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Results tagged with riemann-zeta-function
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The Riemann zeta function is the function of one complex variable $s$ defined by the series $\zeta(s) = \sum_{n \geq 1} \frac{1}{n^s}$ when $\operatorname{Re}(s)>1$. It admits a meromorphic continuation to $\mathbb{C}$ with only a simple pole at $1$. This function satisfies a functional equation relating the values at $s$ and $1-s$. This is the most simple example of an $L$-function and a central object of number theory.
10
votes
2
answers
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$\psi(x)-x$ on average
This is a reference question:
Let $\psi(x)$ be the psi-Chebyshev function. Is there any unconditional result in the literature that proves that there exists $0<a<2$ such that
$$
\int_2^x (\psi(y)-y)^2 …
- 3,062
3
votes
0
answers
239
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Riemann hypothesis and ternary Goldbach
Is there any result of the following shape: There exists an absolute constant $\delta>0$ such that the Riemann hypothesis for some $L$-functions is equivalent to the following estimate for all fixed $ …
- 3,062
11
votes
2
answers
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Is there an explicit expression for the imaginary part of some non-trivial zero of zeta,
Is there an explicit expression for the imaginary part of some non-trivial zero of zeta, in terms of well-known constants, such as say $\gamma$ or $\pi$ say ?
- 3,062
1
vote
2
answers
129
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Bounds of zeta function near $\Re(s)=1$
Richert proved in
https://link.springer.com/article/10.1007/BF01399533
that $$ \zeta(s) =O\left( |\Im(s)|^{100(1-\Re(s))^{3/2}} (\log |\Im(s)|)^{2/3}\right)$$ uniformly in the region $\Re(s)\in [1/2,1 …
- 3,062
1
vote
1
answer
180
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Conjectured error term when counting square-free integers
It is well know that the density of positive square-free integers up to $x$ is asymptotically $x/\zeta(2)$. The error term
$$ E(x)=\sum_{1\leq n \leq x } \mu(n)^2 -\frac{x}{\zeta(2)}
$$ can easily p …
- 3,062