All Questions
Tagged with analytic-number-theory asymptotics
117 questions
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
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 $$\tilde{f}(w)=\int_0^{\infty}f(x)x^{w-1}dx.$$ ...
- 105k
0
votes
0
answers
79
views
Is there an asymptotic expansion for the reciprocal of the classical Euler beta function?
The classical Euler beta function can be defined by
$$
B(p,q)=\int_0^1t^{p-1}(1-t)^{q-1}\operatorname{d\!}t
$$
for $\Re(p),\Re(q)>0$.
The beta function and the classical Euler gamma function $\...
- 1,091
1
vote
0
answers
113
views
Are there any known statistics on the sign of the Stieltjes Constants?
The Stieltjes Constants $\gamma_n$ arise from considering the laurent series of the Riemann Zeta function at $s=1$
$$ \zeta(s) = \frac{1}{s-1} + \sum_{n=0}^{\infty} (-1)^n \frac{\gamma_n}{n!} (s-1)^n $...
- 2,689
6
votes
1
answer
433
views
Asymptotic behavior of partial sums of Dirichlet series
Consider the Dirichlet series:
$$\sum_{n \geq 1} \frac{a_n}{n^s} = \frac{\zeta(s+1/3)}{\zeta(s)}$$
where $\zeta(s)$ is the Riemann zeta function.
Question: Assuming the Riemann Hypothesis (RH), how ...
- 14.6k
1
vote
0
answers
128
views
On Zudilin's linear forms in $1,\zeta(5)$ and $\zeta(7)$
I am reading an article "Well-poised hypergeometric service for diophantine problems of zeta values" by W. Zudilin.
Consider the quantities defined here in pg. $617$
$$\tilde{F_n}:= \frac{1}{...
- 11
2
votes
1
answer
191
views
Sums of multiplicative functions over residue classes
It was stated in this Shiu, P. work, page 169, Theorem 2, that $$\sum_{\substack{n\le x\\ n\equiv a\pmod k}}d_r^{\ell}(n)\ll\frac{x}{k}\left(\frac{\phi(k)}{k}\log x\right)^{r^{\ell}-1}.$$
Here, $d_r(n)...
- 651
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 ...
- 14.6k
2
votes
1
answer
110
views
Asymptotic behavior in a modular color-cycling problem
Consider the following problem: We have $k$ rooms, each equipped with a light that cycles through three colors – red, green, and blue – in a cyclic order. Initially, all lights are set to red. Each of ...
2
votes
0
answers
179
views
A Brun-Titchmarsh type result for divisor sums; asymptotic/improved bound
In Shiu's work ('A Brun-Titchmarsh theorem for multiplicative functions') he proved that if $r\le x$ is a natural number, we have $$\sum_{r<n\le x}d(n)d(n-r)\ll x\log^2x\sum_{d|r}\frac{1}{d}.$$
I ...
CommunityBot
- 1
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
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 ...
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\...
- 105k
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 ...
- 105k
2
votes
0
answers
195
views
Asymptotics of $\vartheta(x+y)-\vartheta(x)$, where $\vartheta$ is the Chebyshev function, when $y\in[x^\alpha, x]$ for some $\alpha\in(0,1)$
Introduction
Consider $\vartheta$ to be the Chebyshev function, that is, $\vartheta(x)$ denotes, for $x\in\mathbb N$, the sum $\sum_{p\le x,\ p\text{ prime}} \ln p$.
I am interested in asymptotics for ...
- 1,326
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
2
votes
0
answers
253
views
Asymptotic behaviour of a sum involving Möbius function
(This is a cross-posted simplification of this question posted in MSE which did not have a complete answer.)
I am trying to get the asymptotic behaviour when $n$ grows to infinity of a partial sum of ...
- 189
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
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 ...
- 4,713
0
votes
0
answers
44
views
Asymptotic counts for imaginary quadratic discriminants with fixed splitting conditions
Let $p$ be prime and $r$ be a positive integer. I am interested in asymptotics for the number of imaginary quadratic discriminants $d$ such that
$p$ does not divide the conductor of $d$,
$p$ splits ...
- 1,354
2
votes
1
answer
198
views
Series with the smallest number whose square is divisible by $n$
I was looking into this sequence. And I'm particularly interested in the asymptotic behavior of the following series (which is stated on the site) $$\sum_{k=1}^n \frac{1}{a(k)} \sim \frac{3(\log n)^2}...
- 3,319
3
votes
1
answer
348
views
On conjectures about the arithmetic function that counts the number of Sophie Germain primes
I've edited this post two years ago on Mathematics Stack Exchange, with identifier 3590406 and same title On conjectures about the arithmetic function that counts the number of Sophie Germain primes, ...
- 148k
0
votes
2
answers
339
views
Error term in França-LeClair approximation of zeta zeros
The imaginary part of the $n$th critical zero of the Riemann zeta function with positive imaginary part (in increasing order) is asymptotically
$$
t_n \sim 2\pi\frac{n}{\log n}
$$
and has been ...
- 7,024
11
votes
1
answer
471
views
Asymptotics for pairs of positive integers whose harmonic (resp. geometric) mean is an integer
How does the cardinality of $$\{(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}\}...
- 316
2
votes
1
answer
740
views
Does the Riemann hypothesis predict a bound for this prime-counting function?
Does the Riemann hypothesis predict an upper bound for
$$\left|f(x)-\left(\operatorname{li}(x)-\frac{x}{\log x} \right)\right|,\quad x\ge 2\tag{1}$$
where
$$f(x)=\sum\limits_{n=2}^x \frac{\Lambda(n)}{\...
6
votes
2
answers
816
views
Probability of large gcd
Is the following statement true?
Let $N$ be sufficiently large, and choose $t$ uniformly randomly in $\{1,2,\ldots,N\}$. Then
$$\Pr[\gcd(t, N)>N^{3/4}] < N^{-1/16}.$$
This is the "dual"...
- 105k
4
votes
0
answers
262
views
Asymptotic number of "modular primes"
We can say that a number $p$ is prime modulo $N$ if for any two numbers $1<a,b<p$, $ab \not\equiv p \pmod N$. We will define $p(n)$ to be the number of primes mod $n$. I'm wondering about the ...
- 3,319
2
votes
1
answer
152
views
The asymptotic behavior of $F(\lambda):=\sum_{k=0}^{\infty}\frac{\Gamma{(a k)}}{\Gamma{(b k)}}{\lambda}^{-k}$, $b>a>0$
Let $b>a>0$. Given $ \lambda>0$, what is the asymptotic behavior of
$$F(\lambda):=\sum_{k=0}^{\infty}\frac{\Gamma{(a k)}}{\Gamma{(b k)}}{\lambda}^{-k}$$
as $\lambda\to 0^{+}$ and as $\lambda \...
- 128k
11
votes
2
answers
1k
views
Mertens-like theorem
Mertens' first theorem states that
$$
\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 -...
- 14.6k
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 ...
- 12.9k
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 ...
- 4,219
0
votes
0
answers
68
views
Around similar inequalities than an inequality due to Nicolas, that involve products of consecutive Ramanujan primes
This is cross-posted (and this post is a version to ask just around the veracity of Conjecture 1) as the post with identifier 3594907 and same title), that I've edited on Mathematics Stack Exchange ...
0
votes
0
answers
80
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
132
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 $n$ ...
- 37.7k
0
votes
0
answers
158
views
Similar problems for the Dedekind psi function than those that are in the literature for the Euler's totient function: reference request or proposals
I know the linked articles from Wikipedia for the Euler's totient function and the Dedekind psi function. I know that are in the literature famous, interesting and difficult problems involving the ...
2
votes
3
answers
343
views
complete estimates of the error for a well-known asymptotic expression of partition p(n,m)
Let $p(n,m)$ be the number of partitions of an integer $n$
into integers $\le m$, we have a well-known asymptotic expression:
For a fixed $m$ and $n\to\infty$,
$$p(n,m)=\frac{n^{m-1}}{m!(m-1)!} (1+...
- 2,821
0
votes
0
answers
89
views
A similar inequality for the Dedekind psi function, than an inequality stated by Schinzel
I would like to ask about the next question that seems to me interesting. I know an article that was written by Andrzej Schinzel in which he stated Lemma 2. In this post we denote the Dedekind psi ...
1
vote
1
answer
96
views
On the behaviour for the quotient involving Fermat numbers of $\frac{\psi(F_m)}{F_m}$ where $\psi(x)$ denotes the Dedekind psi function
In this post we denote the Dedekind psi function as $\psi(m)$ for integers $m\geq 1$. This is an important arithmetic fuction in several subjects of mathematics. As reference I add the Wikipedia ...
- 28.2k
0
votes
0
answers
157
views
Does it make sense to express upper bounds on arithmetic sequences with Dirichlet generating functions?
In order to see what happens when taking the functional equation in this form:
$$\xi(s) := \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s) $$
$$\xi(s) = \xi(1 - s)$$
$$\pi^{-s/2}\ \Gamma\left(\...
- 1,183
3
votes
0
answers
151
views
On an inequality for the arithmetic function counting the number of primes $\lfloor n^c\rfloor$ in the spirit of Ramanujan's prime counting inequality
In page 3 of [1] (please see if you need it the book by Berndt) Axler refers an inequality that involves the prime-counting function $\pi(x)$ and that was deduced by Ramanujan. I'm curious to know if ...
1
vote
1
answer
152
views
Sum of an arithmetic sequence involving Euler factors
I am trying to find an asymptotic formula for the following sum as $T \to \infty$.
$$ \sum_{t = 1}^{T} \prod_{\substack{p \; \textrm{prime} \\ p | t}} \rho(p) \frac{1 - \frac{1}{p^2}}{1 - \frac{\rho(p)...
- 1,990
4
votes
1
answer
235
views
Relation between $\pi$, area and the sides of Pythagorean triangles whose hypotenuse is a prime number
Consider all Pythagorean triangles $a^2 + b^2 = p^2$ in which the hypotenuse $p$ is a prime number. Let $h(x) = \sum_{p \le x}p^2$, $a(x) = \sum_{p \le x}ab$ and $r(x) = \sum_{p \le x}(a+b)^2$. Is it ...
- 188k
5
votes
0
answers
206
views
Number of solutions of linear congruence with bounded variables
Fix some nonzero integers $a_1, \dots, a_k$, with $k \geq 2$, and some real numbers $c_1, \dots, c_k \in (0,1)$.
For every positive integer $m$, let $N(m)$ be the number of $k$-tuple of integers $(x_1,...
- 121
1
vote
1
answer
310
views
Asymptotic lower bound for the number of square free with at least two prime factors
In one of Soundararajan's papers, he claims without proof that it is a standard exercise to show that the number $N(X)$ of positive square-free integers $d \equiv 1 \; \bmod \; 8$ less than $X$, with ...
- 14.6k
5
votes
1
answer
369
views
A bound for the number of integer solutions to a simple inequality
I am interested in proving an upper bound (expressed as a power of $N$, with $N\rightarrow\infty$ ) for the number of elements of the set
$$
A_N=\{(k,l,m,n)\in([N,2N]\cap\mathbb{Z})^4: |k^2+l^2-m^2-n^...
- 1,262
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 ...
- 105k
1
vote
0
answers
172
views
Asymptotics of $\sum \frac{d(n)}{n}$ with generating functions
We can determine the asymptotics of partial sums involving the divisor function accurately by means of, for example, the hyperbola method: $$\sum_{n\leq N}\frac{d(n)}{n}=\frac{1}{2}(\log(N))^{2}+2\...
- 4,829
3
votes
2
answers
546
views
Approximation of $\sum_{\rho}\frac{1}{|\rho|^2}$, over the non-trivial zeros of the Ramanujan's zeta function
I would like to know if it in the literature an approximation for
$$\sum_{\rho}\frac{1}{|\rho|^2}\tag{1}$$
where the sum is over all of the non-trivial zeros of the Ramanujan's zeta function (also ...
- 13.1k
4
votes
1
answer
291
views
A similar lemma to a lemma due to Lagarias, for the partial sums of reciprocal of primes
I was inspired in Lemma 3.1 of [1] and in the Theorem 4.12 of [2] to ask about a similar statement that shows Lagarias in his paper as Lemma 3.1.
The Lemma from Lagarias's paper is that if $H(n)=\...
- 7,193
5
votes
3
answers
1k
views
What is the asymptotic of the irregular blue curve? Is it $(8x)^{1/2}$ or is it something else?
From Terry Tao's post here there is the statement:
"Conversely, if one can somehow establish a bound of the form
$$\displaystyle \sum_{n \leq x} \Lambda(n) = x + O( x^{1/2+\epsilon} ) \tag{1}$$
...
- 1,183