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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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1k views

Lagrange four squares theorem

Lagrange's four square theorem states that every non-negative integer is a sum of squares of four non-negative integers. Suppose $X$ is a subset of non-negative integers with the same property, that ...
3
votes
1answer
104 views

Density of numbers $n$ which are co-prime with their $\phi$-value

Let $n$ be a positive integer. The Euler $\phi$-function is defined by $$\displaystyle \phi(n) = \# \{1 \leq a \leq n-1 : \gcd(a,n) = 1\}.$$ It is in fact a multiplicative function, and one has the ...
6
votes
2answers
1k views

Does this multiplicative function have a name? If so, what is known about it?

It is well-known that the Euler $\phi$-function is multiplicative: that is, for co-prime positive integers $m,n$ we have $\phi(mn) = \phi(m)\phi(n)$. Thus it is defined by its values on prime powers. ...
2
votes
0answers
47 views

Almost-prime values attained by polynomials, with extra conditions

Many results in sieve theory are of the following form: Let $f(x) \in \mathbb{Z}[x]$ be polynomial satisfying certain conditions. There are infinitely many integers $n$ such that $f(n)$ is a ...
2
votes
1answer
206 views

Does $\sum_{n=1}^\infty \frac{\mu(n)}{n^s}$ converge for $\sigma > \frac{1}{2}$?

Looking at @Lucia's answer to this question it appears $\sum_{n=1}^\infty \frac{\mu(n)}{n^s}$ converges for $\sigma > \frac{1}{2}$. Can someone point me to a proof or provide proof for this? If I ...
0
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0answers
159 views

On the connection between Faulhaber's formula and identity $n^{2m+1}=\sum_{k=0}^{n-1}\sum_{j=0}^m A_{m,j}k^j(n-k)^j$

This question is part of series of the questions, as follows: Coefficients in the sum $\sum_{k=0}^{n-1}\sum_{j=0}^{m}A_{j,m}(n-k)^jk^j=n^{2m+1}$, Coefficients $U_m(n,k)$ in the identity $n^{2m+1}=\...
0
votes
0answers
113 views

Typical size of $\infty$ norm of integer points in subspaces associated to a structured linear diophantine equation

Take natural numbers $A_1,B_1,A_2,B_2$ random pairwise coprime in $[n,2n]$ for $n$ large enough and consider the space of solutions to $A_1a+B_1b=0$ and $A_1A_2a+A_1B_2b+B_1A_2c+B_1B_2d=0$ spanned by $...
5
votes
1answer
288 views

consecutive prime gaps and explicit bound

I am aware of the theorem that $p_{n+1}- p_n \leq n^{0.525}$ which is true for all sufficiently large numbers due to Baker, but if i want to make the implicit "for all sufficiently large numbers" ...
1
vote
1answer
122 views

Approximate the following series on the euclidean grid

I had trouble in finding a closed form solution for the following series, so now I am trying to find a good approximation for it. The $\sqrt{i^2 + j^2}$ in the exponent comes from distances on the ...
1
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0answers
182 views

Hardy-Littlewood vs heuristics on the zeta zeros

The first Hardy-Littlewood Conjecture asserts: Conjecture 1: Fix integers $0< a_1 < \ldots < a_k$. The density of those primes $p\le x$ such that $p + 2a_1,\ldots, p+2a_k$ are also primes, ...
4
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0answers
221 views

$\sum_{n=1}^\infty \Lambda(n) e^{-nz}$ and $L(s,\chi)$

Let $$W(z)=\sum_{n=1}^\infty \frac{\Lambda(n)}{n^{1/2}} e^{-nz}, \qquad\Re(z) > 0$$ For $\frac{y}{2\pi}=\frac{a}{q} \in \mathbb{Q}$, as $x \to 0^+$ we have $$W(x-iy) -{\scriptstyle \underset{(n,q) &...
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votes
0answers
28 views

Discrepancy bound of integer tensor product sequence?

Here discrepancy is from $(2.4)$ in https://www.ricam.oeaw.ac.at/files/people/siambook_nied.pdf given by 'The (extreme) discrepancy $D_N(P) = D_N(x_l,\dots,X_N)$ of the point set $P$ of $N$ points in $...
4
votes
2answers
591 views

Numbers whose prime factors all have odd exponent

Let $S$ denote the set of natural numbers $m$ with the property that for all prime powers $p^k || m$ we have $k \equiv 1 \pmod{2}$. What is the asymptotic density of $S$? Note that $S$ contains all ...
0
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0answers
95 views

Difference between Dirichlet Pigeonhole and Exponential sums bound in particular situation?

Dirichlet Pigeonhole says given prime $p$ and vector $(v_1,v_2,\dots,v_n)\in\mathbb Z^n$ there is an integer $m$ such that the vector $m(v_1,v_2,\dots,v_n)\bmod p$ lies in box $[-p^{(n-1)/n}-1,p^{(n-1)...
6
votes
1answer
196 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 ...
0
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0answers
85 views

Quantum ergodicity of Eigenfunctions on PSL2(Z)/H 2

I am trying to read this paper. But any line of it is impossible to read! I have studied Graduate Algebra books, Multiplicative Number Theory, Graduate Real and Complex Analysis, R Zeta fn , ...
8
votes
0answers
101 views

Is there an approximate formula for the discriminant of a sparse polynomial?

Consider integer polynomials $P \in \mathbb{Z}[X] \setminus \{0\}$ of a degree $D \geq 1$ and without multiple complex roots. Let me introduce a notation $$ d(P) := \frac{1}{D} \log{|\mathrm{Disc}(P)|}...
4
votes
1answer
285 views

What is the asymptotic growth of $\sum_{k=1}^n 2^{\omega_k}$?

Question: Let $\omega_k$ be the number of distinct prime divisors of k. What is the asymptotic growth of $C_n := \sum_{k=1}^n 2^{\omega_k}$? Thank you for considering this elementary question. ...
6
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0answers
97 views

Smooth numbers in short intervals

Let $\psi(x,y)$ be the number of positive integers up to $x$ which are $y$-smooth, that is, integers whose prime factors are at most of size $y$. There has been, for a few decades now, a lot of ...
3
votes
1answer
151 views

Almost-Primes in Short Intervals

Let $S$ be the set of integers which are a product of $k$ distinct primes, $k$ a fixed positive integer (the condition that the primes are distinct is not crucial). Landau used the Prime Number ...
2
votes
0answers
171 views

What is the sum of the binomial coefficients ${n\choose p}$ over prime numbers?

What is known about the asymptotics, lower and upper bound of the sum of the binomial coefficients $$ S_n = {n\choose 2} + {n\choose 3} + {n\choose 5} + \cdots + {n\choose p} $$ where the sum runs ...
11
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2answers
283 views

Has the arcsine law been generalised to higher order divisor functions?

The arcsine law for the distribution of the logarithms of the divisors of an integer $n$ states that $$ \frac{1}{x}\sum_{n\leq x}\frac{1}{d(n)}\sum_{\substack{q|n\\q\leq n^{A}}}1\sim \frac{2}{\pi}\...
5
votes
0answers
180 views

Explicit bounds for the Mertens function

It is a consequence of some forms of the prime number theorem that with $\mu$ the Möbius function, for all $A > 0$, there exists $c_A$ such that for all sufficiently large $x$, $$\frac{1}{x}\sum_{n\...
0
votes
0answers
78 views

Fixing quadratic surds

Assume that $n\equiv -1 \text{ }(\text { mod }24 )$ is a positive integer and $(n,48)=1$. Let $$S(n,48)=\bigg\lbrace \begin{pmatrix}r&s\\0& t \end{pmatrix}\colon r>0,rt=n,(r,s,t)=1,48\mid ...
7
votes
1answer
258 views

Riemann sum formula for definite integral using prime numbers

I had asked this question in MSE. It got lot of upvotes but no answer (except one which was too long to be posted as a comment) hence I am posting it in MO. While answering another question in MSE I ...
2
votes
1answer
172 views

Is there a relation between the number of lattice points lie within these circles

Suppose we have a circle of radius $r$ centered at the origin $(0,0)$. The number of integer lattice points within the circle, $N$, can be bounded using Gauss circle problem . Suppose that another ...
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36 views

Does a weighted sequence derived from a sequence matching the Siegel lemma BV bound behave as an uniformly random sequence?

Pick integers $r_1',\dots,r_t'$ that achieve the Bombieri Vaaler bound for the Siegel lemma and find an $m$ (it exists by Dirichlet Pigeonhole) such that given prime $T$ gets $m(r_1',\dots,r_t')\equiv(...
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votes
1answer
161 views

Strong estimates for the zeta function on natural numbers

Let $$\zeta(s) = \sum_{n = 1}^\infty \frac{1}{n^s}$$ be the Riemann zeta function (here we just consider real $s$). We do have a description given by $$\zeta(s) = \frac{s}{s-1}-s\int_{1}^\infty \frac{...
4
votes
0answers
87 views

Injectivity of product functions on natural number sequences

Let $M = \{ a = (a_i)_{i} : a_i \in \mathbb{N}, a_1 \geq 2, a_i > a_j \forall i>j\}$ the set of all ascending natural number sequences, with $a_1$ at least 2. We now define for each $k \geq 2$ ...
12
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2answers
501 views

Show that there exist $k\in\{1,2,\cdots,n\}$ such that $\frac{1}{n}\sum_{i=1}^{n}\left(\{kx_{i}\}-\frac{1}{2}\right)^2>\frac{1}{12}-\frac{1}{6n}$

The following question has post mathsatck :Nice problem, perhaps from a question of expectation.The problem seemed so elementary, but I had tried to solve it for a long time and eventually failed.I 'd ...
6
votes
2answers
267 views

A simultaneous generalization of the Grunwald-Wang and Dirichlet Theorems on primes

By Grunwald-Wang Theorem, if for some odd number $n$ the equation $x^n=a$ has no solutions in $\mathbb Z$, then the equation $x^n=a\mod p$ has no solutions for some prime number $p$. I am interested ...
7
votes
2answers
728 views

A stronger form of the Dirichlet Theorem on prime numbers in arithmetic sequences

Question 1. Let $a,b>1$ be two natural numbers. Is there a prime number $p\in 1+b\mathbb N$ such that $a+p\mathbb Z$ is a generator of the multiplicative group of the field $\mathbb Z/p\mathbb Z$? ...
2
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0answers
108 views

What is the best known upper bound for $( \gamma_{n+1}-\gamma_{n})\max_{\{T\in(\gamma_{n},\gamma_{n+1})\}}(\vert\zeta(1/2+iT)\vert) $?

For $ n $ a positive integer, denote by $ L(n) : =\gamma_{n+1}-\gamma_{n} $ with $ \gamma_{n} $ the imaginary part of the $ n $-th critical zero of the Riemann zeta function and by $ M(n) : =\...
2
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0answers
83 views

Question about the paper of Soundararajan and Radziwill on Selberg's CLT for $\log|\zeta(1/2+it)|$

The paper: http://www.math.mcgill.ca/radziwill/CLTNotes3.pdf. Briefly, I'm having trouble inferring the moments given at the end of Proposition 2 from Lemma 1. I'm going by splitting the integrands ...
0
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0answers
50 views

Best method to compute sum of divisors of bounded evaluation of a bivariate quadratic?

Given a bivariate quadratic polynomial $g(u,v)\in\mathbb Z[u,v]$ and an integer $n$. How fast can we compute $\sum_{i=-\ell}^{\ell}\sum_{j=-\ell'}^{\ell'}\sigma_0(g(i,j))\bmod2$ where $\sigma_0$ is ...
0
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0answers
177 views

On number of squares by a quadratic

Given a bivariate degree $2$ polynomial $$m^2+(ax−by)^2−2x(ma+2br′)−2y(mb+2br)−4rr'\in\mathbb Z[x,y]$$ where $0<r,r'<\max(a,b)<rr'<m<ab$ with $\gcd(a,b)=1$ holds with max coefficient ...
3
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1answer
228 views

Arithmetic properties of a sum related to the first Hardy-Littlewood conjecture

The starting point of this post is an earlier question, where I conjectured (and GH from MO confirmed) that the von Mangoldt function is the limit at $s=1$ of a certain Dirichlet series, $$\Lambda(m)=\...
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0answers
79 views

The sieve formula choosen in Zhang's breakthrough work [duplicate]

In the breakthrough work of the proof of weak twin prime conjecture, Goldstone, Pintz and Yildirim as well as Zhang use the following modified Selberg sieve: $v=\lambda^2$ where $\lambda(n)$takes ...
47
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5answers
2k views

Motivated account of the prime number theorem and related topics

Though my own research interests (described below) are pretty far from analytic number theory, I have always wanted to understand the prime number theorem and related topics. In particular, I often ...
6
votes
2answers
280 views

asymptotic for li(x)-Ri(x)

Is it true that $$\operatorname{li}(x)-\operatorname{Ri}(x) \sim \frac{1}{2}\operatorname{li}(x^{1/2}) \ (x \to \infty),$$ where $$\operatorname{Ri}(x) = \sum_{n = 1}^\infty \frac{\mu(n)}{n} \...
0
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0answers
82 views

How to prove a Mersenne number is not pseudoprime to the base 3?

I start think in finding a small divisor $p$ of $M_n$ then I obtain that $p\mid (3^{M_{n}-1}-1)$ hence $ord_{p}(3)\mid M_{n}-1$ Now can I prove p is the same as $M_{n}$ so that $M_{n}$ is not ...
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86 views

Counting "simultaneous squares' over the Gaussian integers

Let $n$ be a square-free integer. Then for a given integer $m$, $m$ is a square modulo $n$ if and only if the sum $$\displaystyle \sum_{d | n} \left(\frac{m}{d}\right) > 0.$$ In fact one can ...
3
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0answers
135 views

Modular root of $-1$

Take two distinct coprime integers $a,b$ and take $q=a^2+b^2$. Consider the equation $(xy^{-1})^2\equiv-1\bmod q$. The solutions are two roots which correspond to $(x,y)=(a,b)$ and $(x,y)=(b,a)$. Look ...
0
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1answer
141 views

Parity and number of squares taken by polynomials in a range?

I have a polynomial $f(x)=a^2x^2+bx+c\in\mathbb Z[x]$ with $f(x)$ not a constant times a square and $abc\neq0$ and I want to know how many $x$ between $-a$ and $a$ the polynomial is a perfect square. ...
11
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1answer
170 views

Partial product of Euler factors

Let $\mathbb P$ denote the set of prime numbers and for a subset $T\subset \mathbb P$ let $$ \zeta_T(s)=\prod_{p\in T}\frac1{1-p^{-s}}, $$ where $\mathrm{Re}(s)>1$. Is there any $T$ such that $T$ ...
8
votes
1answer
855 views

Are there infinitely many primes of this form?

The semiprime $87 = 3*29$ has a curious property: it's the fact that both $87^2 + 29^2 + 3^2 = 8419$ and $87^2 - 29^2 - 3^2 = 6719$ are prime numbers. This intrigued me and led me to wonder if ...
0
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0answers
120 views

An elementary size bound in number theory?

Given integer $a$ of size $R^\alpha$ with $\alpha\in(0,1)$ and $t$ large primes $R_i$ of roughly same size $R$ what is the smallest $T$ one needs so that there is an integer $0<K<R$ with as ...
0
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0answers
78 views

Infinite difference length of integer subsets

Let $A$ be a set of integers. In our recent researches, we've faced to the following property and definition: We say $A$ has infinite difference length, if (a) For every integer $n$ there exist a ...
2
votes
1answer
120 views

Twisted modular equation

Let $\gamma_2(\tau)=j(\tau)^{1/3}$. The modular equation shows that the functions $$j\left(\frac{a\tau+b}{c\tau+d}\right),\qquad ad-bc=n$$ are integral over $\mathbf Z[j]$. Under what conditions is ...
1
vote
1answer
214 views

Sum of log over friables

Let $x$ and $y$ be two positive real numbers. What is the mean value of the function $\log$ on $y$ friables integers less than $x$ i.e the value of the following sum $$\sum_{\substack{n \leq x \\ P(n)...