Questions tagged [l-functions]
L-functions are meromorphic functions on $\mathbb C$ that are used extensively in number theory.
225 questions
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Technical step in the proof of Linnik's theorem in Iwaniec-Kowalski (18.82)
Going through the proof of Linnik's theorem in Iwaniec and Kowalski's Analytic number theory, I came across an affirmation I don't really understand. On Page 440, starting from the explicit formula ...
- 41
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L-function attached to p-adic Galois representation
Given an Artin representation $\rho$ of a number field $K$, it's known that the attached L-function $L(\rho,s)$ is defined (and even non-vanishing) on $\text{Re}(s)\ge1$.
Now if we start with a ...
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How to define log and arg of a Dirichlet L-function in the case that an exceptional zero exists
I'm reading Multiplicative Number Theory by Montgomery and Vaughan, and I am very confused about Theorem 11.4, which states that if $L(s,\chi)$ has an exceptional zero $\beta_1$ then for $\sigma>1-\...
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Question about induction formula for Artin L-functions
Let $K/\mathbb{Q}$ be a finite not-Galois extension. I want to show that
$$L(\operatorname{Ind}_{G_K}^{G_\mathbb{Q}} 1,s) = \zeta_K(s).$$
By checking the Euler factors agree. I am looking at the ...
- 931
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How to Deduce Trivial Bounds for Derivatives of L-functions on the Critical Line
I've been looking at Titchmarsh's Theroem 13.2 where he proves the equivalence of the classical Lindelof hypothesis and the sharp bound for all $2k$ moments of the Riemann zeta function (https://sites....
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Bound for $L(\sigma+it,\chi)$ when $\sigma\geqslant 1/2$
Let $|t|\geqslant 3$. I don't know the bound for $L$-functions when $\chi$ is a primitive character modulo $q$. It seems to me that
$$\zeta(\tfrac{1}{2}+it)\ll_{\varepsilon} |t|^{1/6+\varepsilon}$$
...
- 1,118
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Dyadic Partition of Unity Argument for Estimating the Central Values of L-functions
I've often see it said in many papers that one can obtain an estimate for an $L$-series $L(s,f) = \sum_{n \ge 1}\frac{a_{f}(n)}{n^{s}}$ (for simplicity assume $L(s,f)$ is entire) at $s = \frac{1}{2}$ ...
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About the Serre's proof of the fact $L(1,\chi)\neq 0, \forall \chi\neq\chi_0$
In A Course in Arithmetic, by Serre, he proves the above fact. At the proof, he says "It follows that $\zeta_m$ has all its coefficients greater than those of the series $$\sum_{(n,m)=1} n^{-\phi(...
- 301
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The denominator of $\frac{L(E,1)}{\Omega_E}$ dividing the conductor of $E$
Let $E$ be an elliptic curve over $\mathbb{Q}$ with conductor $N$, real period $\Omega_E$ and L-function $L$. Then the denominator of
$$\frac{L(E, 1)}{\Omega_E}\in\mathbb{Q}$$
divides $N$.
Where can ...
- 173
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Integral representation of Lambert Series.
Is there a general integral representation for Lambert series, similar to those that exist for various L-series and zeta functions?
For a sequence of complex numbers $\{a_n\}$, its corresponding ...
- 1,023
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If the analytic rank is one then the sign in the functional equation is -1?
Let $E$ be an elliptic curve defined over $\mathbf{Q}$. Let $L(E, s)$ denote its $L$-function over $\mathbf{Q}$. Also $f$ denotes the weight two cusp form associated to $E$, but this shouldn't be ...
- 459
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Dedekind zeta function of abelian number fields is a product of Dirichlet L-functions
Can somebody give an explicit (and self-contained) proof of the fact that the Dedekind zeta function of an abelian number field $K$ is a product of Dirichlet L-functions? I.e.,
$$
\zeta_K(s)=\prod_{\...
- 1,085
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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^{\...
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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 $L(s, χ)$ ...
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Some curios sums of Hurwitz zeta-function and Lehmer's totient problem [closed]
For all squarefree $k\in \mathbb N$
$$\left|\frac{\sum_{n=1}^{k-1}\zeta_H(-1,n/k)}{\sum_{n=1}^{k-1}\chi_0(n)\zeta_H(-1,n/k)}\right|=\left|\frac{\sum_{n=1}^{k-1}1}{\sum_{n=1}^{k-1}\chi_0(n)}\right|=\...
- 97
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q-series Expansion at Cusp and L-Functions
We know that the coefficients at $\infty$ of a modular form that is an eigenfunction of all Hecke operators in, say, $\Gamma_0(q)$, give an $L$-function with an Euler product and a functional equation....
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A Conjecture Relating Modulo Arithmetic and the Riemann Zeta Function.
I recently created a function that has perplexed many of my fellow amateur mathematicians. It goes something like this: $$f\left(g(x)\right)=\frac{1}{N^{2}}\sum_{n=1}^{N}\left(Ng(x)\operatorname{mod}n\...
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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 ...
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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 $\...
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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/\...
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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 ...
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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 ...
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65
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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_{\...
- 53
3
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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 ...
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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-...
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