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.
26
votes
2answers
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
votes
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
vote
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
votes
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 $$W(x-iy) -{\scriptstyle \underset{(n,q) &...
0
votes
0answers
28 views
Discrepancy bound of integer tensor product sequence?
Here discrepancy is from $(2.4)$ in https://www.ricam.oeaw.ac.at/files/people/siambook_nied.pdf given by 'The (extreme) discrepancy $D_N(P) = D_N(x_l,\dots,X_N)$ of
the point set $P$ of $N$ points in $...
4
votes
2answers
591 views
Numbers whose prime factors all have odd exponent
Let $S$ denote the set of natural numbers $m$ with the property that for all prime powers $p^k || m$ we have $k \equiv 1 \pmod{2}$.
What is the asymptotic density of $S$?
Note that $S$ contains all ...
0
votes
0answers
95 views
Difference between Dirichlet Pigeonhole and Exponential sums bound in particular situation?
Dirichlet Pigeonhole says given prime $p$ and vector $(v_1,v_2,\dots,v_n)\in\mathbb Z^n$ there is an integer $m$ such that the vector $m(v_1,v_2,\dots,v_n)\bmod p$ lies in box $[-p^{(n-1)/n}-1,p^{(n-1)...
6
votes
1answer
196 views
Prime plus square equals prime
Furstenberg–Sárközy's theorem states that if a set of positive integers has positive upper density, then there exists (infinitely many) pair of elements of the set, whose difference is a perfect ...
0
votes
0answers
85 views
Quantum ergodicity of Eigenfunctions on PSL2(Z)/H 2
I am trying to read this paper. But any line of it is impossible to read! I have studied Graduate Algebra books, Multiplicative Number Theory, Graduate Real and Complex Analysis, R Zeta fn , ...
8
votes
0answers
101 views
Is there an approximate formula for the discriminant of a sparse polynomial?
Consider integer polynomials $P \in \mathbb{Z}[X] \setminus \{0\}$ of a degree $D \geq 1$ and without multiple complex roots. Let me introduce a notation
$$
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
votes
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
votes
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
$$S(n,48)=\bigg\lbrace \begin{pmatrix}r&s\\0& t \end{pmatrix}\colon r>0,rt=n,(r,s,t)=1,48\mid ...
7
votes
1answer
258 views
Riemann sum formula for definite integral using prime numbers
I had asked this question in MSE. It got lot of upvotes but no answer (except one which was too long to be posted as a comment) hence I am posting it in MO.
While answering another question in MSE I ...
2
votes
1answer
172 views
Is there a relation between the number of lattice points lie within these circles
Suppose we have a circle of radius $r$ centered at the origin $(0,0)$. The number of integer lattice points within the circle, $N$, can be bounded using Gauss circle problem .
Suppose that another ...
0
votes
0answers
36 views
Does a weighted sequence derived from a sequence matching the Siegel lemma BV bound behave as an uniformly random sequence?
Pick integers $r_1',\dots,r_t'$ that achieve the Bombieri Vaaler bound for the Siegel lemma and find an $m$ (it exists by Dirichlet Pigeonhole) such that given prime $T$ gets $m(r_1',\dots,r_t')\equiv(...
-2
votes
1answer
161 views
Strong estimates for the zeta function on natural numbers
Let $$\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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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,
$$\Lambda(m)=\...
0
votes
0answers
79 views
The sieve formula choosen in Zhang's breakthrough work [duplicate]
In the breakthrough work of the proof of weak twin prime conjecture, Goldstone, Pintz and Yildirim as well as Zhang use the following modified Selberg sieve:
$v=\lambda^2$
where $\lambda(n)$takes ...
47
votes
5answers
2k views
Motivated account of the prime number theorem and related topics
Though my own research interests (described below) are pretty far from analytic number theory, I have always wanted to understand the prime number theorem and related topics. In particular, I often ...
6
votes
2answers
280 views
asymptotic for li(x)-Ri(x)
Is it true that $$\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
votes
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 ...
1
vote
0answers
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
votes
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
votes
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
votes
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
votes
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
votes
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)...
