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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2 votes
2 answers
247 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^*=\...
1 vote
1 answer
335 views

On equations with arithmetic functions [closed]

Is this good topic for research: equations with arithmetic functions, for example equations like $\varphi(n)=\sigma(n)$ or $\varphi(n)+\sigma(n)=d(n)$ ? If Anyone here have an advise please tell me ...
14 votes
0 answers
290 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 $e$. ...
5 votes
1 answer
193 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 votes
0 answers
67 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 $[x,x+\log^2(x)]$ ...
3 votes
2 answers
791 views

Estimate about primes

Can anyone give an estimate (upper bound or lower bound) for the number of divisors $d\mid P_r$ such that $\frac{\sqrt{P_r}}{2}< d < \sqrt{P_r}$, where $P_r$ is the product of the $r$ smallest ...
2 votes
0 answers
67 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?
1 vote
0 answers
62 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?
14 votes
0 answers
402 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
113 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|\...
0 votes
0 answers
77 views

Relevance of the deduction of similar theorems than Maier's theorem for other prime constellations

A year ago I asked this question on Mathematics Stack Exchange with identifier 4245823 and same title Relevance of the deduction of similar theorems than Maier's theorem for other constellations of ...
0 votes
0 answers
125 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 $n$ ...
6 votes
0 answers
218 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 ...
5 votes
1 answer
194 views

Results using a certain kind of identity

Recently, I've been reading about asymptotics for smooth numbers as well as smooth numbers in arithmetic progressions. One of the ideas I find especially pleasing among some of these results is the ...
18 votes
1 answer
2k views

A question about Speiser's 1934 result on the Riemann hypothesis

A number of sources concerning Speiser's 1934 result state that the Riemann Hypothesis (RH) implies $\zeta'(s)\neq 0$ for all $0<\text{Re}(s)<1/2$. But I have seen some (possibly less reliable) ...
7 votes
2 answers
602 views

Reference request for the explicit formula for $\sum_{n\leq x} \Lambda(n)n^{-s}$

Denote by $\Lambda(n)$ th e von Mangoldt function, which is equal to $\log p$ if $p\geq 2$ is a prime, and $0$ otherwise. Let $\rho$ denote a complex zero of the Riemann $\zeta$-function. If i recall ...
34 votes
1 answer
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 ...
0 votes
0 answers
125 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 ...
1 vote
1 answer
115 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 ...
6 votes
1 answer
165 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
143 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 $$\gamma =1+ \frac{\zeta'(2)}{\zeta(2)} -\frac{\zeta'(0)}{\zeta(0)}+ \frac{\zeta'(-1)}{\zeta(-1)}.$$ ...
2 votes
0 answers
145 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 ...
38 votes
5 answers
7k views

Is $\zeta(3)/\pi^3$ rational?

Apery proved in 1976 that $\zeta(3)$ is irrational, and we know that for all integers $n$, $\zeta(2n)=\alpha \pi^{2n}$ for some $\alpha\in \mathbb{Q}$. Given these facts, it seems natural to ask ...
17 votes
1 answer
3k views

Values of zeta at odd positive integers and Borel's computations

Someone recently quoted to me this recent article that claims to prove that $\zeta(2n+1) \notin (2\pi )^{2n+1} \mathbb{Q}$. [Edit: published reference: Musha, Takaaki. Negation of the conjecture for ...
7 votes
1 answer
254 views

Upper bound of the analytic rank of the modular Jacobian varieties $J_1(N)$

Does there exist an upper bound of the analytic rank of the modular Jacobian varieties $J_1(N)$? (Or more generally of $J_\Gamma$ for a congruence subgroup $\Gamma_0 \subseteq \Gamma \subseteq \...
4 votes
1 answer
503 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\...
2 votes
0 answers
84 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 $q$,...
1 vote
1 answer
103 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 ...
10 votes
1 answer
2k views

Is new $n$-conjecture as follows correct?

Given a positive integer $P>1$, let its prime factorization be written as$$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}.$$ Define the functions $h(P)$ by $h(1)=1$ and $h(P)=\min(a_1, a_2,\ldots,...
1 vote
0 answers
297 views

What is most current greatest lower bound on gaps between P2 almost primes

What is the current best result on the greatest lower bound on gaps between P2 almost primes where P2 represents a prime or the product of two semi-primes?
4 votes
0 answers
98 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 ...
3 votes
1 answer
668 views

Looking for a paper of Lagarias and Odlyzko

I have been studying about the Chebotarev Density Theorem and have been hunting for the following paper of Lagarias and Odlyzko for quite a while: Effective versions of the Chebotarev density theorem, ...
21 votes
2 answers
3k views

What are the consequences of an ineffective proof of the Riemann Hypothesis?

Suppose a proof came out (and was verified by credible peer review) of the following statement: There is a $T_0$ such that for all $t>T_0$, all zeros $\zeta(\beta+it)=0$ have $\beta=1/2.$ where $...
30 votes
3 answers
3k views

Heuristic argument for the prime number theorem?

Here is a bad heuristic argument for the prime number theorem. Let $n$ be a positive integer and assume that PNT holds up to $n$. Then $n$ itself is prime if and only if for each prime $p<n$ the ...
1 vote
1 answer
259 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 ...
68 votes
3 answers
37k views

Yitang Zhang's 2007 preprint on Landau–Siegel zeros

The recent sensational news on bounded gaps between primes made me wonder: what is the status of Yitang Zhang's earlier arXiv preprint On the Landau-Siegel zeros conjecture? If this result is correct, ...
4 votes
2 answers
183 views

Counting integers with k large prime divisors

If $x \ge y \ge 1$ are real numbers and if $k$ is a positive integer, take $\Phi_k(x, y)$ to be the number of integers $\le x$ with exactly $k$ prime factors and no prime factor $\le y$. If $y$ is ...
1 vote
0 answers
70 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{...
2 votes
1 answer
548 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}+...
8 votes
1 answer
625 views

Absolute convergence of Rankin–Selberg series

Let $\pi$ and $\pi'$ be two general automorphic representations on $\operatorname{GL}(n)$ and $\operatorname{GL}(n')$ over $\mathbb{Q}$. I heard that the Rankin-Selberg $L$-function $L(s,\pi\times\pi')...
9 votes
2 answers
448 views

Distribution $f$ such that (a) $\widehat{f}$ has compact support, (b) $\mathbb{E}(|X|)$ is minimal?

(What follows is motivated by an answer to Fourier optimization problem related to the Prime Number Theorem) Let $f:\mathbb{R}\to [0,\infty)$ be such that (a) $\int_{\mathbb{R}} f(x) dx = 1$, (b) $\...
2 votes
1 answer
207 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}^\...
24 votes
1 answer
2k views

Parity of the multiplicative order of 2 modulo p

Let $\operatorname{ord}_p(2)$ be the order of 2 in the multiplicative group modulo $p$. Let $A$ be the subset of primes $p$ where $\operatorname{ord}_p(2)$ is odd, and let $B$ be the subset of primes $...
2 votes
0 answers
140 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 - ...
2 votes
1 answer
270 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 ...
3 votes
0 answers
112 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 ...
2 votes
0 answers
190 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 ...
0 votes
1 answer
124 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 ...
5 votes
0 answers
113 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 ...
2 votes
0 answers
320 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 $x > 0$, ...

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