Questions tagged [analytic-number-theory]
Questions on the use of the methods of real/complex analysis in the study of number theory.
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Asymptotic formula for sum of $\frac{1}{n^\lambda}$ over square free $n\leq x$
I'd like to get an asymptotic formula for $\sum_{\substack{n\leq x\\ n\text{ square free}}}\frac{1}{n^l}$ for $l>0$.
We know that $\sum_{\substack{n\leq x\\ n\text{ square free}}}1= cx+O(\sqrt{x})$....
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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
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Problems in analytic number theory
There is one computation I am struggling with. I quote "Problems in analytic number theory" page 129:
"It is clear that as $\sigma \rightarrow 0^{+}$ log$\zeta(1+\sigma) = log(\frac {1}{...
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Number of representations of an integer by a binary quadratic form
In a paper by Heath-Brown, upon having to estimate the number of solutions $(x,y)\in\mathbb Z^2\cap[-B,B]^2$ to the equation $Q(x,y)=k$ with $Q$ an integer-coefficient non-degenerate quadratic form ...
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Normal Order of Distinct Prime Factor $\omega(n)$
Define $\omega(n)$ as number of distinct prime factors $n$ has, that is if $n=p_1^{a_1}... p_k^{a_k}$, then $\omega(n)=k$.
It is commonly understood that normal order of $\omega(n)$ is $\log\log(n)$, ...
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lower bound of $\sum_{n=1}^x \frac{\mu(n)}{n}$
Denote by $\mu$ the Mobius function. Poussin showed that
$$
\sum_{n=1}^x \frac{\mu(n)}{n} = O(1/\log x),
$$
and there are further improvements since. I wonder what is known about lower bound of ...
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Non negativity involving sequences
Define for $n\in\mathbb{N}$ $$a_n=\left[n\sum_{k=1}^{n}\frac{1}{k^5}\right]$$ where $[x]$ denotes the greatest integer $\leq x$.
Prove that $$(n+1)a_n-n a_{n+1}+1\geq 0 \ \ \forall n\geq 1$$
$$(n+1)...
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$\lim_{n\to\infty}\left\{n\sum_{k=1}^n\frac{1}{k^5}\right\}$
I need to calculate the limit (if it exist) $$\lim_{n\to\infty}\left\{n\sum_{k=1}^n\frac{1}{k^5}\right\}$$ where $\{x\}$ denotes the fractional part of $x$ and $n\in\mathbb{N}$.
By definition of ...
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How do you parameterize simultaneous solutions to equations with expressions like "$ x +2 \left\lfloor\frac{x}{3}\right\rfloor + 1 - [3 \mid x]$"?
Let all functions be integer functions herein. I.e. $\Bbb{Z}\to\Bbb{Z}$ or $\Bbb{N}\to\Bbb{Z}$ where appropriate.
I found this jewel of floor functions.
So that made me wonder whether, we can solve ...
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Some questions about Fousseraus proof of $\pi(x)=o(x).$
Below is a well known Corollary from Analytic number theory and a proof (excerpt) by G. Fousserau (1892) which I have found here: Narkiewicz. (2000). The Development of Prime Number Theory on page 13.
...
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Write the sum in terms of the Riemann zeta function
I believe it is a question from JHMT.
Write the sum $\sum_{a=1}^\infty\sum_{b=1}^\infty\frac{gcd(a,b)}{(a+b)^3}$ in terms of Riemann zeta function. The answer should be $-Z(2)+\frac{Z(2)^2}{Z(3)}$...
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The sum $f(x)= \sum_{p \le x} e^{\frac{1}{p\log p}}$ is very close to $\pi(x)$. Why is that?
The prime counting function $\pi(x)$ counts the number of primes less than a given $x$. There are other counting functions like Chebyshev's functions which count sums of logarithms of primes up to a ...
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Dedekind zeta function of abelian number fields is a product of Dirichlet L-functions
Can somebody give an explicit (and self-contained) proof of the fact that the Dedekind zeta function of an abelian number field $K$ is a product of Dirichlet L-functions? I.e.,
$$
\zeta_K(s)=\prod_{\...
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A special kind of Poisson summation formula
There is a Poisson summation formula as follows:
Let $V$ be a smooth function with compact support on $\mathbb{R}$. For $X > 1$ and $q > 1$, we have
$$\sum_{n\equiv a~\mathrm{ mod }~q}{V}\left( ...
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If $\zeta(5)=\frac{a}{b}\in\mathbb{Q}$ then $b\neq p^k$
If $\zeta(5)=\frac{a}{b}\in\mathbb{Q}$ then prove that $b\neq p^k$ where $p$ is any prime and $k\in\mathbb{N}$
Take $a,b\in\mathbb{N}$ such that $(a,b)=1$. Now if $b=p^k$ then $$p^k\zeta(5)=a$$ So, by ...
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