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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.
4
votes
Accepted
Series of the inverse quadratic trinomial
This can be expressed in terms of elementary functions, if $p/2$ is an integer. Suppose, for example that it is a positive integer. Then your sum is
$$S:=\sum_{m=p/2+1}^\infty\frac{1}{m^2+c}=\frac{1}{ …
- 91.8k
2
votes
Zeros of polynomial approximations of the Riemann $\zeta$ function
Since $\zeta$ has a single pole, at $z=1$, the radius of convergence of the Taylor series at $c$ is $r=|c-1|$. Moreover, if
$$\zeta(z)=\sum_0^\infty a_n(z-c)^n$$
is the Taylor expansion at $c$, then t …
- 91.8k
4
votes
Accepted
Turan Inequalities
These were first proved by Laguerre himself (They are called sometimes Laguerre's inequalities). The LP class can be characterized as the closure of real polynomials with real zeros, so it is enough t …
- 91.8k
6
votes
Does there exist a rational polynomial $P(x)\in{\mathbb Q}[x]{}$ such that $P(\zeta(s))=\zet...
Here is a simple argument which applies to polynomials in $C[x]$, and even to most rational functions in $C(x)$. $\zeta$ is a meromorphic function in the plane.
So the Nevanlinna characteristic $T(r,\ …
- 91.8k
79
votes
Is the Riemann zeta function surjective?
$\zeta$ function has only one pole at $z=1$. It also has order $1$. If $\zeta$ omits $c\in C$ then $g:=1/(\zeta-c)$ is entire with one simple zero at $1$.
As it is of order $1$, it must be $g(z)=(z-1) …
- 91.8k
11
votes
Accepted
Three questions about three functions similar to $\sin,\cos$
There is no addition formula: functions satisfying an algebraic addition formula have been completely characterized,
Painlevé, P.
Sur les fonctions qui admettent un théorème d’addition, Acta Math. 27, …
- 91.8k
2
votes
Zeros of the derivative of $\xi$
Assuming the Riemann hypothesis, the $\xi(1/2+iz)$-function belongs to the Laguerre-Polya class (defined as the closure of polynomials with all zeros real). This follows from the parametric descriptio …
- 91.8k
4
votes
Books on complex analysis for self learning that includes the Riemann zeta function?
All your requirements (except "geometric beauty") are satisfied by the book of
John Stalker,
Complex Analysis, Springer 1998.
It is written by a mathematical physicist who loves formulas and number …
- 91.8k