All Questions
Tagged with analytic-number-theory asymptotics
117 questions
25
votes
1
answer
2k
views
Wrong asymptotics of OEIS A000607 (number of partitions of an integer in prime parts)?
Sequence A000607 in the Online Encyclopedia of Integer Sequences is the number of partitions of $n$ into prime parts. For example, there are $5$ partitions of $10$ into prime parts: $10 = 2 + 2 + 2 + ...
- 690
21
votes
1
answer
771
views
Covering a set with geometric progressions
Consider the set $S_n=\{1,2,\cdots ,n\}$. What is the minimum number of distinct geometric progressions that cover $S_n$? Let us call this number $a_n$. I was wondering about this number after doing a ...
- 1,090
20
votes
1
answer
1k
views
Quantitative lower bounds related to Zhang's theorem on bounded gaps
Let $\mathcal{H}=\left\{ h_{i}\right\} _{i=1}^{k}$ be an admissible set, and define $$\pi_{\mathcal{H}}(x)=\left|\left\{ n\leq x\ :\ \exists\ i,j\leq k,\ i\neq j\ \text{such that both }n+h_{i},\ n+h_{...
- 11.4k
13
votes
1
answer
524
views
The number of representations of an integer as the inner product of integral lattice points
I was looking through some old notes of mine and I came across a couple lemmas/identities I wrote down in regards to a question I asked about four years ago. In particular I wrote that for an ...
- 2,506
11
votes
2
answers
1k
views
Mertens-like theorem
Mertens' first theorem states that
$$
\sum_{p \leq n} \frac{\log p}{p} = \log n + O(1).
$$
I read in this paper that the following variant is "classical":
$$
\sum_{p \leq n} \frac{\log p}{p -...
11
votes
2
answers
771
views
Tauberian theorem with better error term
This is a fairly vague question.
Suppose we have a sequence of positive numbers $(c_n)_n$ and we want to find an asymptotic formula for $S(x) = \sum_{n \leq X} c_n$. In favorable circumstances, ...
- 1,362
11
votes
1
answer
471
views
Asymptotics for pairs of positive integers whose harmonic (resp. geometric) mean is an integer
How does the cardinality of $$\{(a,b): 1 \leq a,b \leq n, \ 2ab/(a+b) \ \mbox{is an integer}\}$$ grow as a function of $n$? What about $$\{(a,b): 1 \leq a,b \leq n, \ \sqrt{ab} \ \mbox{is an integer}\}...
- 19.7k
11
votes
1
answer
324
views
Does the mean ratio of the largest prime factor in prime gaps to the lower bound of the gap converge?
Posting in MO since this questions has been unanswered in MSE for 3 months.
Let $p_n$ be the $n$-th prime and $q_n$ be largest among all the prime factors of the composite numbers between $p_n$ and $...
- 5,470
9
votes
2
answers
740
views
Asymptotics of product of Euler's totient function (A001088)?
Conjecture:
\begin{align}
\lim_{n\to \infty } \, \frac{\left(\prod _{k=1}^n \phi (k)\right){}^{1/n}}{n}\sim 0.2059\text{...}
\end{align}
The numerical result from 100000 terms is:
My questions are:
...
- 690
7
votes
2
answers
720
views
On the asymptotic behaviour of the series $\sum_{n=1}^{\infty} \left ( \frac{\zeta(ns)}{n}+\frac{s}{1-ns}\right )$ near $s=0$
I am interested in determining the behaviour of the the series/function
$$f(s)=\sum_{n=1}^{\infty} \left ( \frac{\zeta(ns)}{n}+\frac{s}{1-ns}\right )$$
near $s=0$. It is clear that $f(0)$ is undefined....
- 405
7
votes
1
answer
683
views
The Gauss Circle Problem asymptotic in dimension
The circle problem in $k$ dimensions: "For $n>0$, how many points $z\in \ \mathbb{Z}^k$ have $\|z\|^2\leq n$?"
For large $n$, the answer is $\approx n^{k/2}\cdot \operatorname{Vol}(B^k(0,1))+\...
6
votes
2
answers
816
views
Probability of large gcd
Is the following statement true?
Let $N$ be sufficiently large, and choose $t$ uniformly randomly in $\{1,2,\ldots,N\}$. Then
$$\Pr[\gcd(t, N)>N^{3/4}] < N^{-1/16}.$$
This is the "dual"...
- 359
6
votes
1
answer
809
views
Probability that a positive integer is in the range of the Euler phi function
Define $f(n) = |\{m : m\le n, \exists k \text{ s.t. }\phi(k) = m\}|$.
Clearly, $f(n)\le \left\lfloor \frac{n}{2}\right\rfloor + 1$ since $\phi(n)$ is even for all $n > 2$.
Is $\limsup_{n\...
- 1,890
6
votes
1
answer
4k
views
About the logarithmic derivative of the Riemann zeta function
Let $\rho=\beta+i\gamma$ a non-trivial zeros of the Riemann zeta function and $s=\sigma+it$ a complex number. It is possible to prove that $$\frac{\zeta'}{\zeta}\left(s\right)=\sum_{\left|t-\gamma\...
- 219
6
votes
1
answer
433
views
Asymptotic behavior of partial sums of Dirichlet series
Consider the Dirichlet series:
$$\sum_{n \geq 1} \frac{a_n}{n^s} = \frac{\zeta(s+1/3)}{\zeta(s)}$$
where $\zeta(s)$ is the Riemann zeta function.
Question: Assuming the Riemann Hypothesis (RH), how ...
- 611
6
votes
1
answer
369
views
Asymptotic estimate for an integral involving the squared modulus of the Riemann zeta function
For any fixed $\frac{1}{2} < \sigma < 1$, let
$$\int_0^T \frac{|\zeta(\sigma+it)|^2}{\sqrt{1+t^2}} \ dt = O(T^\theta), \qquad T \uparrow \infty. $$
It is clear that $\theta > 0$, since we ...
- 81
6
votes
1
answer
548
views
Equidistribution on the unit circle of particular sequences of finite subsets
Given a strictly convex function $g : [0, 1] \to \mathbb{R}$, I'm curious about the asymptotic distribution of the points $\exp{(2 \pi i N g(n / N))}$ for $n = 1, 2, \dots, N$, counted with ...
5
votes
3
answers
1k
views
What is the asymptotic of the irregular blue curve? Is it $(8x)^{1/2}$ or is it something else?
From Terry Tao's post here there is the statement:
"Conversely, if one can somehow establish a bound of the form
$$\displaystyle \sum_{n \leq x} \Lambda(n) = x + O( x^{1/2+\epsilon} ) \tag{1}$$
...
- 1,183
5
votes
1
answer
750
views
Sum of reciprocals of rough numbers
Let $x$ and $y$ be given real numbers. We may suppose that $2\leqslant x \leqslant y$ and that $u:= \log(y)/\log(x)$ remains bounded in a compact set away from $1$ as $x,y\to\infty$. An integer $n$ is ...
- 589
5
votes
1
answer
427
views
Lindelöf hypotheses for derivatives of zeta
The Lindelöf hypothesis says that if we have:
$$\zeta(\sigma+iT)=\mathcal O(T^a)$$
Then if one considers $\sigma=1/2$ then $\inf a=0$. Further, from convexity and the functional equation this implies ...
- 83
5
votes
2
answers
711
views
Asymptotic estimate of double summation
I would like to determine an asymptotic expansion for the following double summation:
$$\sum_{a=1}^{N/\sqrt {j}} \sum_{b=a}^{ja} \frac{1}{ab}$$
where $j$ is a real number $\geq 1$ and $N$ tends to $\...
- 163
5
votes
1
answer
369
views
A bound for the number of integer solutions to a simple inequality
I am interested in proving an upper bound (expressed as a power of $N$, with $N\rightarrow\infty$ ) for the number of elements of the set
$$
A_N=\{(k,l,m,n)\in([N,2N]\cap\mathbb{Z})^4: |k^2+l^2-m^2-n^...
- 421
5
votes
1
answer
290
views
Overconcentration of Poles on the Circle of Convergence of a Power Series with Bounded Coefficients
Let $V$ be an arbitrary set of infinitely many positive integers, and let: $$\varsigma_{V}\left(z\right)\overset{\textrm{def}}{=}\sum_{v\in V}z^{v}$$ Let $T_{V}$ denote the set of all $t\in\left[0,1\...
- 1,284
5
votes
0
answers
206
views
Number of solutions of linear congruence with bounded variables
Fix some nonzero integers $a_1, \dots, a_k$, with $k \geq 2$, and some real numbers $c_1, \dots, c_k \in (0,1)$.
For every positive integer $m$, let $N(m)$ be the number of $k$-tuple of integers $(x_1,...
- 121
5
votes
0
answers
233
views
What is known about the mode of the number of divisors $\le x$?
Let $d(x)$ be the divisor function. Let $M(x)$ ($x$ a positive integer) be the most frequent value of $d(y)$ for $1 \le y \le x$. Is an asymptotic known for $M(x)$, and failing that, can $M(x)$ at ...
- 251
5
votes
0
answers
355
views
What is the sum of the binomial coefficients ${n\choose p}$ over prime numbers?
What is known about the asymptotics, lower and upper bound of the sum of the binomial coefficients
$$
S_n = {n\choose 2} + {n\choose 3} + {n\choose 5} + \cdots + {n\choose p}
$$
where the sum runs ...
- 5,470
4
votes
2
answers
704
views
Estimate related to the Möbius function
I need to know, or at least have a good bound for, the asymptotic behaviour on $x$ of amount of integers less or equal than $x$ that are square free and with exactly $k$ primes on its decomposition. ...
4
votes
2
answers
517
views
Average involving the Euler phi function
Does
$$\frac{1}{N^2}\sum _{d=1}^N \log d \sum _{n=1}^{N/d} \frac{\phi(n)}{\log (dn)},$$
converges or not when $N$ goes to infinity?
- 43
4
votes
1
answer
271
views
On approximation of $\sum_{a,b=1}^n\gcd(a,b)$
Denote $g(n)=\sum_{a,b=1}^n\gcd(a,b)$, can we prove that
$$g(n)=\frac6{\pi^2}n^2\ln n+Cn^2+O(n\ln n)$$, where $C=-\frac12+\frac{6}{\pi^2}(-\frac12+\gamma-\ln(2\pi)+12\ln A),$ where $\gamma$ denotes ...
- 143
4
votes
1
answer
549
views
Sum over reciprocal of primes times coefficient
I would like to show that
$$
\sum_{p\leq x} \frac{1}{p^{1+2/\log x}}\left(\frac{\log\left(x/p\right)}{\log(x)}\right)^2=\log\log x +\mathcal{O}(1)
$$
What I have tried
Since we know that
$$
\sum_{p\...
- 199
4
votes
1
answer
454
views
Asymptotics for sums of powers of divisor function
It is a well known result that with $\tau(n) := \sum_{d|n} 1$ that
$$\sum_{n\le X}\tau(n)\sim X\log X,\\ \sum_{n\le X}\tau(n)^2\sim C_1X\log^3X$$
for some $C_1 > 0$. In general, it is also known ...
- 1,890
4
votes
1
answer
423
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 ...
user494572
4
votes
2
answers
472
views
Sharp estimates for Meissel-Mertens constant
I wondered if it is possible to get a similar inequality like $(1.1)$ of Michael D. Hirschhorn, Approximating Euler's Constant, The Fibonacci Quarterly, Volume 49, Number 3 (August 2011) for the ...
4
votes
1
answer
235
views
Relation between $\pi$, area and the sides of Pythagorean triangles whose hypotenuse is a prime number
Consider all Pythagorean triangles $a^2 + b^2 = p^2$ in which the hypotenuse $p$ is a prime number. Let $h(x) = \sum_{p \le x}p^2$, $a(x) = \sum_{p \le x}ab$ and $r(x) = \sum_{p \le x}(a+b)^2$. Is it ...
- 5,470
4
votes
2
answers
622
views
What work can be done to study the solutions of $\varphi\left(x^{\sigma(x)}\sigma(x)^x\right)=2^{x-1} x^{3x-1}\varphi(x)$?
For integers $n\geq 1$ I denote the Euler's totient function as $\varphi(n)$ and the divisor function $\sum_{1\leq d\mid n}d$ as $\sigma(n)$, that are two well-known mulitplicative functions. We ...
4
votes
1
answer
954
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,
$$\Lambda(m)=\...
- 1,183
4
votes
1
answer
291
views
A similar lemma to a lemma due to Lagarias, for the partial sums of reciprocal of primes
I was inspired in Lemma 3.1 of [1] and in the Theorem 4.12 of [2] to ask about a similar statement that shows Lagarias in his paper as Lemma 3.1.
The Lemma from Lagarias's paper is that if $H(n)=\...
4
votes
1
answer
270
views
What explains the asymptotic and the pattern in this sequence related to Riemann zeta zeros?
As the starting point for my experiment I assumed that the imaginary parts of the Riemann zeta zeros are of the form:
$$\Im \{ \rho_n \} = \frac{2\pi}{\log x_n}$$
where $x_n$ is unknown. Therefore I ...
- 1,183
4
votes
1
answer
213
views
Asymptotic behavior of weighted sums involving the fractional part function
Currently, I am studying the asymptotic behavior of sums of the form
\begin{equation}\label{eq1}\tag{1}
\sum_{k=1}^{n-1} f(n-k) \left\{ \frac{n}{k} \right\}
\end{equation}
In this context, based on ...
- 611
4
votes
0
answers
262
views
Asymptotic number of "modular primes"
We can say that a number $p$ is prime modulo $N$ if for any two numbers $1<a,b<p$, $ab \not\equiv p \pmod N$. We will define $p(n)$ to be the number of primes mod $n$. I'm wondering about the ...
- 3,319
4
votes
0
answers
412
views
Effective prime number theorem
The prime number theorem implies that for every $ϵ>0$, there is $n_\epsilon$ such that for all $n≥n_\epsilon$ the number of primes in $[n,cn]$ is at least $\frac{(c−1−\epsilon)n}{\log n}$ and at ...
- 13.9k
4
votes
0
answers
306
views
Effective version of the Bombieri-Vinogradov theorem
Is there an effective version of the Bombieri-Vinogradov Theorem, in that have bounds on the implied constant been found?
- 1,890
3
votes
2
answers
546
views
Approximation of $\sum_{\rho}\frac{1}{|\rho|^2}$, over the non-trivial zeros of the Ramanujan's zeta function
I would like to know if it in the literature an approximation for
$$\sum_{\rho}\frac{1}{|\rho|^2}\tag{1}$$
where the sum is over all of the non-trivial zeros of the Ramanujan's zeta function (also ...
3
votes
2
answers
218
views
The graph and sign of $p_n-\operatorname{ali}(n)$, where $p_n$ is the $n$-th prime and $\operatorname{ali}(n)$ the inverse of the logarithmic integral
I'm inspired in [1] to ask the following question. My problem is that I have not an implementation of the inverse of the logarithmic integral $\operatorname{Li}(x)=\int_2^x\frac{dt}{\log t}$, that ...
3
votes
2
answers
201
views
Meinardus theorem at use: problems with conditions
I am working on an enumerative problem related to knot theory, and I have found the following generating function
$$F(z)=\prod_{n\geq 1} \frac{1}{(1-z^{2n+1})^2}.$$
I am interested on getting ...
- 1,561
3
votes
1
answer
348
views
On conjectures about the arithmetic function that counts the number of Sophie Germain primes
I've edited this post two years ago on Mathematics Stack Exchange, with identifier 3590406 and same title On conjectures about the arithmetic function that counts the number of Sophie Germain primes, ...
3
votes
1
answer
177
views
Mellin transform at $0$
Let $f$ be a smooth complex valued function with support on $[0,2]$ and with Mellin transform $\tilde{f}$. Recall that the Mellin transform is defined as $$\tilde{f}(w)=\int_0^{\infty}f(x)x^{w-1}dx.$$ ...
user543269
3
votes
1
answer
516
views
About the asymptotics of LCM
Let $g(x,c)$ be a uniformly random integer in the range $(x,x+c)$ and $LCM[x_1,x_2...x_i]$ the lowest common multiple of the integers $x_i$.
A) Does the limit of (the asymptotics of $LCM[g(3^1,c),g(...
- 61
3
votes
0
answers
167
views
A sharper estimate for a generalization of the sum-of-divisors function
I am interested in the function $f_n(m)$ which can be defined by the Dirichlet generating function $$\zeta(s)\zeta(s - 1) \cdots\zeta(s - n + 1) = \sum\limits_{m = 1}^\infty \frac{f_n(m)}{m^s} $$
This ...
- 31
3
votes
0
answers
105
views
Error function of the second moment of the divisor function
It is easy to show that the second moment of the divisor function has asymptotics:
$$\sum_{n\leq x} d_0(n)^2 = xP(\log(x))+E_2(x)$$
Where $P$ is some polynomial and that:
$$E_2 = o(x)$$
Previously, ...
- 83