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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4 votes
1 answer
320 views

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 ...
3 votes
0 answers
45 views

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 ...
1 vote
0 answers
87 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. ...
3 votes
1 answer
323 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 ...
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 ...
0 votes
0 answers
79 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}...
4 votes
1 answer
146 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)...
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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2 votes
1 answer
70 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_{(\...
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2 votes
2 answers
222 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^*=\...
2 votes
1 answer
103 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(...
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0 votes
0 answers
28 views

'$r$-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_{&...
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14 votes
0 answers
272 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 $e$. ...
0 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 ...
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2 votes
0 answers
48 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 $[x,x+\log^2(x)]$ ...
5 votes
1 answer
169 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 \...
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2 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?
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1 vote
0 answers
53 views

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?
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2 votes
1 answer
148 views

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 ...
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14 votes
0 answers
378 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 ...
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|\...
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2 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 ...
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0 votes
0 answers
119 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 $n$ ...
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5 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 ...
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 ...
0 votes
0 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 ...
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1 vote
1 answer
93 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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6 votes
1 answer
151 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 ...
1 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 $$\gamma =1+ \frac{\zeta'(2)}{\zeta(2)} -\frac{\zeta'(0)}{\zeta(0)}+ \frac{\zeta'(-1)}{\zeta(-1)}.$$ ...
1 vote
0 answers
119 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 ...
2 votes
0 answers
61 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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1 vote
1 answer
82 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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4 votes
0 answers
76 views

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 ...
6 votes
2 answers
161 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, ...
4 votes
1 answer
438 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\...
0 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 ...
0 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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2 votes
1 answer
392 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}+...
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2 votes
0 answers
126 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 - ...
3 votes
0 answers
104 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 ...
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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 ...
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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 ...
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0 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 ...
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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}^\...
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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 ...
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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 $x > 0$, ...
10 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 ...
0 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\...
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