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.
3,068 questions
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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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Some (philosophical) investigations on possible future of proof of Riemann Hypothesis with possible approach : [closed]
We have seen from history of attempted proofs of Riemann Hypothesis(RH) no matter where / how we start, we always end up at theoretical dead end/ barrier.
As there is duality between Zeta zeros and ...
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Lemma in Roth's Theorem for Primes
I am reading Ben Green's paper Roth's Theorem in the Primes and I don't follow the proof of Lemma 6.1. I am not sure where the fact there are no more than $n^{3/4}$ elements $x\in A_0$ with $x\leq n^{...
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Difficulty with "A new elementary proof of the Prime Number Theorem" by Richter
I'm studying Richter's "A new elementary proof of the Prime Number Theorem" paper, and I'm finding some problems understanding some parts of it. For example, I don't see how to get, in Lemma ...
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Uncertainty principle: minimize $\int_{-\infty}^\infty |t| |\widehat{f}(t)|^2 dt$ for $f$ of compact support
This is a question of uncertainty-principle type stemming from Eigenvalue of a convolution and a restriction?
Let $f:\mathbb{R}\to \mathbb{R}$ be even, absolutely continuous and supported in $[-\frac{...
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How to prove the identity $L(2,(\frac{\cdot}3))=\frac2{15}\sum\limits_{k=1}^\infty\frac{48^k}{k(2k-1)\binom{4k}{2k}\binom{2k}k}$?
For the Dirichlet character $\chi(a)=(\frac a3)$ (which is the Legendre symbol), we have
$$L(2,\chi)=\sum_{n=1}^\infty\frac{(\frac n3)}{n^2}=0.781302412896486296867187429624\ldots.$$
Note that this ...
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Whether or not the root number of GL$_3\times$GL$_2$ $L$-function $L(s, F \otimes g)$ contains the coefficients $\lambda_g(n)$ of $g$?
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}$Let $p$ and $q$ be two distinct primes. Let $$\Gamma_0(p)= \left\{ g\in \GL_3(\mathbb{Z}):g \equiv \left(\begin{matrix} \ast &\ast&\...
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Is the upper bound on $H_{1}$ a decreasing function of the proportion of critical zeros of Zeta?
This question stems from https://arxiv.org/abs/2411.19762 and the numerical observation that the best unconditional upper bound for $H_{1}:=\lim\inf_{n\to\infty}p_{n+1}-p_{n}$, namely $H_{1}^{\flat}=...
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First occurrence of formula for $\sum_{n\leq x} \mu(n) \log n$ in terms of $\psi(y)-\lfloor y\rfloor$?
The identity contained in the last two displayed equations in the following passage (from page 110 in Ayoub's An Introduction to the Analytic Theory of Numbers, 1963) gives us right away a simple ...
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Identities to go from $\sum_{n\leq x} \mu(n) \log \frac{x}{n}$ to $M(x) = \sum_{n\leq x} \mu(n)$?
Let $\mu$ be the Möbius function. Say we have a bound on $\check{M}(x) = \sum_{n\leq x} \mu(n) \log \frac{x}{n}$ of the form $|\check{M}(x)|\leq \epsilon x$ for all $x\geq x_0$.
It is then easy to ...
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Multiplicative functions and Dirichlet characters
I am studying Dirichlet characters and modular functions as part of my research, specifically working through concepts found in the paper "An explicit hybrid estimate for $L(1/2 + it)$" by ...
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Deriving inequality (8.9) from (8.8), in Iwaniec–Kowalski “Analytic Number Theory”
I am working through the problem presented in Chapter 8 of Iwaniec and Kowalski’s Analytic Number Theory (specifically inequalities (8.8) and (8.9)) and I am struggling with the transition between ...
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Existence of Finite Amicable Groups
I'm interested in exploring the concept of "amicable groups" as follows:
Definition. Two finite groups $G$ and $H$ are called amicable groups if:
$G$ is the direct sum of proper subgroups ...
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Sieve Method works for variant question?
There are multiple results on the sieve method, and I wanted to ask about the following variant
(to know if it is trivial by one of the current versions of the sieve method, or seems a challenging ...
CommunityBot
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Fourier transform of fat Cantor set
Let $C_n$ be the set obtained in the $n$-th iteration of the construction of the Smith-Volterra Cantor set, obtained by removing at the $n$-th step $2^{n-1}$ middle intervals of amplitude $1/4^n$. ...
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Clarification and Proof of Inequality (8.11) in Analytic Number Theory by Iwaniec and Kowalski
I am studying inequality (8.11) from Analytic Number Theory by Iwaniec and Kowalski. It is found on top of page 200. In bottom of page 199, the authors prove that
$$
|S_f(N)|^2 \leq N + \frac{2N^2}{q} ...
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Voronoi formula on $\mathrm{GL}_4$ in the level aspect with ramification
$\DeclareMathOperator\GL{GL}$Let $f$ be an automorphic form on $\GL_3$ for $\Gamma_0(p)$ with $p$ being a prime (see Bump or Goldfeld's books for definitions). Recall that, in this paper-"The ...
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Possible refinements of the large sieve inequality
Let $a_n$, $1\leq n\leq N$, be complex numbers, and set $S(\alpha)=\sum\limits_{n=1}^{N}a_ne(n\alpha)$, where $e(\alpha)=\exp(2i\pi\alpha)$. Then, Selberg's large sieve inequality says that
$$\sum\...
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Three optimization problems of uncertainty principle/Paley-Wiener type
Let $\phi:\mathbb{R}\to\mathbb{R}$ be an even function with support on $[-1,1]$. Assume that it is in $L^1\cap L^2$ and that its Fourier transform is also in $L^1\cap L^2$. Assume as well that $|\phi|...
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Can we balance factors using the set of arithmetic sequence so as to achieve a product quality on both sides?
Stated simply the question is: given the set of an arithmetic sequence of cardinality $2N$, where $N$ is greater than or equal to $2$, is it possible to choose $N$ integers in such a way that their ...
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Bounding a number-theoretic integral
Find a good upper bound on $$\int_1^T\frac{\zeta'(s)}{\zeta(s)\zeta(1-s)}X^sdt,$$ where $s=c+it$ for a constant $c>1$ and $X>0$ is a parameter. If needed, we can assume RH.
My attempt here is ...
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Approximation on Dirichlet's arithmetic progression by means of central limit theorem
In this video lecture on
Number theory over function fields taught by Will Sawin
is presented a 'conceptional' reason for error estimation
$\#\{p \in \Bbb P: p =a \ \text{mod} \ N, p <x \}
=\frac{1}...
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Prove or disprove that $|(1/\zeta)^{(n)}(x)| \leq \frac{n!}{(x-\frac{1}{2})}$ for all real $x>1$
$|(1/\zeta)^{(n)}(x)| \leq \frac{n!}{(x-\frac{1}{2})}$ for all real $x>1$.
I had this conjecture for a long time. I tried various methods and techniques but they all failed. It might also be wrong ...
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Can one show that $(-1)^{n-1} {(1/\zeta)}^{(n)}(x) >0$ for all real $x>1$?
Is it true that $(-1)^{n-1} {(1/\zeta)}^{(n)}(x) >0$ for all real $x>1$ ?
Or in other words can you show that the higher order derivatives of the reciprocal of the Riemann zeta function ...
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Largest prime factors of integer polynomials
I have a question in analytic number theory which is closely related to the open problem (Bunyakovsky conjecture and more generally, Schinzel's hypothesis H) that asks you if, any irreducible ...
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Primes which are safe and Sophie Germain
If $p$ is a Sophie Germain prime then $2p+1$ is safe prime.
If $2p+1$ is safe prime then $p$ is Sophie Germain prime.
What is their conjectured distribution of primes $p$ which are both Sophie ...
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Bounds of zeta function near $\Re(s)=1$
Richert proved in
https://link.springer.com/article/10.1007/BF01399533
that $$ \zeta(s) =O\left( |\Im(s)|^{100(1-\Re(s))^{3/2}} (\log |\Im(s)|)^{2/3}\right)$$ uniformly in the region $\Re(s)\in [1/2,1]...
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On integral relating logarithm of absolute value of Zeta function
Sorry for such a direct question:
Consider the following integral:
$$I(t)=\int_{1/2}^{1} {\log|\zeta(a+it)|}da.$$
How to find the nature of $I(t)$ as $t\rightarrow\infty$?
CommunityBot
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Fastest decay of Fourier transform of function of (one-sided or two-sided) exponential (or faster) decay
Let $f:\mathbb{R}\to \mathbb{R}$ be a function in $L^2$ satisfying $|f(x)|\ll e^{-a_1 x}$, $a_1>0$, for $x\to \infty$. (Variant: assume as well that $|f(x)|\ll e^{a_2 x}$, $a_2>0$, for $x\to -\...
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What smoothing to use for PNT-like results?
Consider a Dirichlet series $\sum_n a_n n^{-s}$ with desirable analytic properties (e.g., analytic extension to $\Re s>0$); one example would be $a_n=\mu(n)$. Say we want to estimate $\sum_{n\leq x}...
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(Explicit) Tauberian theorems: removing $(\log x/n)$
Say that $\{a_n\}_{n\geq 1}$, $|a_n|\leq 1$, are such that $$\left|\sum_{n\leq x} a_n \log \frac{x}{n}\right|\leq \epsilon x\quad\text{for all $x\geq x_0$.}$$ What sort of bound can we deduce on $S(x)=...
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Asymptotic behavior of weighted sums involving the fractional part function
Currently, I am studying the asymptotic behavior of sums of the form
\begin{equation}\label{eq1}\tag{1}
\sum_{k=1}^{n-1} f(n-k) \left\{ \frac{n}{k} \right\}
\end{equation}
In this context, based on ...
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Conjectured error term when counting square-free integers
It is well know that the density of positive square-free integers up to $x$ is asymptotically $x/\zeta(2)$. The error term
$$ E(x)=\sum_{1\leq n \leq x } \mu(n)^2 -\frac{x}{\zeta(2)}
$$ can easily ...
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Prime number theorem via large sieve type sums
We know that the prime number theorem is equivalent to the statement
$$
M(x)=\sum_{n\le x}\mu(n)=o(x).
$$
By using Ramanujan sums, we can write $M(x)$ as
$$
M(x)=\sum_{q\le x}\sum_{\substack{0\lt a\le ...
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Are all integers up to $x$ but possibly $O_{\varepsilon}(x^{\varepsilon})$ the sum of $a$ squares and $b$ primes with $a+b\leq 3$?
This question is related to https://math.stackexchange.com/questions/3710032/conjecture-all-but-21-non-square-integers-are-the-sum-of-a-square-and-a-prime.
We know since Lagrange that every natural ...
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Comparing sizes of sets of natural numbers
It seems natural to consider $\lim_{q \rightarrow 1^-} \sum_{n \in S} q^n - \sum_{n \in T} q^n$, when it exists, as a way of comparing the sizes of two sets $S,T \subseteq {\bf N}$ that have the same ...
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Residue of Dirichlet series at $s = 1$
Let $(a_n)_{n \ge 1}$ be a sequence of complex numbers, and suppose that the sequence has a well-defined "average", in the sense that
$$ \lim_{N \to \infty} \frac{1}{N}\sum_{i = 1}^N a_i = R$...
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Mellin transform at $0$
Let $f$ be a smooth complex valued function with support on $[0,2]$ and with Mellin transform $\tilde{f}$. Recall that the Mellin transform is defined as $$\tilde{f}(w)=\int_0^{\infty}f(x)x^{w-1}dx.$$ ...
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Beauty of some numbers discovered by Ramanujan
I am a graduate PhD student and my topic is analytic number theory. I am also a mathematics teacher. I am planning to give a course to pupils in high school that motivates them to study arithmetic and ...
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Can one show $h(x)=|2(\zeta'(x))^2-\zeta''(x)\zeta(x)|$ is a decreasing function for $x\in\mathbb{R}\cap [1,\infty)$?
This question is related to This question.
When I tried to approach it I couldn't even proof that the LHS is a decreasing function on the given domain using regular methods. I have tried to write the ...
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Can one show $\left|\frac{2(\zeta'(x))^2-\zeta''(x)\zeta(x)}{\zeta^3(x)}\right|\leq \frac{2}{(x-\frac{1}{2})^2}$ for $x\in\mathbb{R}\cap [1,\infty)$?
I have found that $\left|\frac{2(\zeta'(x))^2-\zeta''(x)\zeta(x)}{\zeta^3(x)}\right|\leq \frac{2}{(x-\frac{1}{2})^2}$ for all real $x$ such that $x>1$ seems to be true. I have plotted the ...
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Examples of function satisfying some bound
Let $f$ be an arithmetic function such that $f(n)\ll n^{\alpha}$ for some Real number $\alpha$ in $[0,1)$.
Can someone give me examples of such functions other than the sum of divisors function or is ...
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Consequences of the Riemann hypothesis
I assume a number of results have been proven conditionally on the Riemann hypothesis, of course in number theory and maybe in other fields. What are the most relevant you know?
It would also be nice ...
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What is the definition of Tr in the context of Hilbert modular forms?
I am currently reading Garrett's book "Holomorphic Hilbert Modular Forms". But I meet trouble at the starting line. Let $F = \mathbb{Q}(\sqrt D)$ be a real quadratic field, $u= a + b\sqrt{D}\...
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Can one show that $|\zeta'(x) / \zeta^2(x)| \leq 1/(x-.5)$ for $x\in\mathbb{R}\cap [1,\infty)$?
I have found that $\left|\frac{\zeta'(x)}{\zeta^2(x)}\right|\leq \frac{1}{x-\frac{1}{2}}$ for all real $x$ such that $x>1$ seems to be true. I have plotted the inequality and got this inequality ...
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Infinite dimensional lattice for integers and the Riemann hypothesis?
It is known that for each finite set of primes $p$ we have: $\log(p)$ are linear independent over the rational numbers.
We have $\log(ab) = \log(a)+\log(b)$ and $\log(n) = \sum_{p |n}v_p(n) \log(p)$.
...
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Minimizing vertical integral of a Mellin transform
Let $\eta:[0,\infty)\to [0,\infty)$ satisfy $\eta(0)=1$ and $\int_0^\infty \eta(x) dx = 1$ (say).
Write $M\eta$ for the Mellin transform of $\eta$. Let $\epsilon>0$ be small.
What is the choice of $...
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Integral formula of quantum dilogarithm
In the paper"Level N Teichmüller TQFT and Complex Chern-Simons Theory" arXiv:1612.06986, the authors study the quantum dilogarithm function:
\begin{equation}
\mathrm{D}_{\rm b}(x,n)=\prod_{...
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Gap results for squares times cubes
In studying the distribution of squarefree numbers in short intervals, Filaseta and Trifonov used some ingenious techniques to obtain various upper bounds for the size of the set
$$
S(X) = \left\{u\in(...
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Primes such that a given number has very small order
The following came up in (a previous version of) this answer.
Question. Let $a > 1$ be a positive integer, and $f \in \mathbf Z[x]$ a polynomial with positive leading term. Does there always exist ...
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