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Results tagged with asymptotics
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user 142929
Asymptotic behavior of functions, asymptotic series and related topics
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On $\sum_{\substack{1\leq d\leq x\\(d,r)=1}}\frac{\mu(d)}{d}W(\frac{x}{d})$, with $\mu(n)$ t...
I wondered if it is possible to posed a similar question than Question 2 by Olivier Ramaré from [1] (page 231), although the computational evidence that I have for my conjecture is very small.
Conject …
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Bound for $\left|\sum_{\substack{1\leq d\leq x\\(d,r)=1}}\frac{G_d}{d^{1+\varepsilon}}\right...
In this post we denote the Gregory coefficients, or reciprocal logarithmic numbers, this Wikipedia Gregory coefficients as $G_k$, for integers $k\geq 1$. I would like to know if it is possible to get …
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Interpretation of an equivalence to the Riemann hypothesis due to de Reyna and Toulisse in t...
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 M …
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129
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Asymptotic of $\sum_{1\leq n\leq x}a_n$ where $\exp(\sum_{n=1}^\infty\alpha\operatorname{rad...
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 pro …
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148
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Asymptotic of $\sum_{k=1}^n \operatorname{rad}(k!)$ and similar deductions
Deduce a statement about the asymptotics of the sequence $$\sum_{k=1}^n \operatorname{rad}(k!)$$ as $n\to\infty$. Many thanks. … Is it possible to get a similar statement about the asymptotics of sequences of the type $$\sum_{k=1}^n \operatorname{rad}(f(k))$$ as $n\to\infty$ for examples of continuous and increasing functions $f …
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217
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The graph and sign of $p_n-\operatorname{ali}(n)$, where $p_n$ is the $n$-th prime and $\ope...
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 wil …
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288
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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)=\sum …
4
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612
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What work can be done to study the solutions of $\varphi\left(x^{\sigma(x)}\sigma(x)^x\right...
of them) should be to create intrincated/artificious equations similar than \eqref{1} involving the sum of divisors functions and the Euler's totient function with the purpose to invoke inequalitites, asymptotics …
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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, …
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What about series involving strong primes?
$p$ the sequence of strong primes A051634 I would like to know if it is possible to do some work about the behaviour of $$\sum_{p\leq X}\left(\frac{1}{p}+\frac{1}{p+2}\right)$$
for $X$ very large (asymptotics …
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253
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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 su …
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A question about a sum that involves gaps between twin primes, on assumption of the First Ha...
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{p …
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130
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On a continuous function as a substitute of the prime-counting function in the second Hardy–...
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 Rieman …
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109
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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 MathOv …
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On $(\prod_{\substack{1\leq s\leq X\\s\text{ semiprime}}}s)(\sum_{\substack{1\leq s\leq X\\s...
I think that this post can be interesting to me as companion of the post of MSE, and I don't know if similar expressions as $(1)$ (I mean inequalities as the Lemma from [1] or asymptotics as our Question …