges for $\sigma< 1$, does its alternating version converge for some $\sigma_0 < 1$, and conversely?
Here $\chi$ is a completely multiplicative function with $\chi(p)\in\{-1,+1\}$ for any prime $p$, but not necessarily a character. Also $s=\sigma + it$ as usual. The series corresponding to $L(s,\chi)$...
- 3,098
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Abscissa of convergence for a very specific Dirichlet series / Euler product
I am interested in the convergence of the following Euler product:
$$
\prod_p \frac{1}{1-\chi(p)\cdot p^{-s}}.
$$
The product is over all primes (in increasing order), with $\chi(p)=+1$ if $p \bmod 4 =...
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Why this plot of Siegel Z function is very close to two halve lines?
Let $Z(s)$ be Siegel $Z$ function.
For real $S,T$, define $fu7(S,T)=|Z((S-\frac12)/i + T)|$.
In other words for $s=S + i T$ we have $fu7(s)=|Z((s-\frac12)/i)|$.
For $S_0 \in (0,1)$ and $T_0=\rho_1= 14....
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Infinitely many primes that split completely in an arithmetic progression
Let $d \geq 1$ be an integer. Dirichlet's theorem on arithmetic progression implies that the arithmetic progression $a, a+d, a+2d, \ldots$ contains infinitely many primes if and only if $\gcd(a,d)=1$.
...
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Ratio between relative totient function and totient function
Define $\varphi(n,x)$ as the number of elements in the interval $[1,x]$ that is relatively prime to $n$.
Is it true that there exists some constant $c_1,c_2>0$, such that if $x\geq c_1\log n$, then ...
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Bailey's lemma in number theory
A pair of sequences $(α_n,β_n)$ is called a Bailey pair if they are related by
$$\beta_n=\sum_{r=0}^n\frac{\alpha_r}{(q;q)_{n-r}(aq;q)_{n+r}}$$
or equivalently
$$\alpha_n = (1-aq^{2n})\sum_{j=0}^n\...
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Small covering of divisors
Let $D_n$ be the set of divisors of $n$.
Does there always exists a $B\subseteq D_n$ such that $D_n = \{\gcd(ab,n) \mid a\leq \sqrt{n}, b\in B\}$ and $\sum_{b\in B} \frac{n}{b}=O(n)$?
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From holes in the image of peculiar functions to new perspective on the Riemann Hypothesis
I am working with the Dirichlet eta function $\eta(z)$, with $z=\sigma+it$, $\sigma > \frac{1}{2}$, and $t>0$. Let us define
$$\eta_n(z,\gamma)= \sum_{k=1}^n (-1)^{k+1}\lambda_k^{-\sigma} e^{-it\...
- 3,098
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Conditional stronger bounds on Linnik theorem with prime power modulus
This post is related to questions asked here and here. However, I include the relevant background on least prime in arithmetic progressions presented here for benefit of the reader.
By Linnik's ...
- 1,042
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"Beautiful convergence" in Hejhal, The Selberg Trace Formula
A simple question:
Hejhal in his volumes on the Selberg Trace formula (particularly volume 2) uses the expression "converges beautifully".
I can't find the definition, even searching the ...
5
votes
1
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Divergence of primes dividing polynomials
Let $Q : \mathbb{Z} \rightarrow \mathbb{Z}$ be a polynomial. Form the set
$$M_{Q} := \{p:\text{ }p\text{ is prime, }\exists n_{p}\in \mathbb{Z}\text{ so that }p|Q(n_{p})\}$$
Is $$\sum_{s \in M_{Q}}\...
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1
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Fricke involution and Atkin operator
Let $f\in S_k(\Gamma_0(N))$ be a cusp form for $N>1$. Consider the following operators acting on $f$ via the natural action of $GL_2^{+}(\mathbb{R})$ :
$$ W_N=\begin{pmatrix}
0 & -1\\
N & 0
...
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1
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On the behaviour for the quotient involving Fermat numbers of $\frac{\psi(F_m)}{F_m}$ where $\psi(x)$ denotes the Dedekind psi function
In this post we denote the Dedekind psi function as $\psi(m)$ for integers $m\geq 1$. This is an important arithmetic fuction in several subjects of mathematics. As reference I add the Wikipedia ...
4
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1
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Understanding inequality in "Small gaps between primes"
I am reading Maynard's paper "Small gaps between primes" and I don't understand how to deduce the second line from the first line of below's inequality. How does $\dfrac{1}{(p-1)^2}$ appear?
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0
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A question on the twisted symmetric square L-functions
Sorry to disturb. I have a puzzle which might be naive for many experts here.
Let $f$ be a Hecke newform of prime level $N$ on $\mathrm{GL}_2$, and $
\chi$ a primitive character of square-free ...
- 353
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1
answer
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A variant of effective Chebotarev theorem
Let $L$ be a Galois extension of a number field $K$ with the Galois group $G$. Let $N$ be the smallest integer with the following property: For any conjugacy class $C$ of $G$ there exists an ...
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