Questions tagged [l-functions]
L-functions are meromorphic functions on $\mathbb C$ that are used extensively in number theory.
226 questions
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What are applications of inprimitive Dirichlet characters and associated L-functions?
For simplicity, I'm gonna call L-functions associated with primitive Dirichlet characters 'primitive' and same with inprimitive.
GRH (Generalized Riemann Hypothesis) says that all non-trivial zeros of ...
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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 ...
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Closed form of the integral $\int_{0}^{i\infty} \frac{E_4(z)}{\sqrt{j(z)}}dz$
When playing around with modular form integrations, one may accidentally come across the following mysterious integral:
\begin{align*}
I=\int_0^{i\infty}\frac{E_4(z)}{\sqrt{j(z)}}dz\approx 0....
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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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Eulerian Integral representations for $GL_n\times GL_m$ $L$-functions
I am familiar with $L$-functions for $\operatorname{GL}_2\times \operatorname{GL}_1$ (e.g. Bump 'Automorphic Forms and Representations'), and currently reading on $\operatorname{GL}_n\times \...
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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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Special values of the twisted L-functions of weight 4 modular forms
Consider two Hecke eigenforms of weight $4$:
\begin{align*}f_5:=f_{5.4.a.a}=&q - 4q^2 + 2q^3 + 8q^4 - 5q^5 - 8q^6 + 6q^7 - 23q^9+O(q^{10}),\\
f_{80}:=f_{80.4.a.d}=&q - 2q^3 - 5q^5 - 6q^7 - ...
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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 ...
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Is the L-function of a primitive Hecke eigenform always a primitive L-function
Suppose $f$ is a primitive Hecke eigenform of level $N$. It is true that $L(s,f)$ is a primitve $L$-function in the sense of not factoring as a product of two $L$-functions of degree $1$? I suspect ...
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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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Exercise 15.1.10 from Multiplicative Number Theory I by Montgomery, Vaughan
I have trouble solving one part of item (d) of exercise 15.1.10 from Montgomery & Vaughan's Multiplicative Number Theory. This exercise is about oscillations of $\psi(x;q,a)- \psi(x;q,b)$, however ...
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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
$$\zeta(\tfrac{1}{2}+it)\ll_{\varepsilon} |t|^{1/6+\varepsilon}$$
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Zero-free Region in Iwaniec-Kowalski (Theorem 5.10)
Can someone please explain to me the real zero case in Theorem 5.10 of Iwaniec-Kowalski? They argue that for $t = 0$, any real zero $\beta$ of $L(s,f)$ is a zero of $L(s,g)$ of order at least $4$. ...
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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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Excercise 11.2.9 from Multiplicative Number Theory I. Classical Theory
I have trouble solving the following excercise. This is a result of Bateman & Chowla 1953 The equivalence of two conjectures in the theory
of numbers, J. Indian Math. Soc. (N.S.) 17, 177–181. ...
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