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11 votes
1 answer
328 views

Critical points of Dirichlet L functions

Let $L(s,\chi)$ denote a Dirichlet $L$-function for a real-valued non-principal character $\chi$. This has limiting value $L(\infty,\chi) = 1$ and we are interested in how this limit is approached ...
user2052's user avatar
  • 1,411
9 votes
3 answers
658 views

Vinogradov-Korobov for Dirichlet L-functions?

Where can one find a Vinogradov-Korobov zero-free region for Dirichlet L-functions? It has to be in a standard reference, but I'm having a non-trivial time finding it.
H A Helfgott's user avatar
  • 20.2k
7 votes
3 answers
826 views

Analytic equivalents for primes in arithmetic progressions

By way of context: it is known that the prime number theorem $\pi(x) \sim x/\log x$ is (nontrivially) equivalent to the statement that $\zeta(s)$ does not vanish on the line $\Re s=1$. I would like ...
Greg Martin's user avatar
  • 12.8k
7 votes
1 answer
564 views

Functional equation for general number fields

When it comes to general number fields beyond $\mathbb{Q}$, the litterature is not so abundant in analytic number theory. For instance over $\mathbb{Q}$, for primitve Dirichlet characters modulo $q$, ...
TheStudent's user avatar
7 votes
1 answer
1k views

The Correlation of the Möbius Function and Dirichlet Characters

Let $\chi$ be a Dirichlet character, and define $\phi_\chi (n)$ so that it satisfies $$\sum_{n=1}^\infty \phi_\chi (n)n^{-s}=\frac{\zeta(s-1)}{L(s,\chi)}.$$ In other words $$\phi_{\chi}(n)=\sum_{d|...
Eric Naslund's user avatar
  • 11.4k
6 votes
1 answer
337 views

"Sub-logarithmic" zero-free regions from Deuring-Heilbronn/Linnik's repulsion theorem

For each $n\in\mathbb{N}$, let: $\chi_n\pmod{q_n}$ a real non-principal Dirichlet character ($q_1 < q_2 < \cdots$), $\beta_n$ the largest real zero of $L(s,\chi_n)$, $\delta_n := (1-\beta_n)\...
Alufat's user avatar
  • 825
5 votes
2 answers
941 views

$B(\chi), L'(1,\chi)/L(1,\chi),\dotsc$

Let $\chi$ be a primitive Dirichlet character of modulus $q>1$. Write, as is customary, $B(\chi)$ for the constant in the expression $$\frac{\Lambda'(s,\chi)}{\Lambda(s,\chi)} = B(\chi) + \sum_\rho ...
H A Helfgott's user avatar
  • 20.2k
5 votes
1 answer
324 views

Symmetry type of non-cohomological automorphic forms

By Katz-Sarnak philosophy a family of $L$-functions would have a symmetry type which would reflect the statistics of $L$-functions, such as low lying zeros and moments. Shin-Templier's paper on Sato-...
Pig's user avatar
  • 809
4 votes
0 answers
63 views

Symmetric square $L$-functions over imaginary quadratic field

Let $F = \mathbb{Q}(\sqrt{-d})$ with class number $h_F = 1$, and $\Gamma = \mathrm{PSL}_2(\mathfrak{O}_F)$. Let $f$ be a Maass cusp form in the $L^2$-cuspidal spectrum of the Laplace operator $\...
Misaka 16559's user avatar
3 votes
2 answers
316 views

Explicit formula: explicit work with general smoothing?

The following is a literature question, in the sense that I already know how to do what I am asking about, and in fact have already done it; now I'd like to write a brief historical overview as an ...
H A Helfgott's user avatar
  • 20.2k
3 votes
1 answer
654 views

Order of vanishing of Artin $L$-functions at $s=1$

Let $E/F$ be a finite Galois extension of number fields with Galois group $G$. Let $S$ be a finite set of places of $F$ containing the infinite places. For $\chi$ an irreducible complex character of $...
Henri Johnston's user avatar
3 votes
0 answers
216 views

Maass--Selberg for any Eisenstein series on higher rank

Does there exist a Maass--Selberg relation for any Langlands Eisensein series on $\mathrm{GL}(n)$? By any I mean an Eisenstein series which is induced from any standard parabolic with any discrete ...
Subhajit Jana's user avatar
3 votes
0 answers
540 views

Questions about the exceptional zeros of Dirichlet $L$-functions

I have couple questions regarding the exceptional zeros of Dirichlet $L$-functions. We have the following result: There is a constant $c_1 > 0$ such that $L(\sigma, \chi) \not = 0$ whenever $$ \...
Johnny T.'s user avatar
  • 3,625
3 votes
0 answers
163 views

Oscillatory integral moments of $L(\frac{1}{2} + it, f \times f)$

Understanding moments and subconvexity bounds for $L$-functions is a big topic with a lot of activity. I'm currently looking at a related problem, bounding $$ \int_0^T L\left(\tfrac{1}{2} + it, f \...
davidlowryduda's user avatar
2 votes
2 answers
308 views

Reference for zero sum estimates of Dirichlet L functions

Let $\chi$ be a primitive character mod $p$ (prime) and $\rho = \beta + i \gamma$ be a non-trivial zero of $L(s, \chi)$. I am reading a paper by Ihara and Murty where they use following estimate : $\...
User1326's user avatar
2 votes
1 answer
508 views

Moments of Dirichlet $L$-functions on the critical line

I'm looking for a reference for some questions related to the moments of Dirichlet characters on the critical line, $$ M_k(T;\chi) = \int_T^{2T} |L(1/2+it,\chi)|^{2k}\,dt, $$ where $\chi$ is a ...
Anurag Sahay's user avatar
  • 1,354
2 votes
0 answers
104 views

The Guinand-Weil explicit formula for Hecke characters

The Guinand-Weil formula for the Riemann zeta function is \begin{aligned}&\Phi (1)+\Phi (0)-\sum _{\rho }\Phi (\rho )\\={}&\sum _{p,m}{\frac {\log(p)}{p^{m/2}}}{\Big (}F(\log(p^{m}))+F(-\log(p^...
LeechLattice's user avatar
  • 9,501
2 votes
0 answers
147 views

Well-known estimate for $L(s,\chi)$ for $\sigma=\text{Re}s\geq 1/2$

This is a very short question. Let $s=\sigma+it$ be a complex number with $\sigma \geq 1/2$. In the paper 'Jutila, Matti. "On the Mean Value of $L(1/2, \chi)$ FW Real Characters." Analysis 1.2 (...
LWW's user avatar
  • 663
2 votes
2 answers
491 views

Summation of certain series

Suppose $f(n)$ is a periodic function with period $q$. Now from this paper we get that if $\displaystyle\sum_{n=1}^{q}f(n)=0$ then $\displaystyle\sum_{n=1}^{\infty}\frac{f(n)}{n}=-\frac{1}{q}\...
Subhajit Jana's user avatar
1 vote
1 answer
74 views

Asymptotic for zeroes of $L(s,\chi)$ in a disk $|s|<R$

In 'Remarks on Weil's quadratic functional..' p.191 Bombieri claims any given $L$-function $L(s,\chi)$ has at least $$\big(\frac{1}{\pi}+o(1)\big)R\log R$$ zeroes in a disk $|s|<R$. Is there a ...
Tian An's user avatar
  • 3,799
0 votes
1 answer
195 views

Are Li's numbers $\lambda_n$ absolutely convergent for $n>1$?

Li's numbers $\{\lambda_n\}$ are defined as $$\lambda_n=\frac{1}{(n-1)!}\frac{d^n}{ds^n} [s^{n-1}\log\xi(s)]_{s=1} $$ for all positive integers $n$. Also $\lambda_n$ is given as a sum over the non ...
user avatar
0 votes
1 answer
116 views

Logarithms of $L$-functions of irreducible characters of Galois group

We know that the $L$ functions of Dirichlet characters $\chi$ of $(\mathbb Z / m\mathbb Z)^\times$ satisfy the property that $\log L(s, \chi)$ is holomorphic for $\Re(s) \geq 1$ if $\chi$ is a ...
asrxiiviii's user avatar