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.

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3 votes
3 answers
444 views

Growth of the coefficients of the inversion of the $j$-invariant function

We have the $j$-invariant defined as I have that $$ j(\tau)=\frac{1}{q}+\sum_{k\geq 0}c_kq^k, $$ where $q=e^{-2\pi t}$ ($\tau=it$) and $c_k\sim e^{4\pi\sqrt{k}}/(k^{3/4}\sqrt{2})$. The inversion ...
5 votes
0 answers
643 views

The Basel problem revisited?

In the Basel problem, the $sinc$ function is considered at the Wikipedia page. Let me try to make an alternative function definition: $$f(x) = \prod_{n=1}^\infty \left ( 1+ \frac{x^3}{n^3} \right ) = \...
14 votes
0 answers
791 views

Growth of residues of $1/\zeta(s)$: conjectures?

Let $\rho$ range over the non-trivial zeroes of the Riemann zeta function. Let $$M(T) = \max_{|\Im \rho|\leq T} \left|\mathrm{Res}_{s=\rho} \frac{1}{\zeta(s)}\right| = \max_{|\Im \rho|\leq T} \frac{1}...
6 votes
1 answer
837 views

What is the best known upper bound for $\frac{1}{\zeta'(\rho)}$ assuming the SZC but not the RH?

Let $\zeta$ denote the Riemann zeta function and let $\rho$ denote one of its complex zeros. What is the best known upper bound for $\frac{1}{\zeta'(\rho)}$ assuming that all zeros are simple (SZC), ...
4 votes
1 answer
310 views

Double sum over zeros of Riemann zeta-function

In a paper by Saffari and Vaughan there appears a complicated-looking double sum $$\Sigma_1=\sum_{\rho_1}\sum_{\rho_2}\frac{(1+\theta)^{\rho_1}-1}{\rho_1}\cdot \frac{(1+\theta)^{\bar{\rho_2}}-1}{\bar{\...
2 votes
0 answers
84 views

Estermann's argument for the binary additive divisor problem

In the paper https://eudml.org/doc/149759 an estimate for the binary additive divisor problem is given with a power saving. I don't get the main bit of the argument - I'm obviously missing something. ...
8 votes
1 answer
340 views

Is the set of powerful numbers piecewise syndetic?

Recall that a subset $A \subset \mathbb Z_+$ of positive integers syndetic if there exists a $d>0$ such that every positive integer has distance at most $d$ to an element of $A$. It is called ...
8 votes
1 answer
304 views

Explicit estimates for $N(T,\chi)$ (not $N(T,\chi)+N(T,\overline{\chi})$)

Let $N(T,\chi)$ denote the number of zeros of $L(s,\chi)$ with imaginary part between $0$ and $T$, with any zero with imaginary part equal to $T$ or to $0$ (not that the latter kind really exists) ...
3 votes
0 answers
175 views

The binary additive divisor problem in arithmetic progressions

I find quite a few results about the binary additive divisor problem, that is evaluating \[ \sum _{n\leq x}d(n)d(n+h)\] for certain ranges of $h$. Are there any known results about the same count ...
14 votes
2 answers
543 views

Equidistribution of CM points in the principal genus

It is well known that as the negative discriminant $-D$ goes to infinity, the number of quadratic forms of discriminant $-D$ belonging to the principal genus also goes to infinity. Can we say ...
12 votes
2 answers
2k views

What are the implications of a zero of zeta off the critical line

So what happens if there is a non-trivial zero of the Riemann zeta function off the critical line? Has there been any work in the following direction: We know from Landaus theorem that there is a ...
3 votes
1 answer
157 views

Literature on analogous arithmetic function of logarithm function

In number theoretical estimations, often we take the logarithms of a natural number to express it properly. A perfect example of this is the von-Mangoldt function. I am looking for an analogous ...
2 votes
1 answer
332 views

Connecting two pictures of the Zeta function

Lets consider two views of zeta functions of curves. For the following, let $\mathbb{F}_p$ be the field with $p$ elements where $p$ is prime, and let $\overline{\mathbb{F}_p}$ be the algebraic closure ...
34 votes
1 answer
1k views

Quaternionic and octonionic analogues of the Basel problem

I asked this question in MSE around 3 months ago but I have received no answer yet, so following the suggestion in the comments I decided to post it here. It is a well-known fact that $$\sum_{0\neq n\...
6 votes
0 answers
392 views

Conditions under which an $\eta$-quotient becomes a **weak** modular form (reference request for theorems similar to Ligozat's theorem)

For any $z \in \mathcal{H}$, let $q = e^{2\pi iz}$; and the eta function is defined as ${\displaystyle \eta (q) =q^{\frac {1}{24}}\prod _{n=1}^{\infty }\left(1-q^{n}\right).}$ By an $\eta$-quotient ...
6 votes
0 answers
209 views

Conditional results on average size of Mertens' function

Let $M(x) = \sum_{n \le x} \mu(n)$ where $\mu$ is the Möbius function. Titchmarsh, in his book on the Riemann zeta function, considers consequences of the hypothesis that $$\int_{1}^{X} \left( \frac{M(...
6 votes
0 answers
263 views

Best explicit bound on $\zeta'(1+it)/\zeta(1+it)$

Assume the Riemann hypothesis. We know that $$\left|\frac{\zeta'(1+it)}{\zeta(1+it)}\right| \leq 2 \log \log t + O(1)$$ (see, e.g., Thm. 13.13 in Montgomery-Vaughan). What is the best explicit bound ...
8 votes
0 answers
189 views

Key ideas behind p-adic Baker's theorem

I'm trying to understand Kunrui Yu's series of papers [1 2 3] on lower bounds of linear forms of p-adic logarithms (i.e., p-adic Baker's theorem). I know the proof of the usual Baker's theorem through ...
1 vote
1 answer
274 views

Least number coprime to a given integer

For a positive integer $n$ let $$f(n):=\min\{m\in \mathbb N: m>1, \gcd(m,n)=1\} .$$ Equivalently, $f(n) $ is the smallest prime not dividing $n$. Is there any upper bound literature for this? It is ...
2 votes
1 answer
156 views

On some claims on cyclic modules over Hecke algebra used in Serre's "Quelques applications du théorème de densité de Chebotarev"

I have been reading section 7 of Serre's "Quelques applications du théorème de densité de Chebotarev" (http://www.numdam.org/item/PMIHES_1981__54__123_0/), and in particular have been trying ...
1 vote
0 answers
424 views

Karatsuba photo [closed]

Can anyone confirm if the following link displays a photo of A. A. Karatsuba? https://commons.m.wikimedia.org/wiki/File:A.A.Karatsuba_in_Crimea.jpg
21 votes
1 answer
1k views

Primes that are sums of two squares with constraints on the squares

It is well known that there are infinitely many primes of the form $a^2+b^2$ (namely all primes congruent to $1$ modulo $4$). On the other hand, Euler raised the problem as to whether there are ...
0 votes
1 answer
192 views

Are Li's numbers $\lambda_n$ absolutely convergent for $n>1$?

Li's numbers $\{\lambda_n\}$ are defined as $$\lambda_n=\frac{1}{(n-1)!}\frac{d^n}{ds^n} [s^{n-1}\log\xi(s)]_{s=1} $$ for all positive integers $n$. Also $\lambda_n$ is given as a sum over the non ...
5 votes
1 answer
466 views

Dynamics in the integers - Floor function

Let $\alpha$ be an irrational with $0<\alpha<1$. Consider the function given by \begin{align*} f: &\mathbb{N}\longrightarrow \mathbb{N}\\ &x\longmapsto [ \alpha\cdot x]\end{align*} where ...
8 votes
2 answers
959 views

Prime plus square equals prime

Furstenberg–Sárközy's theorem states that if a set of positive integers has positive upper density, then there exists (infinitely many) pair of elements of the set, whose difference is a perfect ...
23 votes
1 answer
3k views

More mysteries about the zeros of the Riemann zeta function

Update on 12/26/2020: I added the Appendix at the bottom: simplified formula for $|\zeta(s)|^2$, when $\frac{1}{2}<\Re(s)<1$. Update on 1/5/2020: I added the section "more interesting ...
8 votes
2 answers
2k views

Why is the Simple Zeros Conjecture said to be stronger than the Riemann Hypothesis?

Let the "Simple Zeros Conjecture (SZC)" be the statement that all zeros of the Riemann zeta function are simple. I have often heard of the statement that the SZC is stronger than the Riemann ...
5 votes
2 answers
385 views

Estimating $\sum_{n\leq x: n \in A} d(n)^a$ from below for large sets $A\subset \{1,2,\ldots,x\}.$

I apologise for the long-windedness of this question. Let $a$ be a positive real constant and let $d(n)$ denote the number of divisors of $n.$ Define $$ S_a(x)=\sum_{n\leq x} d(n)^a. $$ For $a=1,$ ...
1 vote
1 answer
258 views

Asymptotic lower bound for the number of square free with at least two prime factors

In one of Soundararajan's papers, he claims without proof that it is a standard exercise to show that the number $N(X)$ of positive square-free integers $d \equiv 1 \; \bmod \; 8$ less than $X$, with ...
5 votes
0 answers
331 views

Smallest prime $p$ such that $2\mid\operatorname{ord}_p(q)$, the multiplicative order of $q$ modulo $p$

$\DeclareMathOperator\ord{ord}$Let $q$ be prime. I want to upper bound the smallest odd prime $p$ such that $2\mid\ord_p(q)$ (where $\ord_p(q)$ is the multiplicative order of $q$ modulo $p$). Using ...
42 votes
3 answers
7k views

Iterated logarithms in analytic number theory

As all analytic number theorists know, iterated logarithms ($\log x$, $\log \log x$, $\log \log \log x$, etc.) are prevalent in analytic number theory. One can give countless examples of this ...
3 votes
0 answers
181 views

Stronger estimates for dynamical analogue of a sum of multiplicative order and primes

Let $a$ be a positive integer and $\mathrm{ord}_{p}(a)$ denote the multiplicative order of $a$ modulo $p.$ We know by a result of Murty, Silverman and Rosen that the sum $$\sum_{p~\text{prime}} \frac{\...
3 votes
0 answers
1k views

On new (purely analytic) perspective towards theory of prime numbers

[I'm going to ask this question very carefully as a question similar to this received a critical response on this platform. I myself am very skeptical about this but I want to know, from the experts' ...
0 votes
0 answers
116 views

The strong twin conjecture can be transformed into the unsolvability of a particular Diophantine equation

Let us consider the strong twin conjecture: For all positive integer $n$ there exist a prime $p$ such that $$n+4<p<2^n2^4$$ and $p$ is a prime and $p+2$ is a prime Since the inequalities and the ...
2 votes
1 answer
315 views

Bounds on error term in prime number theorem directly from exponential sum estimates

Most improvements on the zero-free region for $\zeta(s)$ go through bounds on the exponential sums $$\sum_{n\sim N} n^{it}$$ for $N$ in certain ranges depending on $|t|$. Is there any way to directly ...
4 votes
0 answers
228 views

How big are small inverse powers of 2 mod powers of 3?

Let $a \bmod b$ take value as an integer in $[0, b)$. For any $T \ge 1$, for what $R \in [0, 3^n)$ is $$\min \{2^{-t}\bmod 3^n: t =1, \dotsc, T\} := \min A_T > R?$$ When $T$ is fixed as $n$ ...
2 votes
1 answer
444 views

The number of ideals in a number ring below a given norm

I'm doing some number theory, and my deficient knowledge of algebraic number theory shows itself; I have to ask for help. Suppose $K$ is a number field, and we want to count the ideals in $\mathcal ...
2 votes
1 answer
406 views

Riemann-Von Mangoldt formula, revised question

This is my last question, building off of Riemann-Von Mangoldt formula and Does $\frac{1}{\pi}\operatorname{Arg} \zeta(1/2+2\pi i t)$ have mean value $0$?. I apologize for taking a while to ...
1 vote
0 answers
84 views

Varieties of trees with logarithmic degree function

I am interested in varieties of trees with a logarithmic degree function. I am currently looking at Bergeron, Flajolet, and Salvy's work "Varieties of increasing trees." They discuss exactly ...
10 votes
1 answer
340 views

A question on the period integral of Rankin-Selberg $L$-function

$\DeclareMathOperator\GL{GL}$Let $\Pi$ and $\pi$ be irreducible automorphic representations of $\GL_{n+1}(\mathbb{A}_F)$ and $\GL_n(\mathbb{A}_F)$ respectively, where $n \geq 2$, $F$ is a number field ...
20 votes
4 answers
1k views

Bound on $L^2$ norm of $1/\zeta(1+i t)$?

What sort of bounds (explicit of preference) can one give for $$\int_T^{2 T} \frac{dt}{|\zeta(1+i t)|^2} \;\;\;\;\;?$$ Some obvious points: One can give a pointwise bound $\frac{1}{|\zeta(1+ it)|} \...
12 votes
2 answers
1k views

What is known about the prime number theorem for Beurling generalised primes

Background: Beurling's systems of numbers Beurling considered a sequence of reals $1<x_1<x_2<\cdots <$ as "primes" and then the ordered sequence of all products of these "...
9 votes
3 answers
569 views

Deriving an asymptotic for $\pi(x)$ directly from $\log \zeta(s)$?

Denote by $\pi(x)$ the number of primes $p\leq x$. We generally give approximations for $\pi(x)$ by first approximating $\psi(x) = \sum_{n\leq x} \Lambda(n)$. Part of the reason is presumably that, if ...
7 votes
0 answers
322 views

Does $\frac{1}{\pi}\operatorname{Arg} \zeta(1/2+2\pi i t)$ have mean value $0$?

I'm curious about what is known about the distribution of the function $\frac{1}{\pi}\operatorname{Arg} \zeta(1/2+2\pi i t) \in (-1,1]$, on a linear or logarithmic scale, where $\operatorname{Arg}$ ...
3 votes
3 answers
353 views

Oscillations of $\theta(x)-x$, for the Chebyshev $\theta$ function

Is anything known about the relative "periodicity" of the oscillations of $\theta(x)-x$, that is, how frequent, in general terms, are the sign changes? Here, $\theta(x)$ is the Chebyshev $\theta$. ...
5 votes
3 answers
700 views

Good books on the divisor sum function $\sigma(n)$?

I would like gain detailed knowledge about properties of the divisor sum function $\sigma(n)$, special equation that have been studied (e.g. $\sigma(n) = 2n$ perfect numbers, ...) and progress that ...
10 votes
1 answer
706 views

What is known about sums of the form $\sum_{n=2}^{\infty}[\zeta(n)-1]^{p} $?

A fair bit is known about rational zeta series. This includes identities like $$ \sum_{n=2}^{\infty} [\zeta(n) -1] = 1 . $$ Many more identities can be found in articles by e.g. Borwein and Adamchik &...
2 votes
0 answers
263 views

Generalized Siegel Weil formula

I am studying the following Poincare-like series, \begin{equation} F_k(\tau,\bar{\tau})=\sum_{\gamma\in\Gamma_{\infty}\backslash\Gamma}\sqrt{\text{Im}\gamma\tau}(q_{\gamma}\bar{q}_{\gamma})^k, \end{...
1 vote
2 answers
323 views

Estimating the size of set of primes $p$ for which the polynomial is bijective in $\mathbb{F}_p[X]$

Let $f(x)\in \mathbb{Z}[X]$ be a polynomial of degree at least $2$. We denote the set of primes $p$ for which $f(x)$ is injective modulo $p$ as $\mathcal{T}$. Then, can we say something about the ...
16 votes
1 answer
4k views

Order of magnitude of $\sum \frac{1}{\log{p}}$

Question: What is the order of magnitude of the following sum? $$ \sum_{\substack{p<n\\\text{$p$ prime}}} \frac{1}{\log{p}} $$ Additional information: Since $$ \sum_{\substack{p<n\\\text{...

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