The tag has no wiki summary.

learn more… | top users | synonyms

2
votes
1answer
227 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 ...
9
votes
0answers
209 views

L-Functions of Varieties, Zeta Functions of Their Models

Let $k$ denote a number field, with algebraic closure $\bar{k}$. Take a smooth, projective variety $X$ over $k$. If $\mathfrak{p}$ is a prime of $k$, and $l$ is a rational prime different to the ...
6
votes
0answers
264 views

Multiple zeta values at negative integers

For a tuple of integers $\underline{s}:=(s_1,\dots, s_d)$, the multiple zeta value (MZV) at $\underline{s}$ is defined as: $$ \zeta(s_1,\dots, s_d):= \sum_{n_1>\dots>n_d>0}\frac 1 ...
4
votes
0answers
237 views

Zeta functions for infinite abelian extensions?

Are there any reasonable ways to define zeta function of a maximal abelian extension $\mathbb{Q}^{ab}$ of the rationals, or at least zeta functions of some its infinite-dimensional subfields? In ...
4
votes
0answers
296 views

Does the Riemann hypothesis for liftable varieties over a finite field imply the Riemann hypothesis for all varieties over a finite field

The Riemann hypothesis for varieties over a finite field has been proven by Deligne. Still I would like to ask the following question. A variety $X$ over a finite field $k$ is liftable if there ...
3
votes
0answers
212 views

Alternating sums of powers of the lngamma ($\small f_p(x) = \sum \log(k!)^p x^k $ at $\small x=-1$)

I'm still fiddling with this recent question and come to a detail, whether I can find closed forms for the sums of the $\small lngamma() $ function. Precisely $$ \small f_p(x) = \sum_{k=0}^{\infty} ...
3
votes
0answers
283 views

Is there a notion of a zeta function of a morphism?

The Hasse-Weil zeta function is defined only for arithmetic schemes. By an arithmetic scheme I will mean a scheme $X$ together with a morphism of finite type $X\rightarrow S$, where $S$ is an affine ...
3
votes
0answers
387 views

How looks the “land of Tamagawa numbers”?

Jonah Sinick's question here, other interesting ideas he mentioned, and Franz Lemmermeyer's remark make one think at Bloch and Kato's drawing + question. What's known or guessed about that "land" by ...
2
votes
0answers
280 views

The Riemann Zeta Function summing over the Gamma Function

Has anyone studied a function of the form $$\eta(s) = \sum_{n=1}^{\infty} \frac{1}{\Gamma(n)^{s}} = \sum_{n=0}^{\infty}\frac{1}{k!^s}$$ This series is appearing in my research on the volumetric ...
1
vote
0answers
36 views

Hasse-Weil L-Functions of CM Abelian Varieties

In Shimura's paper "On the Zeta Function of an Abelian Variety With Complex Multiplication", in his terminology, the `one-dimensional part' of the zeta function is identified with a Hecke $L$-function ...
1
vote
0answers
160 views

Automorphicity of L-Factors of Zeta Functions

Associated to a variety over a number field $K$, one has a family of ``Hasse--Weil'' L-functions, which can be combined (as an alternating product) to give the Hasse--Weil zeta function of the ...
1
vote
0answers
71 views

binomial transform, Hurwitz zeta function

For $j,n\in\mathbb Z_+$, let $$ L_{j,n}^{(t)}= \sum_{m=0}^{n} \Bigl(-\frac 12\Bigr)^{n-m}{n\choose m}{m+j+1\choose m+1} \left( \frac {1}{t+\frac 12}\right)^{m+j+2} $$ and $$ L_{j,n} ...
1
vote
0answers
707 views

Proof that derivative of Hurwitz Zeta by the first argument is not expressable in terms of Hurwitz Zeta

The set of elementary functions is defined so that it to be closed against operation of differentiation. It is also evidently close against discrete differentiation. In the discrete calculus there is ...
1
vote
0answers
150 views

reference-request for connection between numerator of zeta-function and characteristic polynomial of Frobenius on hyperelliptic curves over finite fields

Let $H$ be a hyperellipitic curve of genus $g$ defined over $\mathbb{F}_q$. The Frobenius endomorphism operates on the divisor class group of $H$ and satisfies a characteristic polynomial ...
1
vote
0answers
217 views

(why) Are the following two constructions of zeta functions equal?

Let $X$ be a variety defined over $\mathbb{Q}$. One has the usual Hasse-Weil zeta function. Now, let's do a different construction. Base change $X$ to $\mathbb{C}$: $X_{\mathbb{C}}$. Now look at its ...
1
vote
0answers
386 views

Has the Weierstass transform been used to give Hermite series representations of the Riemann zeta function?

The inverse of the Weierstrass transform expands a function as a series of Hermite polynomials $H_{n}$. There are several ways to invert the Weierstrass transform which led me to the following ...
0
votes
0answers
174 views

Analytical continuation of the reciprocal of the Zeta function

Is the reciprocal of the Zeta function analytically continuable? As $1/\zeta(n) = \sum_{n=1}^\infty \mu(n)/{n^s}$, this does not look obvious.
0
votes
0answers
134 views

Special values of zeta functions and extensions of base fields.

Let $X$ be a scheme of finite type over a finite field $k=\mathbb{F}_{q}$ of $q$ elements. Then, one can define the zeta function $Z_{X/k}(T)$ of $X$ ovet $k$ as $\prod_{x\in ...
0
votes
0answers
215 views

Does there exist a Weierstrass/Hadamard factorization for $\chi(s)-1$ ?

Would like to build once more on this question. Take $s=\sigma + ti, s \in \mathbb{C}, 0<\Re(\sigma)<1$. Let's assume it is proven that: $$\zeta(1-s) - \zeta(s)$$ has all its zeros on the ...