All Questions
Tagged with cv.complex-variables zeta-functions
30 questions
106
votes
6
answers
19k
views
Why does the Riemann zeta function have non-trivial zeros?
This is a very basic question of course, and exposes my serious ignorance of analytic number theory, but what I am looking for is a good intuitive explanation rather than a formal proof (though a ...
- 29k
74
votes
10
answers
18k
views
Why does the Gamma-function complete the Riemann Zeta function?
Defining $$\xi(s) := \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s)$$ yields $\xi(s) = \xi(1 - s)$ (where $\zeta$ is the Riemann Zeta function).
Is there any conceptual explanation - or ...
- 12.3k
26
votes
3
answers
2k
views
Universality of zeta- and L-functions
Voronin´s Universality Theorem (for the Riemann zeta-Function) according to Wikipedia: Let $U$ be a compact subset of the "critical half-strip" $\{s\in\mathbb{C}:\frac{1}{2}<Re(s)<1\}$ with ...
- 7,127
14
votes
1
answer
1k
views
On meromorphic continuation of zeta function(s) and special values at negative integers
Euler developped (at least) two different approaches in order to calculate the values $\zeta(-m)$ of the zeta function $$\zeta(s) = \sum_{n\geq 1} \frac{1}{n^s}$$ at non-positive integers.
In one ...
- 2,029
12
votes
4
answers
916
views
non-trivial zeros of partial zeta functions
Let $N,a\in\mathbf{Z}_{\geq 1}$. Define a partial $\zeta$-function as
$$
\zeta(s;N,a):=\sum_{\substack{n\geq 1\newline n\equiv a\pmod{N}}} \frac{1}{n^s}
$$
where $Re(s)>1$. Let $\omega$ be either ...
- 7,609
12
votes
1
answer
2k
views
Wick rotation and the Riemann zeta function
The goal of this question is to conceptualize in some way the fact that the Riemann zeta function $\zeta(s)$, and other zeta functions like it, have analytic continuations.
Background
I have by now ...
- 118k
11
votes
1
answer
368
views
Sets with zeta functions that are not the primes
Does there exist a set $S \subset \mathbb N$ such that the Dirchlet density of $S$ is well-defined and positive, the Dirchlet density of $S \cap \operatorname{PRIMES}$ is well-defined and zero, and:
$...
- 148k
8
votes
2
answers
1k
views
Is the maximum domain to which a Dirichlet series can be continued always a halfplane?
Let $f(s)=\sum_n a_n n^{-s}$ be a Dirichlet series whose coefficients satisfy $\lvert a_n\rvert\leq n^{C}$. Then $f(s)$ converges absolutely in some halfplanes, and is conditionally convergent in (...
- 7,836
8
votes
3
answers
2k
views
Evaluating the integral $\int_0^\infty \frac{\psi(x)-x}{x^2}dx.$
Let $\psi(x)=\sum_{n\leq x} \Lambda(n)$ be the weighted prime counting function. I am trying to evaluate the integral $$\kappa:=\int_{1}^{\infty}\frac{\psi(x)-x}{x^{2}}dx$$ in several different ways. ...
- 11.4k
4
votes
1
answer
293
views
Convergence of the summation 1/p^(1+iy) (over all primes p with y a nonzero real number)
For $z\in\mathbb{C}$ with real part greater than $1$ the sum $$\sum_{p}{\frac{1}{p^z}},$$ where the sum is taken over all primes $p$, converges absolutely. It is also well known that the same sum with ...
- 43
4
votes
0
answers
134
views
Converse theorem for zeta universality
Voronin's Universality Theorem for $\zeta(s)$ is that the zeta function can uniformly approximate any non-vanishing holomorphic function to any degree of accuracy in the right-half of the critical ...
- 141
3
votes
1
answer
702
views
Derivative of the Riemann zeta function at $z=-2$
I have a question regarding the derivatives of the Riemann zeta function. It is known that $\zeta'(-1)=\frac{1}{12}-\ln A$, where $A$ is the Glaisher-Kinkelin constant (which is an elegant ...
- 463
3
votes
1
answer
902
views
Is the integral always nonzero?
Let
$$I_{n,p,a}:=\int_0^{\infty-} \frac{g_{n,a}(t)}{t^{p+1}} \, dt,$$
where
$$(*)\qquad\qquad\qquad n\in\mathbb N,\quad -\infty<a<\infty,\quad p_{n,a} < p < n,\qquad\qquad\qquad\qquad\...
- 128k
2
votes
1
answer
224
views
How to compute the following integral $I_{\alpha,\beta}$
We have the following identity (see Bateman, H. (1953). Higher Transcendental Functions [Volumes I], p. 25.)
$$(*)\quad \Gamma(\mu)\, \zeta(\mu,\nu) = \int_{0}^{1} x^{\nu-1} \,(1-x)^{-1} \Bigr(\log ...
- 650
2
votes
1
answer
2k
views
Books on complex analysis for self learning that includes the Riemann zeta function?
I am searching for an introductory book in the field of complex analysis for self learning, that would contain the following:
Analytic number theory : the connection between complex analysis and
...
- 29
2
votes
0
answers
217
views
Zeta function associated with a function $f$
Let the function $f(t) = \cos(at)$, where ($0 < a < 1$). Let us define
$$\zeta(z, f) = \frac{1}{\Gamma(z)} \int_0^{+\infty} \frac{t^{z-1}\cos(at)}{e^t-1}\, dt.
$$
Is there a general formula that ...
- 463
2
votes
0
answers
79
views
Reference request for literature on the following function--power counting zeta function
I'll start by writing the character of interest, and describing some properties of it, before I get to the meat of the question. Any help is greatly appreciated, even an offhand suggestion/comment/...
- 388
2
votes
0
answers
279
views
computing a certain contour integral [closed]
I want to compute an integral along a vertical line segment. The function I'm integrating involves the zeta-function, and usually the way such integrals are done treats the line segment as one side ...
- 139
1
vote
1
answer
284
views
Residues of Zeta-like Function
I'm looking for the residues of the following function $$s \mapsto\sum^\infty_{m,n =1} (m+n) \left[ amn + (m-n)^2 \right]^{-s}$$
at $s=\frac{1}{2}$ and $s=\frac{3}{2}$, where $a$ is some real positive ...
- 145
1
vote
1
answer
159
views
Real points $a∈ℝ$ such that the equation $f^{(k)}(s)=a$ have a finite number of real solutions $s$ for some $k$
Let $$L(C,s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ be the Dirichlet series of the Hasse--Weil L-function of an elliptic curve $C$ over $ℚ$. The modularity theorem implies that $L(C,s)$ is the $L$-...
- 1,197
1
vote
0
answers
102
views
Characterization of turning points for the Ramanujan's zeta function in the spirit of a definition by Arias de Reyna and van de Lune
In [1] the authors provided a definition and characterization of turning points for the Riemann's zeta function. In this post I denote the Ramanujan's zeta function as $$\varphi(s)=\sum_{n=1}^\infty\...
1
vote
0
answers
266
views
Does the property (P) holds true for the derivatives of $L$?
Let $$L(C,s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ be the Dirichlet series of the Hasse--Weil L-function of an elliptic curve $C$ over $ℚ$. As $s$ takes on real negative values, there are trivial zeros ...
- 1,197
0
votes
1
answer
169
views
Analytic extension of the Hurwitz ζ function
For the purpose of formalisation in a theorem prover, I am looking for a simple definition of the analytic extension of the Hurwitz ζ function $\zeta(s,q)$ valid for all $s\in\mathbb{C}\setminus\{1\}$ ...
- 1,241
0
votes
1
answer
168
views
Residue calculation for Eulerian expansion of the cotangent
I am looking for ideas on proving the Eulerian expansion of the cotangent using residue calculation: $$\pi\cot(\pi z)=\frac{1}{z}+\sum_{n=1}^{\infty}\left(\frac{1}{z+n}+\frac{1}{z-n}\right), \ z\in\...
- 463
0
votes
1
answer
277
views
Analytic Continuation of Zeta-like function
Reading a paper about eta invariants I came across a zeta-like function.
I'm looking for the analytic continuation of $$\sum_{k=1}^\infty k(k+a)^{-s}$$ at $s=0$, where $a$ is positive.
In the paper ...
- 145
0
votes
2
answers
682
views
On integral relating logarithm of absolute value of Zeta function
Sorry for such a direct question:
Consider the following integral:
$$I(t)=\int_{1/2}^{1} {\log|\zeta(a+it)|}da.$$
How to find the nature of $I(t)$ as $t\rightarrow\infty$?
- 790
0
votes
1
answer
274
views
Relation between infinite product and regularized product
For a positive sequence $0\le\lambda_{1}\le\lambda_{2}\le\cdots$, consider an infinite product
\begin{equation*}
\prod_{i=1}^{\infty}\lambda_{i}:=\lim_{n\rightarrow\infty}\prod_{i=1}^{n}\lambda_{i}...
-3
votes
2
answers
317
views
When is $\Re(\zeta(s)) - \Im(\zeta(s)) = 0 $ with $\Re(\zeta(s))\neq 0$ and $\Im(\zeta(s))\neq 0$? [closed]
When is $\Re(\zeta(s)) - \Im(\zeta(s)) = 0 $ for $0<\Re(s)<1$. Here $\zeta$ denotes the Reimann zeta function. Does the solution live on a vertical line? Or is this another coincidence when both ...
- 317
-5
votes
0
answers
57
views
Every smooth function contains a bijection [closed]
Let $f:\mathbb{D}\rightarrow \mathbb{R}$ be a continuous non-constant over $\mathbb{D}$. Is there always a subset $\mathbb{A}\subseteq \mathbb{D}$ such that $f:\mathbb{A}\rightarrow \mathbb{R}$ is a ...
-6
votes
1
answer
138
views
Is the real part of the Eta function bounded by $2 \sum_{n=1}^{\infty}{\dfrac{(-1)^{n-1}}{n^{\alpha}}} $ [closed]
Consider the series defined by
\begin{equation}
f(\alpha,\beta) := \sum_{n=1}^{\infty}{\dfrac{(-1)^{n-1}}{n^{\alpha}}\cos(\beta\ln(n))}
\end{equation}
is it true that $$f(\alpha,\beta) \le 2\sum_{n=1}...
