All Questions
Tagged with analytic-number-theory l-functions
61
questions
2
votes
0
answers
66
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.,
$$
\zeta_K(s)=\prod_{\...
- 787
1
vote
1
answer
51
views
Proposition 16.5.4 in Ireland-Rosen
We aim to show that if $\chi$ is a complex Dirichlet character mod $m$, then $L(1, \chi) \neq 0$. Assuming otherwise, we easily prove that if $F(s) = \prod_{\chi}L(s, \chi)$, where the product is over ...
- 4,429
2
votes
0
answers
53
views
Reconciling different ideal-theoretic definitions of Hecke Characters
I'm reading Chapter 7 of Neukirch's book on algebraic number theory, where the author defines a Größencharakter (6.1) as:
Let $\mathfrak{m}$ be an integral ideal of the number field $K$, and let $J^{\...
- 787
1
vote
0
answers
53
views
M/V Multiplicative NT : Theorem 11.3 and the Siegel zero
Two questions regarding Theorem 11.3 in the book of Montgomery & Vaughan Multiplicative Number Theory on the section "Case 4. Quadratic $χ$, real zeros.":
First the book supposes there ...
- 281
0
votes
1
answer
73
views
Zeros of L function on the 0.5 line
Could someone please tell what results are known about the zeros of L function, $L(s,\chi)$ on the 1/2 line, where $\chi$ is a character mod $q$?
Is there an upper bound for this count when we count ...
- 1,132
1
vote
0
answers
46
views
Is there a useful/meaningful notion of a multi-variable L-function in number theory?
I recently encountered multi-variable generalizations of various classical zeta functions. For example, the multi-variable Riemann zeta function
$$
\zeta(s_1, \ldots , s_r) := \sum_{0 < n_1 < \...
- 470
0
votes
0
answers
167
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 ...
- 111
6
votes
1
answer
165
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
$$\sum_{n\leq x}\...
- 5,601
3
votes
0
answers
110
views
Would c != 0 actually disprove the Birch and Swinnerton-Dyre conjecture rather than prove it?
Background:
The Birch and Swinnerton-Dyer conjecture states that the Taylor expansion around the point $s=1$ of the L-function of an elliptic curve $E$ has the form
$$c(s-1)^r+\text{higher order terms}...
- 89
2
votes
0
answers
57
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 ...
- 135
2
votes
1
answer
113
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{...
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 ...
- 161
6
votes
1
answer
146
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 ...
- 173
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 ...
- 1,043
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 ...
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