Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [l-functions]

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

223 questions
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votes
1 answer

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 zeros of its zeta function on the critical line? I…
Kevin Buzzard
  • 4,908
19
votes
1 answer

Roadmap to Iwasawa Theory

I haven’t found any posts on this, so I figured I would ask. Beyond learning basic algebra (rings, groups, fields) and complex analysis, what must one study if they want to start learning a good amount of iwasawa theory? In what sequence should they…
JDivision
  • 390
15
votes
1 answer

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 $k \in \mathbb{N}$, where $\zeta_p(s) = (1-p^s)\zeta(s)$.…
KevinDL
  • 1,856
12
votes
3 answers

Why does the Dedekind zeta function of a number field have a pole at $s=1$?

The analytic class number formula tells us that the Dedekind zeta function $\zeta_K$ of a number field $K$ has a pole at $s=1$ with residue $$\frac{2^{r_1}(2\pi)^{r_2}\text{Reg}_Kh_K}{w_K\sqrt{|\Delta_K|}}.$$ Is there a quick way to see that…
justintomb
  • 281
9
votes
1 answer

What's the idea of Dirichlet’s Theorem on Arithmetic Progressions proof?

Dirichlet’s Theorem on Arithmetic Progressions says that if $a, m$ are natural numbers such that $gcd (a,m) = 1$, then there are infinitely many prime numbers in the arithmetic progression $a + km, k \in \mathbb{N}$. I would like to know what is the…
Nicolás A.
  • 153
9
votes
0 answers

Question on paper of Mazur, Tate, Teitelbaum and $p$-adic L functions of modular forms

I'm trying to fill in the details in proposition 14 of this paper by Mazur, Tate, and Teitelbaum. In particular, I'd like to understand the following. Let $f$ be a cuspidal eigenform of weight $k$ and level $\Gamma_1(N)$, with nebentypus $\epsilon$.…
Arbutus
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8
votes
1 answer

$L$-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 $L$-function. What about the converse ? If two…
Klaus
  • 4,265
7
votes
1 answer

$L$-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 elliptic curve $E$ over a number field $\mathbb{Q}$,…
Robert
  • 189
7
votes
1 answer

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)^{s}}=2^{1-t}\sum_{\mu|P}L_{\pm \mu}(s)L_{\mp…
Shobhit
  • 1,090
6
votes
3 answers

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

Consider the two Dirichlet characters of $\mathbb{Z}/3\mathbb{Z}$. $$ \begin{array}{c|ccr} & 0 & 1 & 2 \\ \hline \chi_1 & 0 & 1 & 1 \\ \chi_2 & 0 & 1 & -1 \end{array} $$ I read the L-functions for these series have special values $ L(2,\chi_1)…
cactus314
  • 25,084
6
votes
1 answer

Show that $\sum_{n\leq x}\frac{f(n)}{\sqrt n}=2L(1,\chi)\sqrt{x}+O(1)$, where $f(n)=\sum_{d\vert n}\chi(n)$.

Exercise 2.4.4 from M. Ram Murty's Problems in Analytic Number Theory asks us to show that for $\chi$ a nontrivial Dirichlet character $(\operatorname{mod} q)$, we have the estimate $$\sum_{n\leq x}\frac{f(n)}{\sqrt…
Alann Rosas
  • 6,553
6
votes
1 answer

Positivity of partial Dirichlet series for a quadratic character?

Let $\chi\colon(\mathbb{Z}/N\mathbb{Z})^\times\rightarrow\{\pm1\}$ be a primitive quadratic Dirichlet character of conductor $N$. For any integer $m=1,2,\cdots,\infty$, consider the partial Dirichlet…
Zhan
  • 173
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6
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2 answers

What is the relationship between these two versions of BSD?

The BSD conjecture is usually formulated like this. If $E/\mathbf{Q}$ is an elliptic curve, then $$ \text{rank }E/\mathbf{Q} = \text{ord}_{s=1} L(E,s), $$ where $L(E,s)$ is the Hasse-Weil $L$-function of $E$. But in Anthony Knapp's book Elliptic…
Adithya Chakravarthy
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6
votes
0 answers

Reference for $L$-functions

What will be a good reference to study $L$-functions for a beginner? Is there any book/lecture note in complex analysis that covers it?
learning_math
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6
votes
1 answer

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.
graveolensa
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