Skip to main content

Questions tagged [l-functions]

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

Filter by
Sorted by
Tagged with
2 votes
0 answers
53 views

Dedekind zeta function of abelian number fields is a product of Dirichlet L-functions

Can somebody give an explicit (and self-contained) proof of the fact that the Dedekind zeta function of an abelian number field $K$ is a product of Dirichlet L-functions? I.e., $$ \zeta_K(s)=\prod_{\...
Sardines's user avatar
  • 769
1 vote
1 answer
42 views

Proposition 16.5.4 in Ireland-Rosen

We aim to show that if $\chi$ is a complex Dirichlet character mod $m$, then $L(1, \chi) \neq 0$. Assuming otherwise, we easily prove that if $F(s) = \prod_{\chi}L(s, \chi)$, where the product is over ...
Johnny Apple's user avatar
  • 4,337
2 votes
0 answers
49 views

Reconciling different ideal-theoretic definitions of Hecke Characters

I'm reading Chapter 7 of Neukirch's book on algebraic number theory, where the author defines a Größencharakter (6.1) as: Let $\mathfrak{m}$ be an integral ideal of the number field $K$, and let $J^{\...
Sardines's user avatar
  • 769
1 vote
0 answers
49 views

M/V Multiplicative NT : Theorem 11.3 and the Siegel zero

Two questions regarding Theorem 11.3 in the book of Montgomery & Vaughan Multiplicative Number Theory on the section "Case 4. Quadratic $χ$, real zeros.": First the book supposes there ...
Ali's user avatar
  • 279
2 votes
1 answer
42 views

Why $L(s, χ)$ is nonzero for $s$ real and $χ$ complex?

A quick question... In Section 11.1 of the book of Montgomery & Vaughan's Multiplicative Number Theory when studying the case $χ$ complex it doesn't suppose there can be a real zero for $L(s, χ)$ ...
Ali's user avatar
  • 279
1 vote
0 answers
26 views

An estimate on GRH for symmetric power L-function

I have a question in an article of K. Soundararajan and Matthew P. Young : The second moment of quadratic twists of modular L-functions. In their article, one says that using the GRH for $L\left(s, \...
cauchy Max's user avatar
2 votes
0 answers
93 views

Functional equation of the Hecke L function in ideal term and "ideal number" term (Neukirch Chapter VII)

$\def\A{\mathbb{A}} \def\B{\mathbb{B}} \def\C{\mathbb{C}} \newcommand{\Cx}{\mathbb{C}^{\times}} \def\F{\mathbb{F}} \def\G{\mathbb{G}} \def\H{\mathbb{H}} \def\K{\mathbb{K}} \def\M{\mathbb{M}} \def\N{\...
user682141's user avatar
  • 1,016
-1 votes
1 answer
55 views

Some curios sums of Hurwitz zeta-function and Lehmer's totient problem [closed]

For all squarefree $k\in \mathbb N$ $$\left|\frac{\sum_{n=1}^{k-1}\zeta_H(-1,n/k)}{\sum_{n=1}^{k-1}\chi_0(n)\zeta_H(-1,n/k)}\right|=\left|\frac{\sum_{n=1}^{k-1}1}{\sum_{n=1}^{k-1}\chi_0(n)}\right|=\...
user714's user avatar
  • 67
2 votes
1 answer
175 views

Explicit translation of Ramanujan’s condition for the Selberg Class?

According to many sites such as Wikipedia, the Ramanujan condition of the Selberg class can be stated as $$\boxed{\forall\epsilon>0:a_n\ll_\epsilon n^\epsilon}$$ My question is: what exactly does ...
tripaloski's user avatar
2 votes
0 answers
40 views

q-series Expansion at Cusp and L-Functions

We know that the coefficients at $\infty$ of a modular form that is an eigenfunction of all Hecke operators in, say, $\Gamma_0(q)$, give an $L$-function with an Euler product and a functional equation....
Riobaldo's user avatar
0 votes
1 answer
70 views

Zeros of L function on the 0.5 line

Could someone please tell what results are known about the zeros of L function, $L(s,\chi)$ on the 1/2 line, where $\chi$ is a character mod $q$? Is there an upper bound for this count when we count ...
math is fun's user avatar
  • 1,142
1 vote
1 answer
115 views

A Conjecture Relating Modulo Arithmetic and the Riemann Zeta Function.

I recently created a function that has perplexed many of my fellow amateur mathematicians. It goes something like this: $$f\left(g(x)\right)=\frac{1}{N^{2}}\sum_{n=1}^{N}\left(Ng(x)\operatorname{mod}n\...
Gabriel Turner's user avatar
0 votes
0 answers
28 views

Definition of Artin $L$-functions

I am reading notes of Samuel Marks on ”Galois Representations”. In this paper, on page 14 (definition 3.1), it is said that given a representation $\rho$ of $Gal(L/K)$ it yields another representation ...
confused's user avatar
  • 489
1 vote
0 answers
45 views

Is there a useful/meaningful notion of a multi-variable L-function in number theory?

I recently encountered multi-variable generalizations of various classical zeta functions. For example, the multi-variable Riemann zeta function $$ \zeta(s_1, \ldots , s_r) := \sum_{0 < n_1 < \...
xion3582's user avatar
  • 470
1 vote
0 answers
53 views

Does the relationship between the L function of the elliptic curve and a quadruple series hold?

Let $E_{X_0(11)}$ be the elliptic curve (over $\bf Q$) of conductor $11$ defined by $$y^2+y=x^3-x^2-10x-20.$$ First, some theorems and formulas are introduced as follows. The modularity theorem (Slow ...
Eufisky's user avatar
  • 3,237
0 votes
0 answers
161 views

Rankin-Selberg Convolution of newforms with different levels

Let $f \in \mathscr{S}_{k}(N,\chi)$ and $g \in \mathscr{S}_{k}(M,\psi)$ be newforms with $(N,M) = 1$. For $\Re(s) > 1$, I was able to derive an integral representation for the Rankin-Selberg ...
Laan Morse's user avatar
1 vote
0 answers
69 views

Classifying all Hecke Characters of a given field and a given conductor [closed]

I'm rather very new to this topics and in the hopes of understanding Tate's Thesis I have come to the issue of Hecke Character. Given the following definition: Let $F$ be a number field and let $\...
blueinfinity's user avatar
6 votes
1 answer
158 views

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}\...
Alann Rosas's user avatar
  • 5,532
2 votes
1 answer
95 views

The equivariant BSD conjecture (and the $\rho$-isotypical component)

I am trying to understand the statement of the equivariant BSD conjecture. Let $E/\mathbb{Q}$ be an elliptic curve. Let $\rho$ be a finite-dimensional irreducible Artin representation, and let $K/\...
Math-Alt's user avatar
3 votes
0 answers
109 views

Would c != 0 actually disprove the Birch and Swinnerton-Dyre conjecture rather than prove it?

Background: The Birch and Swinnerton-Dyer conjecture states that the Taylor expansion around the point $s=1$ of the L-function of an elliptic curve $E$ has the form $$c(s-1)^r+\text{higher order terms}...
Jeff's user avatar
  • 89
2 votes
0 answers
56 views

Bounds for Dirichlet $L$-functions on the critical line [closed]

I am interested in bounds on the constants $A,B$ such that $$L\big(\tfrac{1}{2}+it,\chi\big)\ll_\varepsilon q^{A+\varepsilon}(|t|+1)^{B+\varepsilon},$$ and was curious if any developments have been ...
Troy W.'s user avatar
  • 135
2 votes
1 answer
111 views

Residue of a Dirichlet Series at $s=1$

I have encountered this problem of determining the leading term in the Laurent expansion of a Dirichlet series. Let $d(n)$ be integers and consider the Dirichlet series $$D(s)=\sum_{n=1}^{\infty}\frac{...
Gabrielle Rodriguez's user avatar
2 votes
1 answer
71 views

Converting polylogarithms to Dirichlet L functions

When trying to simplify polylogarithms evaluated at some root of unity, namely $\text{Li}_s(\omega)$ for $\omega=e^{2\pi i ~r/n}$, it is reasonable to convert it to Hurwitz zeta functions or Dirichlet ...
Po1ynomial's user avatar
  • 1,616
2 votes
0 answers
96 views

Arithmetic meaning of certain periods of modular symbols of elliptic curves

I'm following the notation in Chapter II of Cremona's book on modular symbol algorithms. Let $f$ be a weight two cusp form for a congruence subgroup $G$. For any two points $\alpha, \beta \in \mathcal{...
Multramate's user avatar
1 vote
0 answers
30 views

How to understand $x^{-1}$ acts on the mahler transform over $\mathbb{Z}_p$

I am reading about the value of p-adic L-function $L_p(\theta,s)$ at $s=1$. Someone claims the following formula: \begin{equation} L_p(\theta,1):=\int_{\mathbb{Z}_p^{\times}}x^{-1} \cdot \mu_{\...
Taozipeter's user avatar
2 votes
0 answers
42 views

The p-adic value of Dirichlet L-function at 1

I'm reading the book An introduction to p-adic L-functions recently. The book can be found in https://warwick.ac.uk/fac/sci/maths/people/staff/cwilliams/lecturenotes/lecture_notes_part_i.pdf. In 4.4, ...
Taozipeter's user avatar
3 votes
1 answer
66 views

Fourier Expansion of modular form : Constant term using the other ones

I am reading Serre's article ( Formes modulaires et fonctions zêta p-adiques ). At some point, it is written that for a modular form $f$, we can find $a_0(f)$ in terms of $a_n(f)'s$. The procedure to ...
user031197's user avatar
7 votes
1 answer
1k views

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 \...
Nicolás A.'s user avatar
3 votes
1 answer
136 views

On the Iwasawa Algebra

I am reading Joaquin Rodrigues Jacinto's and Chris Williams' notes on $p$-adic $L$-functions http://www.ltcc.ac.uk/media/london-taught-course-centre/documents/Number-Theory---Full-Lecture-Notes-2017-...
Maty Mangoo's user avatar
1 vote
0 answers
80 views

Definition $L$ function of elliptic curve

Let $E$ be an elliptic curve. My book reads, we denote '$L$ series of $E$ regrading $l$ adit representation of $Gal(\overline{\Bbb{Q}}/\Bbb{Q})$' as $L(E/\Bbb{Q}):= \prod_{p}\frac{1}{1-a_pp^{-s}+...
Poitou-Tate's user avatar
  • 6,266
1 vote
1 answer
130 views

Normalisation of $L$-function for classical modular forms and automorphic representations

I found that the normalisation of $L$-functions of classical modular forms and corresponding automorphic representations is somewhat confusing for me. Recall that if $f\in S_k(\Gamma_0(N))$ is a ...
youknowwho's user avatar
  • 1,479
2 votes
0 answers
42 views

First few coefficients of $\zeta_p$ as an element of an Iwasawa algebra

One way of introducing the $p$-adic Riemmann zeta function is to first define a $p$-adic pseudomeasure $\zeta_p$ via interpolation. Namely, $\zeta_p$ is uniquely defined by the property that $$\int_{\...
Milo Moses's user avatar
  • 2,517
3 votes
2 answers
301 views

Selberg Class- Ramanujan Conjecture

The wikipedia for Selberg class L-functions (https://en.wikipedia.org/wiki/Selberg_class) states 4 conditions: Analyticity, Ramanujan conjecture, Functional equation, Euler product. I would like to ...
JohnAnt's user avatar
  • 161
6 votes
1 answer
145 views

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's user avatar
  • 173
0 votes
1 answer
130 views

Do $L$-functions of varieties over function fields have analytic continuation / functional equations?

Suppose that $K$ is a function field (i.e: a finite extension of $\mathbf{F}_q(t)$) and $X$ is a smooth projective variety over $K$. I have a few questions about the $L$-function of $X/K$: How do you ...
Adithya Chakravarthy's user avatar
3 votes
1 answer
120 views

Root Number is root of unity

In his thesis Tate deduces a functional equation for Hecke L-Functions associated to a unitary idele class character $\chi$ of the form $$ L(s,\chi)=\epsilon(s,\chi)L(1-s,\overline{\chi}).$$ Here the ...
Lucky_77's user avatar
1 vote
0 answers
116 views

Interpretation of order of vanishing of Modular L-function

The BSD conjecture gives an interpretation to the order of vanishing of the central value the L-function of an elliptic curve (it is supposed to be the rank of that elliptic curve), while the ...
Rodrigo's user avatar
  • 1,031
6 votes
2 answers
156 views

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 $L$-...
Adithya Chakravarthy's user avatar
1 vote
0 answers
60 views

Proving that order of pole of L-function is greater than or equal to 0 by Artin's Reciprocity Law

I have been studying the proof of Chebotarev's Density Theorem from the book "Problems in Algebraic number Theory" by Ram Murty and Esmonde and encountered a problem. Let $n(H, \psi)$ be the ...
Dishant's user avatar
  • 65
2 votes
0 answers
103 views

One identity about Eisenstein series and L functions of modular forms

Currently, I'm reading Zagier's notes 'Modular forms whose Fourier coefficients involve zeta-function of quadratic fields'. There is an equation (72) in page 42: $$\sum_{n=1}^{\infty} \frac{\sigma_{r}(...
Strange's user avatar
  • 61
0 votes
1 answer
111 views

A question on the Riemann zeta function

Question: Consider a $L$-shaped path $L_\epsilon:\frac{1}{2}+\epsilon\to \frac{1}{2}+\epsilon+i\ (H+\epsilon)\to \frac{1}{2}+i\ (H+\epsilon)$ where $H>0$ is fixed and $\epsilon>0$ is arbitrarily ...
user avatar
2 votes
1 answer
184 views

Approximate functional equation of Selberg Class L-functions

I was reading the paper "Integral moments of $L$-functions" by Conrey et al. They have used the following sharp cutoff version of the approximate functional equation for "Selberg class&...
Anamika Sen's user avatar
1 vote
1 answer
102 views

Whittaker at Archimedean Test vector

Let $\pi$ be a generic irreducible Casselman Wallach representation of $GL_n(\mathbb{R})$. Let $\phi$ be a test vector for the archimedean local L function $L(s,\pi)$. Is there an explicit expression ...
Akash Yadav's user avatar
  • 1,187
1 vote
0 answers
30 views

Why are L-factors polynomials in $q^{-s}$?

I am interested on integrals of "Tate style", say of the form $$\int_{F^\times} \Phi(a) \kappa(a) |a|^s d^\times a$$ Here, $F$ is a local field, $|\cdot|$ the associated absolute value, $\...
TheStudent's user avatar
  • 1,283
1 vote
0 answers
103 views

Davenport, the usage of mean value theorem

In Davenport, Multiplicative Number Theory, page 6, the author asserts that: "Let $\omega$ be a complex character. Suppose $L_{\omega}(1)=0$, then it would imply that $L_{\omega}(s)=L_{\omega}(s)-...
mertunsal's user avatar
  • 498
3 votes
1 answer
105 views

Why are Rankin-Selberg convolutions different when $n=m$?

Let $\pi$ an automorphic cuspidal representation on $GL_n(\mathbb{A})$ (and similarly $\pi'$ on $GL_m$). For $m=n-1$, it is standard to introduce the Rankin-Selberg L-function as (the gcd of the ...
TheStudent's user avatar
  • 1,283
1 vote
0 answers
91 views

What is the importance of RH for $L$-functions of modular forms?

The Reimann Hypothesis (RH) for $L$-functions of modular forms states that all the non-trivial zeroes of an $L$-function of a modular form must lie on the critical line. My question is: why is this ...
Adithya Chakravarthy's user avatar
2 votes
0 answers
65 views

*Why* the converse theorem (conjecture) should be true?

Currently, the following $\mathrm{GL}(n)$ converse theorem due to Cogdell and Piatetski-Shapiro is known: For an admissible irreducible representation $\pi$ of $\mathrm{GL}(n, \mathbb{A})$, $\pi$ is ...
Seewoo Lee's user avatar
  • 15.2k
3 votes
1 answer
207 views

Logarithmic average of the Legendre symbol

My question is simple: can we show that the sum $$ \sum_{k=1}^{p-1} \frac{\left( \frac{k}{p} \right)}{k} $$ is positive for all primes $ p $ where $ (k/p) $ denotes the Legendre symbol modulo $ p $, i....
Ege Erdil's user avatar
  • 17.8k
4 votes
1 answer
180 views

Relations between different zeta functions for a simple algebra

I'm trying to understand the classical works of Eichler, Shimura and many others (especially Shimizu and Tamagawa's annals papers) on the "classical" (I'm a newcomer and I'm not sure whether ...
youknowwho's user avatar
  • 1,479

1
2 3 4 5