1
vote
0answers
15 views

Asymptotic behavior of sums of consecutive powers (bivariate)

Are there some (bivariate) closed form formulas for the asymptotic behaviour of the sum: $\sum_{k=1}^{n} k^d$ where $n$ and $d$ are large integers? I am especially interested in a lower bound of ...
1
vote
0answers
24 views

Question about finite sums and integer recursions.

Let $n$ be a positive integer and let $g(n)$ be a given strictly increasing integer function such that $0<g(n)<n$ for all $n$. Also the sequence $ |g(n) - n|$ is unbounded as $n$ grows. Let ...
1
vote
0answers
14 views

asymptotic approximation for number of partitions of integer that do contain 1 nor 2

Hardy and Ramanujan provided a famous asymptotic approximation to $P(n)$ the number of partitions of an integer $n$ when $n$ gets large. I wonder if there is an asymptotic approximation to ...
0
votes
1answer
44 views

Upper bound for the sum $ \sum_{k=1}^N \frac{1}{\varphi(k)}$

Is there an upper bound for the sum $$ \sum_{k=1}^N \frac{1}{\varphi^{\alpha}(k)} $$ where $\varphi(n)$ is the Euler totient function and $\alpha\geq 1$ a real constant? In particular, I'm interested ...
3
votes
1answer
77 views

Asymptotic of a sum evaluation as $ x \to \infty $

Let be the sum $$ \sum_{n\le x}[x/n]=g(x) $$ where $ [x] $ means floor function. My best try for asymptotic is $ g(x) \sim x\log (x)+\gamma x +1$ where I have used the asymptotic $ [x] \sim x $ ...
2
votes
0answers
22 views

Asymptotics of the differences between successive zeta zeros

Does anyone know what the asymptotic of the differences between successive zeta zeros is? Update It appears that $\zeta(n)$ is not a bad asymptotic, when the data range is stretched: ...
2
votes
1answer
51 views

What is the order of the product of $ \frac{p-1}{p} $ under the square root of a prime?

Is there any known asymptotical order for $$ \prod_{p_k\ \text{prime}}^{\sqrt{p_n}} \frac{p_k-1}{p_k} $$
4
votes
1answer
50 views

On the sum of relatively prime number $<N$

Let $A(N)$ be a function which is the sum of all numbers relatively prime and $<N$ and $B(N)$ the sum of remaining $N−\phi(N)$ numbers. Then I have the following questions- Q-1 For what values of ...
2
votes
0answers
269 views

First disagreement in PROUHET THUE MORSE exponentially big?

Let two sequences of integers be $a_1, \cdots, a_n$ and $b_1, \cdots, b_n$ such that with $a_i \in \{1, \cdots n\}$ and $b_i \in \{1, \cdots, n\}$. Let $k$ be the min integer such that $\sum_{i=1}^n ...
3
votes
2answers
79 views

Big O/little o true/false

These are all from Sipser's book, second edition. I was just hoping someone could verify/explain those that are more difficult for me. $2n = O(n)$: true $n^2 = O(n)$: false $n^2 = O(n\log^2 n)$: I ...
1
vote
1answer
141 views

A question about prime gaps

Recently, I have been reading the Wikipedia article about prime gaps (http://en.wikipedia.org/wiki/Prime_gap) and I came across the following: Hoheisel was the first to show that there exists a ...
0
votes
1answer
59 views

Counting function for sums of three squares

Legendre showed that an integer is the sum of three squares if and only if it is not of the form $4^n(8m + 7)$ for some nonnegative integers $n$ and $m$. However, I have been unable to find any ...
2
votes
1answer
76 views

Selberg's Symmetry Formula

I'm going through a proof of the Prime Number Theorem and the derivation of Selberg's Symmetry Formula. However, in it there is one step that is perplexing me. Would anyone be able to help explain why ...
1
vote
1answer
59 views

Proofs of Asymptotics

I'm going through a few proofs to make sure I understand them and in two of the proofs, there is a step I don't understand. 1) In the first, we have that $f(x) = g(x) + ln\left(\frac{\lfloor x ...
2
votes
0answers
41 views

Asymptotics for Mertens function

It seems that the cumulative mean of the Mertens function is very similar in behaviour to $x$ raised to the power of the first zeta zero. I tentatively notate it as: ...
3
votes
0answers
58 views

Which series of numbers effectively translates the factorial to the exponential function?

We have the relation of the Bernoulli numbers $$B_{2n} = (-1)^{n+1}\frac {2(2n)!} {(2\pi)^{2n}} \left(1+\frac{1}{2^{2n}}+\frac{1}{3^{2n}}+\cdots\;\right).$$ For $n>1$, the right hand sum ...
4
votes
0answers
59 views

prime zeta function when $0<s<1$ [duplicate]

I would like to know if there is a good estimate for the sum which concerns all primes not exceeding $x$: $$\sum\limits_{p\leq x}\frac{1}{p^s}$$$$0<s<1$$. Only this. Thanks in advance!
1
vote
1answer
62 views

Showing $\sum_{n\in\mathbb{N}}\frac{\Lambda(n)}{n}=\log (x)+O(1)$ using a given result.

I'm stuck on the following problem. Use the fact that $$\sum_{p\le x}_{p\,\text{prime}}\frac{\log p}{p}=\log (x)+O(1)$$ to show that $$\sum_{n\le x}_{n\in\mathbb{N}}\frac{\Lambda(n)}{n}=\log ...
3
votes
1answer
140 views

Explanation for Terry T. post

I read here that : " If one inserts these inequalities into the Legendre sieve and optimises the parameter, one can improve the upper bound for the number of primes in $[N,2N]$ to $$O \left(\frac{N ...
4
votes
1answer
123 views

Estimating integrals involving $\pi(x)$

While solving an exercise in analytic number theory, I ran into difficulty of estimating an integral of the form $\displaystyle\int_{1}^{x} \frac{\pi(t)}{t} dt$ where $\pi(x)$ is the prime counting ...
1
vote
1answer
56 views

asymptotic of a product

So the question that I'm working on is the following. Show that $\Pi_{p\leq z}(1-\dfrac{1}{p})=\dfrac{C(1+\mathcal{o}(1))}{\log z}$. First off I take logs and just work with the sum and thisis what ...
3
votes
1answer
97 views

Bound of the sum $\sum_{p\le n}\frac{1}{\log(p)}$

While doing a sum I came to the sum $\displaystyle\sum_{p\le n}\dfrac{1}{\log(p)}$. Where the $\log$ is the natural logarithm. It was easy to prove that $\displaystyle\sum_{p\le ...
1
vote
0answers
41 views

Asymptotics for prime factors

Am I correct in assuming that the same result: $$ N_k(x):=\ \mid\{n\leq x : \Omega(n)=k\}\mid \ \sim \frac{x}{\log x}\frac{(\log_2 x)^{k-1}}{(k-1)!}\ (x \rightarrow \infty) $$ also holds for: $$ ...
7
votes
1answer
178 views

Asymptotic formula for almost primes

I have developed a formula for almost primes which is far more accurate asymptotically than Landau's well known $$\pi_k(n) \sim \left( \frac{n}{\log n} \right) \frac{(\log\log n)^{k-1}}{(k - 1)!}$$ ...
4
votes
0answers
165 views

Double harmonic summation

I am interested in determining an asymptotic formula for the double summation of $1/(ab)$, where $a$ is an odd integer ranging between 1 and $k/\sqrt{j}$, $b$ is an odd integer ranging between $a$ and ...
1
vote
0answers
30 views

Simplifying products

Sorry for the very general title, but I don't even know how to name my question. I got a formula which is: $f(n)=\prod_{i = 0}^{\infty} ((n \; \mathrm{rem} \; p^{i + 1}) \; \mathrm{div} \; p^i + 1) ...
1
vote
0answers
95 views

Questions about the superfactorial function.

N superfactorial or $n\$$ is defined as - $$n\$=\prod_{k=1}^n k!$$ Then is there any asymptotic formula for this? Are there any infinite series , integrals related to this function? Is there a ...
0
votes
0answers
45 views

Asymptotic for Taxicab number.

The taxicab numbers are sums of 2 cubes in more than 1 way. First few are - 1729, 4104, 13832, 20683, 32832, 39312, 40033, 46683, 64232, 65728, 110656, 110808, 134379, 149389, 165464, 171288, ...
0
votes
0answers
45 views

Formula numerators of Bernoulli numbers.

I have been working long on Bernoulli numbers but am unable to find a formula to calculate the numerator of Bernoulli Numbers. I have for the denominator - $Denominator(B_{2n})=\prod_{\omega}p$ Where ...
2
votes
1answer
141 views

Asymptotics for the divisor function

I am attempting to understand Tao's post of 23 September 2008 given here concerning the divisor bound. My troubles are when he uses the big-O notation in proving what he lists as bound (4) $$ d(n) ...
7
votes
5answers
427 views

The asymptotic expansion for the weighted sum of divisors $\sum_{n\leq x} \frac{d(n)}{n}$

I am trying to solve a problem about the divisor function. Let us call $d(n)$ the classical divisor function, i.e. $d(n)=\sum_{d|n}$ is the number of divisors of the integer $n$. It is well known that ...
3
votes
2answers
107 views

Order of a function related to divisors

Let $f(n)=\max(\{d(ab):\ a,b\le n\})$ where $d(m)$ is the number of divisors of $m.$ What is the order of $f$? In particular I'm looking for an asymptotic upper bound.
4
votes
0answers
149 views

Understanding Newman's proof of the prime number theorem

I am trying to understand D.J. Newman's proof of the prime number theorem, as presented by D. Zagier. I am not too familiar with analysis, and so there are some things I don't understand. In (III), ...
4
votes
2answers
345 views

Using sum of logarithms of primes to prove the number of primes up to $n$ is $O(n/\log n)$

I need to show that the number of primes up to $n$ (i.e. $\pi(n)$) is $O(n/\log n)$. In the previous exercise of this question I proved that ${\displaystyle \sum_{i=1}^{\pi(n)}\log p_{i}} \leq Cn$ for ...
2
votes
2answers
124 views

Stirling's Approximation

A sharp Stirling's approximation form states that $$n! \sim \left(\frac{n}{e}\right)^n\sqrt{2\pi n}.$$ Use that form to show that: $$\binom{2m}{m} = \Theta\left(\frac{2^{2m}}{\sqrt{m}}\right).$$
3
votes
1answer
123 views

What happened to the Mertens constant in the strong prime twins conjecture ??

To estimate the amount of primes in an interval $\left(2,x\right)$ one might naively sieve by computing $ x \left(1-\dfrac{1}{2}\right)\left(1-\dfrac{1}{3}\right)...\left(1-\dfrac{1}{p_i}\right)$ ...
2
votes
3answers
68 views

Growth of $\frac{1}{m-1} \sum_{t | m-1} t \varphi(t)$

I am interested in bounds for $$S_m = \frac{1}{m-1} \sum_{t | m-1} t \varphi(t)$$ as $m$ gets large.
1
vote
1answer
110 views

Is the sum always bigger than $n^2$?

Let $s(n)$ an arithmetical function defined as $$s(n)=(p_1+1)^{e_1} (p_2+1)^{e_2} \cdots (p_m+1)^{e_m}$$ where prime factorization of $n$ is $n=p_1^ {e_1} p_2 ^{e_2} \cdots p_m^{e_m}$. (For example, ...
4
votes
3answers
245 views

Number of representable as sum of 2 squares

How to find asymptotically (or some reasonable bound, at least $ o(n) $) number of numbers, representable as a sum of squares of 2 numbers? (in case of bound I am interested in both: lower and upper ...
1
vote
1answer
146 views

Inequality for binomial coefficients

Let $m \leq n, n \leq N$ and $0\leq k \leq m$. I am wondering what is the dependence of $n$ and $N$ that for all $m, k$ $$ \frac{{N-m \choose n-k}}{{N \choose n}}\leq 1. $$ Thank you for your help.
0
votes
1answer
63 views

Establishing an Inequality and Possible Circular Reasoning.

Let $0<\varepsilon \ll \delta$. Fix $\delta$. For any $k_0 \in \mathbb{N}$, I can deduce that $$1<\frac{\log n_k}{(1+\delta)^{k-k_0}\log n_{k_0}}<1+\frac{\log7}{\delta\log n_{k_0}}$$ ...
1
vote
1answer
42 views

Growth Rate of the Sequence of Denominators of the Sequence of Principal Convergents of an Irrational Number.

Let $\delta >0$. Take $\theta \in [0,1]-\mathbb{Q},$ let $\lbrace \frac{m_k}{n_k}\rbrace$ be the sequence of principal convergents to $\theta$, obtained from the continued fraction representation ...
4
votes
1answer
208 views

How to show how primorials grow asymptotically?

The primorial $p_n\# $ is defined as the product of the first $n$ primes: $$p_n\# = \prod_{k = 1}^n p_k.$$ Asymptotically, primorials grow like $$p_n\# = e^{(1 + o(1))n\ln n)}.$$ How does one derive ...
3
votes
2answers
219 views

The geometric mean of primes less than or equal to $x$

I want to show that the limit of the geometric mean of primes less than or equal to $x$ is $e$ as $x \to \infty$. Is this correct? Using the product law of logarithms we have $$\ln \prod\limits_{p ...
0
votes
2answers
138 views

Almost certainly incorrect proof about $\prod p$

Let p be prime. Assume (1): $\hspace{10mm} (\prod_{p\leq n} p)^{1/n} \sim e.$ Then $$(e^{\ln \prod p})^{\frac{1}{n}} = e^{(\sum \ln p)/n} \sim e \implies \lim_{n=1}^\infty \frac{e^{(\sum \ln ...
2
votes
2answers
221 views

Estimation of sums with number theory functions

Problem 1. Let $d(k)$ denote the number of divisors of $k\in\mathbb{N}$. Prove that: $$\sum_{k=1}^{n}d(k)=n\ln n +O(n)$$ Problem 2. Show estimation below: ...
0
votes
1answer
146 views

Simple clarification - deduction using big-O notation

A set of lecture notes I'm reading on Halasz's theorem makes the following statement in a proof, which I can't quite follow - I was hoping someone might be able to clear up what I'm missing: ...
4
votes
1answer
104 views

Numbers of the form $a^m-b^n$

Can all positive integers $k$, be written as a difference of two perfect powers $k=a^m-b^n$, with $m,n>1$ and $a,b$ positive integers? A number is imperfect if it can not, which numbers are ...
3
votes
0answers
92 views

Products of primes of the form $an + b$

What is the asymptotic order of numbers divisible by no primes except those of the form $an+b$ ($a$, $b$ fixed)? Surely (except for the trivial cases) they are of order strictly between that of he ...
25
votes
1answer
755 views

How many primes does Euclid's proof account for?

This is a passing curiosity, and I haven't found any duplicates, so I thought I'd share my thoughts. In the most basic (or at least the most famous) proof of the infinitude of prime numbers, due to ...