All Questions
Tagged with analytic-number-theory number-theory
1,945
questions
1
vote
1
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140
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Given the primes, how many numbers are there?
I have a concrete question and a more abstract version of the same. There are lots of theorems that take information on the number of elements and translate it to information on the primes; I’m ...
- 32.3k
-1
votes
1
answer
70
views
How to give this sum a bound?
Let $x,y\in\mathbb{Z},$ consider the sum below
$$\sum_{x,y\in\mathbb{Z}\\ x\neq y}\frac{1}{|x-y|^{4}}\Bigg|\frac{1}{|x|^{4}}-\frac{1}{|y|^{4}}\Bigg|,$$
is there anything I could do to give this sum a ...
- 329
7
votes
1
answer
711
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What's the easiest way to prove the Riemann Zeta function has any zeros at all on the critical line?
I learned in my intro to complex analysis class that the Riemann Zeta has trivial zeros at the negative even integers; this follows from the Riemann Zeta Reflection identity and the behavior of the $\...
- 1,290
2
votes
1
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53
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The "Euler Product formula" for general multiplicative functions
For the totient function $\phi$, we have the well known "Euler's product formula" (as named on Wikipedia) $$\phi(n) = n \prod_{p | n} \left( 1 - \frac{1}{p} \right)$$
This is easy to show ...
1
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0
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57
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Periodic zeta function
Let $e(x)=e^{2\pi ix}$ and let $$F(x,s)=\sum _{n=1}^\infty \frac {e(nx)}{n^s}$$ be the periodic zeta function.
What is the functional equation for the periodic zeta function ?: I can find a statement ...
- 1,662
1
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1
answer
55
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$H(x)$ approximates $\pi(x)$ pretty well. But what are the drawbacks, when compared with Riemann's $R(x)$?
I'm aware of the Gram series which is equivalent to $R(x)$ (Riemann prime counting function):
$$ R(x)=\sum_{n=1}^\infty \frac{\mu(n)}{n}li(x^{1/n}). $$
Over the interval $x=2$ to $x=10^4$ the average ...
- 754
3
votes
0
answers
45
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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 ...
- 157
0
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39
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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 $\log\log(n)$, ...
- 43
1
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1
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63
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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 $-Z(2)+\frac{Z(2)^2}{Z(3)}$...
- 17
1
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1
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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
1
vote
1
answer
72
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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
$$\sum_{n\equiv a~\mathrm{ mod }~q}{V}\left( ...
- 39
4
votes
0
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127
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If $\zeta(5)=\frac{a}{b}\in\mathbb{Q}$ then $b\neq p^k$
If $\zeta(5)=\frac{a}{b}\in\mathbb{Q}$ then prove that $b\neq p^k$ where $p$ is any prime and $k\in\mathbb{N}$
Take $a,b\in\mathbb{N}$ such that $(a,b)=1$. Now if $b=p^k$ then $$p^k\zeta(5)=a$$ So, by ...
- 928
4
votes
1
answer
140
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Closed form of $\int_{(0,1)^5} \frac{xyuvw}{(1-(1-xy)u)(1-(1-xy)v)(1-(1-xy)w))}\, dx\,dy\,du\,dv\,dw$
I need closed form for the integral $$I:=\int_{(0,1)^5} \frac{xyuvw}{(1-(1-xy)u)(1-(1-xy)v)(1-(1-xy)w))} \, dx\,dy\,du\,dv\,dw$$
Then $$0<I<\int_{(0,1)^5} \frac{1}{(1-(1-xy)u)(1-(1-xy)v)(1-(1-xy)...
- 928
4
votes
1
answer
192
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Does this function in $3$b$1$b has a name?
I was watching this video from $3$Blue$1$Brown channel, at minute 21:00 he introduced the following function:
$$
\chi(n)=
\begin{cases}
0 & \text{if } n=2k \\
1 & \text{if } n=4k+1\\
-1 & \...
- 6,100
0
votes
0
answers
94
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Expressing a function in terms of the nontrivial zeros of the Riemann zeta function
Consider the function $\phi(x)$:
$$ \phi(x)=\sum_{n=1}^{\infty}\frac{\mu(n)}{n^{2}}\left(\left\{ \frac{x}{n}\right\}-\frac{1}{2}\right)$$
$\left\{\cdot\right\}$ being the fractional part function. ...
