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Results tagged with riemann-zeta-function
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user 37555
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.
5
votes
Analogues of the Riemann zeta function that are more computationally tractable?
Let $p_k$ be the $k$-th prime number, and pick a sequence of primes $q_k$, such that $q_k\sim p_k^{3/2}$. Let $G$ be the arithmetic semigroup consisting of all integers not divisible by one of the $q_ …
4
votes
On an observation which relates to the exponential sum $\sum_{n=1}^{[\sqrt{t/2\pi}]} n^{-\fr...
The assertion is generally believed to be true. In fact, much more is conjectured: For every $\epsilon>0$ we have $\left|\zeta(\frac{1}{2}+it)-\sum_{n\leq t^\epsilon} n^{-\frac{1}{2}-it}\right|= \math …
5
votes
Accepted
Can there be more than two zeta zeros in between a Gram point and a França-LeClair point?
It is generally believed that a positive proportion of zeros of $\zeta$ satisfy your condition. In fact, for each fixed $k$, random matrix theory predicts a distribution of the renormalized tuples $(\ …
1
vote
Special values of $\zeta$ outside the real line and the critical strip
There is a well known conjecture that the ordinates of the non-trivial zeros of $\zeta$ are $\mathbb{Q}$-linearly independent.
There are two major motivations for this conjecture. First all numbers …
1
vote
Estimating the height required to find a given small value of $|\zeta(s)|$ near the line $\s...
It suffices to construct points in the rectangle with $|\zeta(s)|\leq\delta$. This can be done, even to the right of 1, by diophantine approximation: Pick some $t$, such that for the first $k$ primes …