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.
2,662
questions
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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 ...
- 41
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votes
0
answers
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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 ...
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vote
0
answers
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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. ...
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3
votes
1
answer
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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 ...
- 2,690
3
votes
0
answers
69
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 ...
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votes
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views
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}...
- 2,690
4
votes
1
answer
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views
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
$$
L_4^* (s,\chi_4)...
- 2,690
3
votes
1
answer
160
views
Positivity of partial Dirichlet series for a quadratic character?
Let $\chi\colon(\mathbb{Z}/N\mathbb{Z})^\times\rightarrow\{\pm1\}$ be a primitive quadratic Dirichlet character of conductor $N$. For any integer $m=1,2,\cdots,\infty$, consider the partial Dirichlet ...
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votes
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answer
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views
A sum related to the first moment of quadratic $L$-functions at $s=1$
Let $(\frac{m}{n})$ be the Jacobi quadratic symbol defined for positive squarefree odd integers $n,m$. Does the following sum go to infinity?
$$
\sum_{1\leq n \leq (\log x)^{100} } \mu^2(2n) \sum_{(\...
- 2,735
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votes
2
answers
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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^*=\...
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votes
1
answer
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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 $g(...
- 2,735
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votes
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'$r$-prime factor number' distribution centering an '$l$-prime factor number'
Let '$k$-prime factor integer' $n$ be an integer of form
$$2^t\prod_{i=1}^{k'} p_i^{e_i}$$
where $1\leq k'\leq k$ and at every $i\in\{1,\dots,k\}$ $p_i$ is a distinct odd prime and $e_i\in\mathbb Z_{&...
- 13.1k
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votes
0
answers
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views
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 $e$.
...
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votes
0
answers
42
views
Absolute error in linearly approximating the sum of sum-of-divisors function
The sum-of-divisors function is defined as $\sigma(k):=\sum_{\ell\mid k}\ell.$ It is well-known that $$\sum_{k\le x}\sigma(k)=\frac{\pi^2}{12}x^2+O(x\log x),$$ and therefore it seems natural to study ...
- 587
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votes
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answers
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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 $[x,x+\log^2(x)]$ ...
- 41
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votes
1
answer
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views
Does there exist a sequence $(x,y) \in \mathbb{Z}^2$ such that $|\alpha x - y| \sqrt{x^2 + y^2}$ approaches a given real number?
Let $\alpha > 0$ be a real irrational algebraic number and $c > 0$.
I am interested in the following question.
Does there exist a sequence $(x_i,y_i) \in \mathbb{Z}^2$ such that
$$
\lim_{i \...
- 3,221
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votes
0
answers
52
views
Twin prime distribution centering twice a semiprime
What is the conjectured distributional behavior of semiprimes $pq$ ($p$ and $q$ are primes) having the property $2pq+1$ and $2pq-1$ are primes?
- 13.1k
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vote
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answers
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Distribution of number of prime factors of $p^k\pm1$
What is the behavior of number of prime factors of integers of form $p^k\pm1$ where $p$ is a fixed odd prime or $2$ and $k$ varies over positive integers?
- 13.1k
2
votes
1
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Exponential sum vs. exponential integral via Poisson summation
When we want to estimate an exponential sum
$$
\sum_{M<m\le M'}e(f(m))
\quad\text{with}\quad
1\le M\le M'\le 2M
\quad\text{and}\quad
e(x):=\exp(2\pi ix)
$$
where $e(x):=\exp(2\pi ix)$
and the phase ...
- 323
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votes
0
answers
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views
Is every prime $q$ of the form $x^2 + py^2$ for some prime $p<q$?
For every odd prime $q \geq 3$, does there exist a prime $p < q$ and integers $x,y$ such that
$$\displaystyle x^2 + py^2 = q?$$
One can easily show that all primes $q \not \equiv -1 \pmod{3}$ can ...
- 22.5k
1
vote
0
answers
96
views
Integral points in smooth cubic curves
Let $X$ be a smooth affine cubic curve in $\mathbb A^2$ defined by $f(T_1,T_2)\in\mathbb Z[T_1,T_2]$ (of course $\deg(f)=3$ by definition), and
$$n(f, B)=\{(x_1,x_2)\in\mathbb Z^2| |x_1|\leq B, |x_2|\...
- 393
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votes
0
answers
167
views
Possible regularisation for sum of function of primes
Consider the following sum of function of primes:
$$-\sum_{p}\ln\left( 1 - \frac{1}{(ep)^{1/2}} \right){\ln(p)}$$
Here $p$ runs through all primes and $e$ is Euler's constant.
We can see that the sum ...
- 139
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votes
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views
Asymptotic bound of some number theoretic function
I asked this in stack exchange but did not get anything so I am posting it here.
I am self-studying asymptotic behavior of some number theoretic function and the following question comes up.
Let $n$ ...
- 169
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votes
0
answers
187
views
A bias for runs in Legendre symbols?
$\newcommand\Legendre[2]{\genfrac(){}{}{#1}{#2}}$An odd prime $p$ defines the sequence $\Legendre1 p,\Legendre2 p,\dotsc,\Legendre{p-1}p$
of values of the Legendre symbol describing the quadratic ...
- 15.6k
9
votes
3
answers
909
views
Proving Mertens' theorem using the prime number theorem
Mertens' Theorem states that
$$\sum_{p \leq x}\frac{1}{p} = \log \log x + M + O(1/\log x).$$
This is weaker than the prime number theorem; in fact according to the Wikipedia page, the prime number ...
- 19k
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votes
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answers
80
views
Exact value of $0<\wp^{-1}(2^{-{2/3}})<2$ with $\wp$ the Weierstrass elliptic function
I am investigating solutions to the differential equation
$$\ddot{y}(t)=6y(t)^2,\dot{y}(0), y(0)=y_0>0.\tag{1}\label{1} $$
Let $\wp(t)$ be the Weierstrass elliptic function with elliptic invariants ...
- 528
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vote
1
answer
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views
Cardinality of $\{ n_i + i^k: i \in \mathbb{N} \} \cap [1,T]$ where $\{n_i \}$ is all natural numbers in some order
Let $n_1, n_2, ...$ be a sequence of natural numbers such that $\{n_i: i \in \mathbb{N}\}$ as a set is all of natural numbers. Let $k$ be a positive integer. Is is possible to obtain a lower bound of ...
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votes
1
answer
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views
Is the set of all solutions $x > 0$ to $ \pi(x) = \operatorname{li}(x)$ unbounded?
Is the set of all solutions $x > 0$ to the equation $\pi(x) = \operatorname{li}(x)$ unbounded? Is $\liminf_{x \to \infty} |\pi(x)-\operatorname{li}(x)|$ equal to $0$?
Here, $\pi(x)$ denotes the ...
- 3,480
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vote
0
answers
76
views
An identity among values of the logarithmic derivative of $\zeta(s)$
From some known special values of the Riemann zeta function and its derivative, one can show that
$$\gamma =1+ \frac{\zeta'(2)}{\zeta(2)} -\frac{\zeta'(0)}{\zeta(0)}+ \frac{\zeta'(-1)}{\zeta(-1)}.$$
...
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vote
0
answers
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views
Error or gap in "Modular Functions and Dirichlet Series", by Apostol
My question concerns Apostol's Chapter 7, Kronecker's Theorem with Applications. It's Theorem 7.11, page 156.
I’m attaching the proof in question. There is a lot going on, but I’ve highlighted the ...
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votes
0
answers
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views
Second moment of $S(T)$ for Dirichlet L-functions
Let $S(T)$ denote the argument of the Riemann zeta function. Selberg established that $$\int_0^T |S(t)|^2 \text{d}t\sim\frac{T}{2\pi^2}\log \log T.$$ Let now $\chi$ be a Dirichlet character modulo $q$,...
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vote
1
answer
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views
Why is $\sum_{m=1}^{n}\frac{(\nu(m)-\log\log n)^2}{n\log\log n}=\int_{-\infty}^{\infty}\omega^2\, \mathrm{d}\sigma_n(\omega)$?
I was reading an article on Probabilistic Number Theory by M.Kac where I am not able to understand why a particular equation mentioned here in page $657$ equation $(7.7)$ is true?
I do understand that ...
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L_infinity norm of signed sums of Fourier characters and discrepancy of Fourier matrices
Consider signed sums $\displaystyle A_f(x) =\sum_{\chi} (-1)^{f(\chi)} \chi(x)$ for some set $S$ of characters of an abelian group $G$, and signing $f$ of the characters. For a fixed set $S$ what is ...
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votes
2
answers
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views
Average value of the prime omega function $\Omega$ on predecessors of prime powers
For a positive integer $n$, the prime omega function value $\Omega(n):=\sum_{p\mid n}{\nu_p(n)}$ counts the number of prime divisors of $n$ with multiplicities. A result of Hardy and Wright, [1, ...
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votes
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answer
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views
Proving $\zeta_K\left(\frac12\right)\neq 0 \implies \zeta_K'\left(\frac12\right)\neq0?$
For an algebraic number field $K$, let $\zeta_K(s)$ be the Dedekind zeta function associated to $K$, and let $\zeta_K'(s)$ be its derivative.
I believe that the following statement is true:
$$\zeta_K\...
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votes
1
answer
194
views
GRH and the Euler product
Let $L(\chi, s)$ be the Dirichlet L-Function of a primitive character $\chi$. I believe, if I’m not mistaken, the convergence of the Euler product of $L(\chi, s)$ in the critical strip is known to be ...
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votes
0
answers
66
views
On number of shifted integer solutions to a linear system
Let $M$ be a random diagonally dominant non-singular $3\times 3$ integer matrix
$$\begin{bmatrix}
m_{11}&m_{12}&m_{13}\\
m_{21}&m_{22}&m_{23}\\
m_{31}&m_{32}&m_{33}
\end{...
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votes
1
answer
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views
The nontrivial zeros of the zeta function and the prime counting function
The truncated explicit formula has the shape
\begin{equation}\label{1}
\psi(x) =x-\frac{\zeta^{\prime}(0)}{\zeta(0)}-\sum_{|\rho|\leq T}\frac{x^{\rho}}{\rho}+\sum_{n=1}^{\infty}\frac{x^{-2n}}{2n}+...
- 21
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votes
0
answers
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views
Expected error term in the distribution of Friedlander-Iwaniec primes
In 1998 John Friedlander and Henryk Iwaniec famously proved the asymptotic formula
$$\displaystyle \mathop{\sum \sum}_{a^2 + b^4 \leq x}\Lambda(a^2 + b^4) = \frac{4x^{\frac{3}{4}}}{\pi} \int_0^1 (1 - ...
- 22.5k
3
votes
0
answers
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views
Bounding number of solutions of a congruence
Let $d$ be a positive integer. Let $f(d,a)$ be the number of values of $x$ in $[1,d]$ such that $$x^{a}\equiv 1\pmod{d}.$$ I wanted to know if for some $0<\epsilon<1$, we can prove the following ...
- 1,000
3
votes
0
answers
162
views
Degree four polynomials with no real roots
Consider a degree four polynomial
$$
f = a_4x^4 + a_3x^3 + a_2x^2 + a_1x+ a_0 \in \mathbb{R}[x]
$$
with real coefficients. The discriminant $\Delta_f$ of $f$ is a homogeneous polynomials of degree six ...
- 107
2
votes
1
answer
210
views
Explicit bounds on number of primes of given size
How many prime numbers of $b$ bits are there?
Beyond the prime number theorem, one can give explicit bounds on the number of primes below some integer $n$, or in a given interval. For instance, Rosser ...
- 456
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votes
1
answer
103
views
Asymptotic for a sum involving GCD and Euler totient function
Let $\varphi$ denote the Euler's totient function. Is there any reference in literature for the value of sum $$\sum_{\substack{r\le x\\ d\mid r}}\gcd(\phi(d),r)$$ where $d$ is some fixed positive ...
- 1,000
5
votes
0
answers
92
views
Asymptotics for a sum involving GCD and multiplicative order
Let $n$ be a positive integer and $\mathrm{ord}_{n}(a)$ be the least positive integer $d$ such that $n\mid a^{d}-1$. I wanted to know if for some choice of $y=y(x)$, one can obtain asymptotics for ...
- 1,000
2
votes
1
answer
158
views
Voronoï summation for cusp forms with characters
In an attempt to solve an unrelated problem, I was led to the task of estimating/bounding from above sums of the form
$$\sum_{m=1}^\infty\lambda(m)e\left(-\frac{am}{q}\right)h(m)$$
where $\sum_{m=1}^\...
- 413
22
votes
0
answers
8k
views
Philosophy behind Zhang's 2022 preprint on the Landau–Siegel zero
Now that a week has passed since Zhang posted his preprint Discrete mean estimates and the Landau–Siegel zero on the arXiv, I'm wondering if someone can give a high-level overview of his strategy. In ...
- 10.3k
2
votes
0
answers
259
views
An approximation for the prime counting function
NOTE: I've edited the question one last time, to be much simpler, in the hopes of getting more responses.
SETUP: Let $p_n$ denote the $n$th prime, let $p_x = p_{\lceil x \rceil}$ for all $x > 0$, ...
- 3,480
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votes
0
answers
376
views
Are prime numbers among sums of prime numbers distributed as $\frac n{2\ln(n)}$?
Let $(s_n)_{n\in\mathbb N}$ be defined as follows:
For $n\in\mathbb N$, $s_n:=2+3+5+\cdots+p_n$ is the sum of the first $n$ prime numbers (e.g.: $s_1=2$, $s_2=5$, $s_3=10$, $s_4=17$, $\ldots$).
Let $\...
7
votes
0
answers
150
views
The spectrum of the Banach algebra of certain arithmetic functions under Dirichlet convolution
I was thinking about using the tools of functional analysis to study some subring of arithmetic functions under Dirichlet convolution. If I let $D_s$ be the ring of arithmetic functions with finite ...
- 141
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votes
0
answers
43
views
Does this approximate linear Diophantine Equation have bounded number of solutions?
Consider the linear diophantine equation
$$\alpha u+\beta v =r+ \delta$$
where $\alpha,\beta,r\in\mathbb Q$ are known and their binary expansion has $O(k)$ bits to exactly represent them and $\delta\...
- 13.1k