All Questions
Tagged with analytic-number-theory asymptotics
333
questions
2
votes
2
answers
111
views
Asymptotic Formula of Selberg
I'm new to asymptotic operation so I need help to understand it. As I know $\mathcal{O(x)}$ is a set of functions. In Selberg's paper about elementary proof of prime number theorem there is that ...
user1127460
1
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1
answer
87
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The sum $f(x)= \sum_{p \le x} e^{\frac{1}{p\log p}}$ is very close to $\pi(x)$. Why is that?
The prime counting function $\pi(x)$ counts the number of primes less than a given $x$. There are other counting functions like Chebyshev's functions which count sums of logarithms of primes up to a ...
- 754
0
votes
1
answer
60
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Lower bound for the prime zeta function
The prime zeta function is defined for $\mathfrak{R}(s)>1$ as
$P(s)=\sum_{p} \ p^{-s}$, where $p \in \mathbb{P}$.
It is well-know this series converges whenever $\mathfrak{R}(s)>1$.
Now, ...
- 85
2
votes
0
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Does the limit of the exponential mobius exponential series asymptotically equal its regularized power series?
Context:
Consider the function $\sum_{n=0}^{\infty} e^{nx}$. An extremely unrigorous manipulation of this series would yield
$$ \sum_{n=0}^{\infty} e^{nx} = \sum_{n=0}^{\infty} \sum_{k=0}^{\infty} \...
- 17.5k
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1
answer
40
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On $- \dfrac{\zeta'}{\zeta} (1-c+it) - \dfrac{\zeta'}{\zeta} (c-it)$
In Gonek's paper the following is claimed: $$ - \dfrac{\zeta'}{\zeta} (1-c+it) = \dfrac{\zeta'}{\zeta} (c-it) + \log \dfrac{t}{2 \pi} + O(\dfrac{1}{t}) \tag{$*$} $$
According to Titchmarsh's book Ch4,...
- 281
1
vote
2
answers
134
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Dyadic sum of $\frac{x}{\ln x}$ (i.e. dyadic asymptotic for prime number theorem)
For $x\geq 10$, denote $j=j_x$ s.t. $4 \geq \frac{x}{2^j}\geq 2$. I want to prove that
$$\sum_{i=0}^j \frac{x/2^i}{\log(x/2^i)} \sim \frac{2x}{\log x}$$
The reason I care is because the prime number ...
- 8,945
1
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1
answer
106
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How to show that $O(\sum_1^{\infty} n^{1-{2\sigma}} e^{-\delta n} \sum_1^{n/2} 1/r )=O({\delta}^{2 \sigma -2} \log \dfrac{1}{\delta})$?
The following lemma is from Titchmarsh's The Theory of the Riemann Zeta-Function:
I have difficulties in getting both the estimates:
1- $O((\sum_1^{\infty} n^{-{\sigma}} e^{-\delta n})^2) = O((\int_1^...
- 281
1
vote
1
answer
65
views
Sum over primes in a paper of Selberg
The above is from paper of Selberg on the elementary proof of prime number theorem. Prior to eq.$(2.11)$ the author derives an asymptotic expression for $\sum_{pq≤x}\ln(p)\ln(q)$, where $p,q$ are ...
- 1,521
9
votes
2
answers
319
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An accurate, rapidly converging estimate of $\sum_{j = 1}^{\infty}\frac 1 {j^2}$ (the Basel Problem) using only elementary calculus
Although the Basel Problem $$\zeta(2) = \sum_{j = 1}^{\infty}\frac 1 {j^2}$$ took a hundred years to solve analytically, using elementary calculus we can easily get strikingly accurate bounds: $$\zeta(...
- 4,450
1
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1
answer
137
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Inequality from Selberg's proof of the PNT
Selberg states the incorrect inequality
$$\sum_{p\leq x}\frac{\log p}{x}=\log x+O(1)$$
that of course refers to the inequality
$$\sum_{p\leq x}\frac{\log p}{p}=\log x+O(1)\tag*{(1.4)}$$
and says that ...
0
votes
1
answer
70
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A multiparameter inequality
Let $x\ge 2$ be a real number and $\nu=:\nu(x)$ an integer-valued function. Let $I$ be an interval of $\mathbb{R}$ such that $\lambda(I)<(\log\nu)^{-1/2}$ where $\lambda$ denotes the Lebesgue ...
- 11
3
votes
1
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83
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What is the order of $\mathrm{Z}(x)-\pi(x)?$
Consider the offset logarithmic integral which approximates the number of primes up to a given $x$ quite well $$ \mathrm{Li}(x)=\int_2^x\frac{1}{\log t}~dt$$
And consider the alternating series
$$\...
- 754
2
votes
2
answers
169
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Cauchy sequences converge
Statement (taken from here):
Let ${f: {\bf N} \rightarrow {\bf C}}$, ${F: {\bf R}^+ \rightarrow {\bf C}}$ and ${g: {\bf R}^+ \rightarrow {\bf R}^+}$ be functions such that ${g(x) \rightarrow 0}$ as ${...
- 2,008
4
votes
2
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166
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Asymptotics of $p_k$-adic valuation of the sum of the divisors of the $n$-th primorial
Given this product:
$$a(n) = \prod_{k=1}^{n} (1+p_k)$$
where $p_k$ is the $k$-th prime number and which can be interpreted also as the sum of the divisors of the $n$-th primorial (OEIS A054640), is ...
- 2,103
1
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0
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A question from Titchmarsh's book " The Theory of the Riemann zeta function, Theorem 9.16, page 231
I am studying about upper bounds for $N(\sigma, T)$ (zero-density estimates), and while going through Theorem 9.16 in Titchmarsh's book (2nd edition) (page 231), I got a bit stuck in understanding the ...
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