All Questions
Tagged with analytic-number-theory asymptotics
117 questions
3
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0
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151
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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 ...
3
votes
0
answers
106
views
Friedlander-Iwaniec Flipping moduli
I am reading section 12 (Flipping Moduli) of the paper "The polynomial $X^2+Y^4$ captures its primes" by Friedlander and Iwaniec.
At page 997, just below equation (12.7) we start estimating the ...
- 31
3
votes
0
answers
320
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On sets of coprime numbers
We know that from prime number theorem that the number of primes below $n$ and above $\frac n2$ (denoted by $\pi_{n,\frac n2}$ is approximately $$\pi_{n,\frac n2}\approx\frac{n}{2\ln n}.$$
Denote by $...
user76479
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
$$f(x)=\sum\limits_{n=2}^x \frac{\Lambda(n)}{\...
- 1,126
2
votes
1
answer
198
views
Series with the smallest number whose square is divisible by $n$
I was looking into this sequence. And I'm particularly interested in the asymptotic behavior of the following series (which is stated on the site) $$\sum_{k=1}^n \frac{1}{a(k)} \sim \frac{3(\log n)^2}...
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 $n\to\infty$,
$$p(n,m)=\frac{n^{m-1}}{m!(m-1)!} (1+...
- 199
2
votes
1
answer
230
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On $\sum_{\substack{p\leq x\\p,p+2\text{ twin primes}}}\frac{(\log p)^m}{p}$, on assumption of the first Hardy–Littlewood conjecture
I wondered, inspired in a result from [1] (Proposition 17) what should be the asymptotic behaviour of the sequence, on assumption of the First Hardy–Littlewood conjecture,
$$\sum_{\substack{\text{...
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 \...
- 852
2
votes
1
answer
235
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How to get asymptotic expansion of the sum of modified Bessel function $\sum_{n=1}^\infty K_0(s\, n)$ as $s\to 0^+$?
I guess for the modified Bessel funcion $K_0(z)$,
$$\sum_{n=1}^\infty K_0(s\, n)
\sim
\frac{-2\log 2 - \log \pi + \gamma}{2} + \frac{\log s}{2} + \frac{\pi}{2\, s}, \quad s\to 0^+,$$
if taking
$$\...
- 21
2
votes
1
answer
110
views
Asymptotic behavior in a modular color-cycling problem
Consider the following problem: We have $k$ rooms, each equipped with a light that cycles through three colors – red, green, and blue – in a cyclic order. Initially, all lights are set to red. Each of ...
2
votes
1
answer
197
views
Interpretation of an equivalence to the Riemann hypothesis due to de Reyna and Toulisse in the spirit of a formula from an article
In [1] the authors present an equivalence to the Riemann hypothesis that is the Theorem 6.2.
On the other hand I know a statement from [2], in English this is the article Andrew Granville and Greg ...
2
votes
2
answers
385
views
What is the growth rate of the sum of powers of distinct primes closest to a given a integer?
Let $n$ be a positive integer, and
$$2 = p_1 < p_2 < \dots < p_m \le n$$
be the sequence of all primes less than or equal to $n$.
For each index $j$ let $p_j^{e_j}$ be the largest power of $...
- 557
2
votes
1
answer
338
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Bombieri-Vinogradov in short intervals
In 1985 Perelli, Pintz & Salerno proved a short-interval form of the Bombieri-Vinogradov theorem with $\theta \in (7/12, 1]$. Have there been any improvements on this, in particular with the ...
- 338
2
votes
1
answer
191
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Sums of multiplicative functions over residue classes
It was stated in this Shiu, P. work, page 169, Theorem 2, that $$\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}.$$
Here, $d_r(n)...
user536681
2
votes
2
answers
590
views
Asymptotic formula for the number of ways to write a number as the sum of $k$ triangular numbers
How would one derive an asymptotic formula for the number of representations of a number $n$ as the sum of $k$ numbers of the form $\frac{m(m + 1)}{2}$
I think that one could use the circle method, ...
- 1,890
2
votes
0
answers
179
views
A Brun-Titchmarsh type result for divisor sums; asymptotic/improved bound
In Shiu's work ('A Brun-Titchmarsh theorem for multiplicative functions') he proved that if $r\le x$ is a natural number, we have $$\sum_{r<n\le x}d(n)d(n-r)\ll x\log^2x\sum_{d|r}\frac{1}{d}.$$
I ...
user344548
2
votes
0
answers
195
views
Asymptotics of $\vartheta(x+y)-\vartheta(x)$, where $\vartheta$ is the Chebyshev function, when $y\in[x^\alpha, x]$ for some $\alpha\in(0,1)$
Introduction
Consider $\vartheta$ to be the Chebyshev function, that is, $\vartheta(x)$ denotes, for $x\in\mathbb N$, the sum $\sum_{p\le x,\ p\text{ prime}} \ln p$.
I am interested in asymptotics for ...
- 1,326
2
votes
0
answers
253
views
Asymptotic behaviour of a sum involving Möbius function
(This is a cross-posted simplification of this question posted in MSE which did not have a complete answer.)
I am trying to get the asymptotic behaviour when $n$ grows to infinity of a partial sum of ...
- 189
2
votes
0
answers
422
views
Sequences with high densities of primes: how to boost them to get even more and larger primes
I propose a methodology to help find large prime numbers with a much higher probability than picking up random numbers and testing them for primality. This would help speed up prime number generators ...
- 3,098
2
votes
0
answers
203
views
Asymptotics on a double sum over primes
I am attemping to find asymptotics of
$$\sum_{p \leq n}\ln p \left( \sum_{k=1}^\infty \left(\left\{\frac{n}{p^k} \right\} - \left\{\frac{n-1}{(p-1)p^k} \right\} \right) - \left\{\frac{n-1}{p-1} \...
- 1,549
2
votes
0
answers
110
views
On variations of a claim due to Kaneko in terms of Lehmer means
This post is cross posted from Mathematics Stack Exchange, due that there was a mistake from my part (see the excellent partial answer and my thread of edits of my question on MSE) this post on ...
2
votes
0
answers
167
views
What about series involving strong primes?
I know about the importance in analytic number theory of the sutdy of series involving prime numbers or constellations of prime numbers, for example, if I am not wrong, major theorems are Mertens' ...
2
votes
0
answers
163
views
Generalization of regularly varying functions
A continuous function $L :\mathbb{R}_+ \to \mathbb{R}_+$ is called regularly varying (at infinity) if for all $a > 0$,
$$
\lim_{x\to \infty} \frac{L(ax)}{L(x)}= g(a)
$$
for some function $g(a)&...
- 3,223
2
votes
0
answers
210
views
A sum with integer parts
Let $ \mathcal{A} $ be a set of reals such that $ \sum_{a \in \mathcal{A} } \frac{1}{a} = \infty $ and $ \sum_{a \in \mathcal{A} } \frac{1}{a^2} < \infty $. For instance, $ \mathcal{A} = \mathbb{N}^...
- 593
2
votes
0
answers
571
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Integrating a product of integrals involving Bessel functions
I have asked similar questions on Math Stack Exchange, but not been able to receive many helpful responses. Therefore, I am posting this problem here, and any input would be extremely valuable.
I ...
- 161
2
votes
0
answers
110
views
Bounds on the number of zeros of a quadratic form
Let $Q(x_1, \dots, x_n)$ be a non-degenerate indefinite quadratic form with integer coefficients. Let $N(Q,T)$ be the set of vectors $x=(x_1, \dots, x_n) \in {\mathbb Z}^n$ such that $|x|<T$ and $Q(...
- 6,224
1
vote
1
answer
867
views
$n$th prime: a better approximation
Let $p_n$ be the $n$-th prime, then from Wikipedia I got that
$p_n \approx n \left(\ln n + \ln \ln n -1 + \frac{\ln \ln n-2}{\ln n}+\frac{6\ln \ln n-( \ln \ln n)^2-11}{\ln^2 n} \right)$.
What is a ...
user95470
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 ...
1
vote
2
answers
191
views
On non-singularity of integer matrices with bounded entries
Given $B>0$ and $n\in\Bbb N$ what is the probability that a given $n\times n$ integer matrix with all entries bound by absolute value $<B$ is non-singular? I am looking for precise scaling.
- 13.9k
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 ...
- 577
1
vote
1
answer
356
views
Asymptotics for a special sum with a divisor function
Is there an asymptotic for: $$\sum_{n\leq x\atop n \not\equiv 0(mod\ 2)}d(n)\ \mathrm{\mathbf{and}}\ \sum_{n\leq x\atop n \not\equiv 0(mod\ 2,3)}d(n)$$
I need this for this https://mathoverflow.net/...
user102007
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 $T \to \infty$.
$$ \sum_{t = 1}^{T} \prod_{\substack{p \; \textrm{prime} \\ p | t}} \rho(p) \frac{1 - \frac{1}{p^2}}{1 - \frac{\rho(p)...
- 577
1
vote
1
answer
258
views
Sum of divisors of Stirling numbers of the second kind
In this post we denote the Stirling number of the second kind as ${n\brace k}$, I add as reference the article Stirling numbers of the second kind from the encyclopedia Wikipedia. And we denote the ...
1
vote
1
answer
196
views
A question about a sum that involves gaps between twin primes, on assumption of the First Hardy–Littlewood conjecture
I wondered, inspired in a result mentioned from [1] (page 45), what should be the asymptotic behaviour of the sequence on assumption of the First Hardy–Littlewood conjecture
$$\sum_{\substack{\text{...
1
vote
1
answer
141
views
Compare $\operatorname{rad}(an+b)$ and $\varphi(cn+d)$ in a simple and interesting inequality, for some choice of integers $a,b,c$ and $d$
We denote for an integer $n>1$ its square-free kernel as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p,$$
with the definition $\operatorname{rad}(1)=1$. You can see this ...
user75829
1
vote
1
answer
230
views
Asymptotic for a number theoretic sequence and its Dirichlet series' convergence
I would like to know the asymptotic behaviour at large $n$ for $t\in\mathbb{R}$, $t\neq0$ of the following function:
\begin{align*}
A_n(t)&=\sum_{q=\frac{a}{b}\in \mathbb{Q}^+|\gcd(a,b)=1 \& ...
- 634
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 $s=1$
$$ \zeta(s) = \frac{1}{s-1} + \sum_{n=0}^{\infty} (-1)^n \frac{\gamma_n}{n!} (s-1)^n $...
- 2,689
1
vote
0
answers
128
views
On Zudilin's linear forms in $1,\zeta(5)$ and $\zeta(7)$
I am reading an article "Well-poised hypergeometric service for diophantine problems of zeta values" by W. Zudilin.
Consider the quantities defined here in pg. $617$
$$\tilde{F_n}:= \frac{1}{...
- 11
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
1
vote
0
answers
86
views
Asymptotic behaviour of $\sum_{k=1}^{n}\frac{R_{k+1}+R_k}{R_{k+1}-R_k}$, where $R_k$ are the Ramanujan primes
I was inspired in the following post from this Math Overflow that was asked yerterday Asymptotic behavior of a certain sum of ratios of consecutives primes to ask about a similar question for ratios ...
1
vote
0
answers
133
views
On a continuous function as a substitute of the prime-counting function in the second Hardy–Littlewood conjecture satisfying certain asymptotics
It it well-known that the prime-counting function $\pi(x)$ satisfies the prime number theorem and that were in the literature two related conjectures to this arithmetic function, these are: the ...
1
vote
0
answers
315
views
From an inequality for the Euler's totient function to a combination of Firoozbakht's conjecture and Nicolas' criterion for the Riemann hypothesis
In this post we ask about the veracity of an inequality deduced from a combination of Firoozbakht's conjecture (see [1] or [2]) and Nicolas' criterion for the Riemann hypothesis (see for instance [3])....
1
vote
0
answers
129
views
Asymptotic of $\sum_{1\leq n\leq x}a_n$ where $\exp(\sum_{n=1}^\infty\alpha\operatorname{rad}(n)n^{-s})=\sum_{n=1}^\infty\frac{a_n}{n^s}$
Yesterday I tried to study the article [1] in wich were showed incredible expressions related to Dirichlet series. In the same way I wondered about next question.
We denote for integers $m>1$ the ...
1
vote
0
answers
28
views
Asymptotic size for the number of terms not exceeding $n$ in the class $r$ for a classification of the type Erdös-Selfridge for square-free integers
It is possible to define a classification similar than the Erdös-Selfridge classification of primes for different sequences. Please ee [1], section A18 and the references cited in this book. Because ...
1
vote
0
answers
101
views
Size of a set defined by divisor function
After some computations, I guessed the following conjecture.
How can I prove or disprove it? thanks!
Let
$$
A(k)=\#\left\{\left(t,\frac{k+t+a}{4t-1}\right):1\leq t\leq k,\ 1\leq a\leq k+t,\ a\mid(k+...
- 841
1
vote
0
answers
195
views
Asymptotics for certain sum involving the divisor function, Ramanujan sum
Let $c_q(n)$ be the Ramanujan sum, and let $\tau(n)$ be the divisor function. Is there an asymptotic formula for $$\sum_{n\le x}\tau(n)c_q(n)$$ with error terms that do not depend on $q$?
- 1,890
1
vote
0
answers
78
views
Related to derivative of Modified Bessel I function wrt the order
I recently met some problems related to the modified Bessel I funtions. Let $I(\nu,x):=I_\nu(x)$, and $I'_\nu(\nu,x):=\dfrac{\partial}{\partial \nu}I(\nu,x)$.
Using maple, it seems that $Re(\dfrac{I'...
0
votes
1
answer
404
views
Asymptotics of "ugly" function elucidate Goldbach's conjecture?
Question
We now define the following "ugly" function:
$$ A_c(s,r,n,m) =
\begin{cases}
1 & \text{ if only $sr+nm=2c$ } \\ 0 & \text{otherwise}
\end{cases}
$$
How does the "ugly"...
- 89
0
votes
1
answer
149
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Asymptotic of $\sum_{k=1}^n \operatorname{rad}(k!)$ and similar deductions
We denote for integers $m>1$ the product of the distinct prime numbers dividing $m$ as $$\operatorname{rad}(m)=\prod_{\substack{p\mid m\\p\text{ prime}}}p,$$
with the definition $\operatorname{rad}(...
0
votes
1
answer
204
views
On $(\prod_{\substack{1\leq s\leq X\\s\text{ semiprime}}}s)(\sum_{\substack{1\leq s\leq X\\s\text{ semiprime}}}\frac{1}{s})$ as $X\to\infty$
Few weeks ago an user from Mathematics Stack Exchange answered my question On an inequality that involves products and sums related to the sequence of semiprimes (asked May 26). It seems that for ...