L-Functions are meromorphic functions on $\mathbb{C}$ that are used extensively in number theory

learn more… | top users | synonyms

5
votes
1answer
62 views

L-series through integrals of rational functions

Recently I stumbled upon this short proof here: $$L(1,\chi_2)=\sum_{j=0}^{+\infty}\left(\frac{1}{3j+1}-\frac{1}{3j+2}\right)=\int_{0}^{1}\frac{1-x}{1-x^3}\,dx=\int_{0}^{1}\frac{dx}{1+x+x^2}$$ so: $$\...
0
votes
1answer
40 views

How do we evaluate this Dirichlet L-series

In this answer, David Speyer, whose answer is magnificent, states that "The sum $\sum \chi_3(n)/n$ is only slightly less well known; it is $\pi/(3 \sqrt{3})$.", where $\chi_3(n)$ is the character ...
0
votes
0answers
24 views

Mock gamma factors

I wonder if analogues of gamma factors as used to defined a complete L-function of the form $\displaystyle{\prod_{j=1}^{\infty}\Gamma(\lambda_{j}s+\mu_{j})}$ with a possibly non integral value of $d=\...
1
vote
1answer
35 views

Integral formula for the local L factor of a base changed automorphic representation

Let $\Bbb A$ the ring of rational adeles and let $\pi=\bigotimes_{p\leq\infty}\pi_p$ be an automorphic (cuspidal) representation of ${\rm GL}_2(\Bbb A)$. Fix a quadratic extension $K\supset\Bbb Q$. ...
0
votes
0answers
12 views

Factorization of Dirichlet characters

A Dirichlet character $\chi$ of modulus $q$ is called primitive if it cannot be factored as $\chi=\chi'\chi_0^{''}$, where $\chi_0^{''}$ is a principal character and $\chi'$ is a character of modulus ...
1
vote
0answers
42 views

What is an L-Function and how can it be used for categorization?

On hackernews, I came across this article, describing the LMFDB, the database of L-functions, modular forms, and related objects which aims at the categorization of mathematical objects where ...
0
votes
1answer
25 views

L-functions identically zero

Let $f:\mathbb{N}\to\mathbb{C}$ and assume that the L-function $$L(f;s)=\sum_{n=1}^\infty \frac{f(n)}{n^s}$$ converges absolutely on some right half-plane $Re(s)>k$. Is it true that $L(f;\cdot)$ ...
0
votes
0answers
13 views

Do equivalent elements of the Selberg class commute with the field automorphisms of C?

This question is a follow-up to Are all Dirichlet coefficients of any element of the Selberg class necessarily algebraic?. Let's associate to a given element $F$ of the Selberg class the field $K_{F}$ ...
1
vote
1answer
24 views

Are all Dirichlet coefficients of any element of the Selberg class necessarily algebraic?

The title says it all: do we know at least one element of the Selberg class having at least one transcendental coefficient in its development in a Dirichlet series for $\Re(s)>1$? Or are such ...
0
votes
0answers
25 views

Covariance group of the functional equation of an L-function

These last few days, I've been wondering whether one could consider the parameters/variables $\chi$ and $s$ a Dirichlet L-function depends on as coordinates such that the pair of transformations $(\...
0
votes
0answers
33 views

Do UF+PNT+SMO+GRH imply SOC?

The title may sound esoteric, but let's make it explicit. Suppose that the conjunction of unique factorization (UF, still open), prime number theorem (PNT, proved by Yoshikatsu Yashiro), Strong ...
2
votes
0answers
44 views

Which mathematical objects generate the zeroes of $L$-functions?

I've studied analytic and algebraic number theory for years and years, and I encountered a hard question about Riemann zeta function and other kinds of $L$-functions - which might be one of the most ...
2
votes
2answers
83 views

What do the zero's of L-functions entail?

I don't know exactly how, but I've read the Riemann Zeta function's nontrivial zero's imply something about an error term for an approximation function thing for the Prime Counting Function. I found ...
0
votes
0answers
31 views

Arithmetic data in an elliptic curve over a field $\mathbb K$

Note: In this context, $E(K)$ denotes an elliptic curve $E$ over a number field $K$, and $L(E,s)$ denotes the Hasse-Weil $L$ function. Is the rank of the abelian group $E(K)$ of points of $E$ the ...
1
vote
0answers
7 views

Does $F\otimes G\in\mathcal{M}$?

Let $\mathcal{M}$ be the class of automorphic L-functions which belong to the Selberg class. Let $F$ and $G$ be elements of this class, and define $F\otimes G$ by $a_{p}(F\otimes G)=a_{p}(F).a_{p}(G)$ ...
2
votes
1answer
37 views

Analytic formulas for special values of $L$-functions

In The Princeton Companion to Mathematics, IV.1, “Algebraic Numbers”, the conditionally convergent series \begin{equation} (1)\qquad\frac{\log(\sqrt{2}+1)}{\sqrt{2}}=1-\frac{1}{3}-\frac{1}{5}+\frac{1}...
4
votes
1answer
67 views

Sum of reciprocals of primes diverges

I can show that $$\log(\zeta (s)) = \sum _{p\in\Bbb P} \frac{1}{p} + R(s)$$where $$R(s) = \sum _{m\geq 2} \sum_{p\in\Bbb P} \frac{1}{m} \frac{1}{p^{ms}}$$ where $\Bbb P$ is the set of all primes, ...
2
votes
1answer
52 views

elliptic curve isogeny class 14.a $L$-function Dirichlet coefficients

Are the Dirichlet coefficients $a(n)$ of the $L$-function associated with isogeny class 14.a the irrationals that the inverse symbolic calculator suggests they are? The Lcalcfile suggests that they ...
3
votes
0answers
54 views

is this map necessarily a field automorphism?

Let $M$ denote the intersection of the Selberg class and the class of automorphic L-functions and let's define the automorphism group of $M$, denoted by $Aut(M)$, as the group (under composition) of ...
0
votes
1answer
50 views

dimension of a scheme and degree of an L-function

Disclaimer: I first asked this question on Mathoverflow but I was told it was off-topic for that site, so I ask it here. I try to understand correctly the notion of scheme, as Serre in the second ...
0
votes
0answers
46 views

definition of the L-function $L(f, \chi, s): \mathbb{A}_K \rightarrow \mathbb{C}$, what is smoothness and what is $f$?

To summarize the question I'm going to ask: for those who have studied L-functions and class field theory, I am confused about the definitions of some things and haven't found a good reference for ...
1
vote
0answers
44 views

Effective bounds on L(1,Chi) for Chi a Dirichlet Character

I have $\chi$, a Dirichlet Character $\bmod n$, and I have established that $L(1,\chi) \geq C / \log \log n$ for some constant C under the generalized Riemann Assumption. I'll call this proposition $A$...
1
vote
1answer
53 views

Complex Galois Representaions

I'm trying to understand 1 and 2 dimensional complex representations, induced representations and associated Artin L-functions for a project. I'm finding it hard to find appropriate material to help ...
1
vote
0answers
23 views

a question on upper bound for Bessel function $K_{2it}(x)$

Can we have $$K_{2it}(x)\sinh(t)\ll_{x} 1$$ for $1<x< (1+|t|)^3,$ where $K_{2it}(x)$ deotes the ordinary K-bessel function and $t>1$. This is true when $x\ge (1+|t|)^3$ from some references. ...
1
vote
0answers
47 views

On the partial zeta function

Let $F$ be a number field and $S$ be a finite set of places of $F$ including archimedian places. Let $\zeta^S(s)$ be the partial L-function, that is the meromorphic continuation of the product of ...
2
votes
1answer
139 views

expository articles on special values of L functions

While searching for some notes on L functions i have seen the following statement... In mathematics, the study of special values of L-functions is a sub field of number theory devoted to generalizing ...
6
votes
3answers
151 views

Analogue of $\zeta(2) = \frac{\pi^2}{6}$ for Dirichlet L-series of $\mathbb{Z}/3\mathbb{Z}$?

Consider the two Dirichlet characters of $\mathbb{Z}/3\mathbb{Z}$. $$ \begin{array}{c|ccr} & 0 & 1 & 2 \\ \hline \chi_1 & 0 & 1 & 1 \\ \chi_2 & 0 & 1 & -1 \end{...
2
votes
2answers
104 views

Quadratic twist of Elliptic curves with complex multiplication

Suppose $E/\mathbb{Q}$ is an elliptic curve that has complex multiplication by $\mathcal{O}_K$, where $K=\mathbb{Q}(\sqrt{D})$, for $D<0$ and squarefree. In "The main conjectures of Iwasawa ...
3
votes
1answer
98 views

Identities for L-series and Euler product

It is a mabe a stupid question for many experts here. There is something wrong in the following reasoning, and now I could not find it. Could someone help me out? Any advice will be highly appreciated....
3
votes
1answer
181 views

$L$-function of an elliptic curve and isomorphism class

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. We have a $L$-function $$L(E,s)$$ built from the local parameters $a_p(E)$. If two elliptic curves are isomorphic, they clearly have the same $...
3
votes
3answers
314 views

Inverse of Dirichlet series equality

I stumbled across a formula in here and tried to prove it for myself: $$\frac{1}{L(s,\chi)}=\sum\limits_{n=1}^{\infty}\frac{\mu(n)\chi(n)}{n^s}$$ However I got stuck. In my attempt I tried to show ...
4
votes
1answer
89 views

Motivation for using $L(1,\chi)$ in the proof of Dirichlet's Theorem

Having read the proof of Dirichlet's Theorem on the infinitude of primes in arithmetic progressions, I am left wondering what his motivation for studying $L(1,\chi)$ was and why it is reasonable that ...
1
vote
0answers
56 views

Why are L-functions called such? [duplicate]

L-functions are discussed here: http://www.lmfdb.org/intro/tutorial... So why are they called L-functions...? Is there a reason for the L?
1
vote
0answers
52 views

Dirichlet L function

The function is defined here - http://en.wikipedia.org/wiki/Dirichlet_L-function If $\chi$ is primitive and $\chi(-1)=1$ how do I show that $L$ has infinite number of zeros in the critical strip
1
vote
1answer
130 views

Computing the analytic $p$-adic $L$-function via modular symbols in MAGMA

I need to compute the analytic $p$-adic $L$-function of an elliptic curve at a prime $p$ via modular symbols using MAGMA. In SAGE...
2
votes
0answers
57 views

On the special value of Hecke L function.

For a nontrivial Hecke character $\chi:A_Q^{\times}/Q^{\times}\to S^1$, we know $L_Q(s,\chi)$ is nonzero. Is this true for number field $F$? I know is is holomorphic at $s=1$ by Artin conjecture, but ...
2
votes
1answer
137 views

$L$-functions of elliptic curves over $\mathbb{Q}$

How to find out the $P_{v}(E/\mathbb{Q},X)$ theoretically given below in the definition of $L$-functions for elliptic curves over $\mathbb{Q}$ $?$ Please cite some references for the same. For an ...
4
votes
1answer
130 views

Dirichlet series experiment - computing the rational coefficient

Let consider the sequence of numbers $a_n = 0,1,-1,0,1,-1,0,1,-1, ...$ extended periodically ( so it has period $9$, $a_{n+10}=a_n$. In fact, this is a Dirichlet character $a_n = \chi_9(n)$ modulo 9. ...
7
votes
1answer
147 views

Evaluation of $\sum_{m,n=-\infty}^{\infty} (m^2+Pn^2)^{-s}$ where $(m,n)\neq 0$

I was trying to learn about evaluating certain double sums and came across this formula: $$\sum_{\begin{matrix}m,n=-\infty \\ (m,n)\neq (0,0)\end{matrix}}^{\infty} \frac{1}{\left( m^2+Pn^2\right)^{...
3
votes
1answer
75 views

$L$-function of character in terms of other character

Let $F/K$ be a finite Galois field extension and $\varphi : \mathfrak{I}_K \rightarrow \mathbb{C}^{\times}$ a Hecke character of $K$.Define $\psi = \varphi \circ N_{F/K}$ as a Hecke character of $F$. ...
10
votes
1answer
376 views

What are the branches of the $p$-adic zeta function?

I'm reading the book $p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions by Neal Koblitz. In it, Koblitz wants to iterpolate the Riemann Zeta function for the values $\zeta_p(1-k)$ with $k \in \...
3
votes
1answer
139 views

Dirichlet series 'shifted' by a polynomial

Let $F(x) \in \mathbb{Z}[x]$ and $$ \xi(s) = \sum^\infty_{n=1}g(n)n^{-s} $$ be the Dirichlet series associated an arithmetic function $g(n)$. Define a new Dirichlet series $$ \xi_F(s) = \sum^\...
4
votes
1answer
62 views

L-function at s=5 with D=-4?

I want to know the value of $L(5,-4)$. Recall that $$ L(s,D)=\sum_{n=1}^\infty \left(\frac{D}{n}\right) n^{-s}. $$ I would like a reference with computations of $L(5,D)$, or more generally, of $L(s,...
2
votes
0answers
106 views

Riemann-Weil formula question

given a sum over the imaginary part of the zeros $ \sum_{t} h(t) $ is this version of the Riemann-Weyl formula correct? $ \sum _{t} h(t) -2h(i/2)= -2 \sum_{n=1}^{\infty}\Lambda (n)g(\log n)n^{-1/2}+...
23
votes
1answer
540 views

Computing (on a computer) the first few (non-trivial) zeros of the zeta function of a number field

If $K$ is a number field, whose Galois closure over the rationals has degree 24 or so, and whose discriminant is around $163^4$, then what is a numerically efficient way of computing the first few ...
5
votes
1answer
654 views

what does the “L” in “L-function” stand for?

I haven't been able to find a reference that tells what word (if a word) the L is short for.