All Questions
Tagged with analytic-number-theory nt.number-theory
2,517 questions
3
votes
1
answer
275
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 ...
- 105k
3
votes
0
answers
127
views
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^{...
- 105k
6
votes
1
answer
2k
views
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 ...
- 2,060
2
votes
0
answers
95
views
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{...
- 20.2k
37
votes
3
answers
2k
views
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 ...
- 215
1
vote
0
answers
111
views
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&\...
- 41
-3
votes
0
answers
70
views
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}=...
- 6,974
4
votes
1
answer
255
views
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 ...
- 63.9k
4
votes
0
answers
151
views
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 ...
- 63.9k
3
votes
0
answers
239
views
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 ...
- 63.9k
4
votes
0
answers
513
views
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 ...
- 111
1
vote
1
answer
186
views
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 ...
- 6,367
2
votes
1
answer
226
views
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
- 1
3
votes
0
answers
190
views
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 ...
- 41
1
vote
2
answers
295
views
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\...
- 341
2
votes
0
answers
187
views
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|...
- 20.2k
-3
votes
1
answer
194
views
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 ...
- 1,381
3
votes
2
answers
359
views
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 ...
- 302
3
votes
1
answer
596
views
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 ...
- 4,985
1
vote
2
answers
225
views
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]...
- 465
3
votes
0
answers
192
views
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}...
- 20.2k
6
votes
1
answer
494
views
(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)=...
- 20.2k
4
votes
1
answer
213
views
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 ...
- 611
1
vote
1
answer
229
views
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 ...
- 465
6
votes
1
answer
292
views
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
$$
M(x)=\sum_{q\le x}\sum_{\substack{0\lt a\le ...
- 13k
1
vote
1
answer
171
views
Are all integers up to $x$ but possibly $O_{\varepsilon}(x^{\varepsilon})$ the sum of $a$ squares and $b$ primes with $a+b\leq 3$?
This question is related to https://math.stackexchange.com/questions/3710032/conjecture-all-but-21-non-square-integers-are-the-sum-of-a-square-and-a-prime.
We know since Lagrange that every natural ...
- 1,028
5
votes
2
answers
237
views
Residue of Dirichlet series at $s = 1$
Let $(a_n)_{n \ge 1}$ be a sequence of complex numbers, and suppose that the sequence has a well-defined "average", in the sense that
$$ \lim_{N \to \infty} \frac{1}{N}\sum_{i = 1}^N a_i = R$...
- 105k
3
votes
1
answer
177
views
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 $$\tilde{f}(w)=\int_0^{\infty}f(x)x^{w-1}dx.$$ ...
- 105k
6
votes
1
answer
835
views
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 ...
- 19.6k
157
votes
7
answers
74k
views
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 ...
- 50.6k
0
votes
2
answers
223
views
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}\...
- 185
3
votes
1
answer
855
views
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 $\log(n) = \sum_{p |n}v_p(n) \log(p)$.
...
CommunityBot
- 1
4
votes
0
answers
130
views
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(...
- 1,990
11
votes
1
answer
637
views
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 ...
- 12.9k
0
votes
0
answers
92
views
Arithmetic properties of exceptional moduli
For a fundamental discriminant $d$, let $\chi_d = \left(\frac{d}{\cdot} \right)$ be the quadratic character associated with the field $\mathbb{Q}(\sqrt{d})$. Consider the associated Dirichlet $L$-...
- 26.9k
15
votes
0
answers
417
views
Quasiperiodic continued fractions
Is anything known about continued fractions in which the sequence of integers is quasiperiodic?
Quasiperiodic is meant here in the sense of 1D quasicrystals. For example, draw an irrationally-sloped ...
CommunityBot
- 1
1
vote
1
answer
199
views
Weights in the proof of Chen's theorem in Nathanson's "Additive Number Theory The Classical Bases"
I was reading Professor Nathanson's graduate texts in mathematics 164: "Additive Number Theory The Classical Bases" (http://www.alefenu.com/libri/nathansonbases.pdf), and I was wondering if ...
- 13
2
votes
1
answer
354
views
Problem in Uchida's Theorem
I m studying relative class number of CM fields and i found a very interesting theorem from Kόji Uchida's article: class numbers of imaginary abelian fields (1970) but one argument of the ...
- 103
8
votes
1
answer
366
views
Why do we have fewer distinct Gauss sums over a field of characteristic $2$?
Let $p$ be a prime number and $q$ be a power of $p$. Fix a non-trivial additive character $\psi\colon \mathbb{F}_q\to \mathbb{C}^\times$ and, for each non-trivial multiplicative character $\chi\colon \...
- 79.9k
8
votes
1
answer
867
views
Brauer–Siegel's Theorem and application
$\newcommand\underto[1]{\xrightarrow[#1]{}}$Brauer–Siegel's Theorem says that if we consider an infinite sequence of number fields $K_i/\mathbb{Q}$ such that $$\frac{[K_i:\mathbb{Q}]}{\log{|D_i|}}\...
- 12.9k
8
votes
1
answer
356
views
Average bounds on Rankin-Selberg coefficients for modular forms
Let $f$ be a cuspidal Hecke newform of weight $k$ and level $N$, and denote by $a_f(n)$ its $n$-th Fourier coefficient. The newform $f$ is normalized so that $a_f(1) = 1$. As a consequence of Rankin-...
CommunityBot
- 1
1
vote
0
answers
107
views
Circle Method applications only for Quadratic & Cubic Form
So I have studied multiple applications of Circle Method.
1- Initially I read about the Classical Hardy Littlewood circle method, where they use generating functions, and determine hte coefficent ...
- 133
1
vote
1
answer
373
views
$\{ x/p\} $ on average
This is a vague question: Lemma 2.2 of Friedlander and Lagarias' "On the distribution in short intervals of integers having no large prime factor" says that $$\sum _{p\leq w}\left (\{ x/p\} -...
- 14.6k
1
vote
0
answers
108
views
Asymptotics for sums of two sets of positive integers
Assume that $A$ and $B$ are subsets of $\mathbb N$, with counting functions verifying $A(x)\gg x^\alpha$ and $B(x)\gg x^\beta$, with $\alpha+\beta<1$. Let $C=A+B$ and $C(x)$ its counting function.
...
- 3,319
3
votes
1
answer
189
views
Brun-Titchmarsh for sum over square divisors
Let $f(n)$ be a nonnegative arithmetic function satisfying
$f(p^l) \leq A_1^l$ for all primes $p$, integers $l\geq 1$, and some constant $A_1$;
$f(n) \leq A_2 n^\varepsilon$ for all $\varepsilon > ...
- 10.1k
6
votes
1
answer
431
views
Asymptotic behavior of partial sums of Dirichlet series
Consider the Dirichlet series:
$$\sum_{n \geq 1} \frac{a_n}{n^s} = \frac{\zeta(s+1/3)}{\zeta(s)}$$
where $\zeta(s)$ is the Riemann zeta function.
Question: Assuming the Riemann Hypothesis (RH), how ...
- 14.6k
0
votes
0
answers
268
views
Do plots $(5)$ and $(6)$ go to infinity not at the same rate but at similar rates?
The following has been proven by joriki and GH from MO (see here): assuming that $n>1$, then the von Mangoldt function
$$
\Lambda(n)=\lim\limits_{s \rightarrow 1} \zeta(s)\sum\limits_{d|n} \frac{\...
- 1,183
4
votes
0
answers
266
views
How dense are quotients of smooth numbers?
As usual, call a positive integer $y$-smooth if it has no prime factors greater than $y$. Write $S(x,y)$ for the set of $y$-smooth integers $\leq x$. Write $R(x,y)$ for the set of quotients $\{a/b: a,...
- 20.2k
11
votes
3
answers
765
views
Uniform distribution of sequence mod 1
Is it known whether "for most $r$" the sequence $$r \cdot 2^k \bmod 1, \qquad k \in \mathbb N $$ is uniformly disributed in $[0,1]$?
- 12.3k
1
vote
0
answers
127
views
Some property of the greatest prime factor
Let $n$ be a positive integer $\geq 2$ et denote by $ P^{+}(n)$ the greatest prime factor of $n$ my question is as follows:
If $a$ and $b$ are two numbers, is there any method to express or to bound $...
- 10.4k