28

Are L-functions uniquely determined by their values at negative integers? No. The rescaled Riemann zeta function $$ \zeta(2s) = \sum_{m=1}^\infty \frac{1}{m^{2s}} = \sum_{n=1}^\infty \frac{a_n}{n^s}, $$ corresponding to the coefficient sequence $$ a_n = \begin{cases} 1 & \textrm{if $n$ is a square}, \\ 0 & \textrm{otherwise}, \end{cases} $$ is an ...


22

See the work of Lamzouri, Li, and Soundararajan (I link the arXiv version; the paper appeared in Math. Comp.). Assuming that $\chi$ is a primitive quadratic character (as the title suggests) then Theorem 1.4 of that paper gives an explicit bound on the least prime quadratic residue on GRH. (Indeed that theorem gives an explicit bound on the least prime in ...


21

This is an elaboration on ACL's answer, way too long for a comment, which highlights a technical ingredient (well-known to all experts) that underlies the precise sense in which the $\ell$-adic etale cohomology of the geometric generic fiber provides a "uniformity" in $p$: the good properties of constructible $\ell$-adic sheaves. In particular, I think it is ...


21

That is not what the link says. To quote (emphasis mine): ... in which this conjecture was reduced to a special instance of the Bloch-Kato conjecture for the symmetric square motive of an elliptic curve. That means something quite different. You could equally say that Wiles "reduced" the proof to the fact that $X(3)$ and $X(5)$ have genus zero,...


19

There are examples that don't come from number theory, although it's not much simpler. Specifically, the Lefschetz zeta function. Let $X$ be a compact manifold of dimension $d$ and let $f: X\to X$ be a map. Let $L(f^n)$ be the number of fixed points of $f^n$, counted with appropriate multiplicity. Then if we define $$\zeta_f(t) = e^{ \sum_{n=1}^{\infty} L(...


19

A spirit. A general approach that may interest you is the following: counting laws can be obtained by studying suitable generating functions, via Tauberian arguments: the rightmost pole (resp. residue) of the generating function gives the growth order (resp. leading constant) in the counting law. In the case of Schanuel's theorem, the generating function is ...


15

This is not exactly an answer, but should illustrate how tentative the opinion of experts on L-functions is, when it comes to explain what L-function really are. Two quotes from the first volume on number theory by Kazuya Kato, Nobushige Kurokawa and Takeshi Saito: It seems as if the homeland where $\zeta$ functions originally come form is an unknown ...


15

I think few people would argue claims that Langlands' SLN 544 was written "in a different time" (e.g., pre-rigidity-theorems), and never really edited carefully (although the retyping is a relief), nor that Moeglin-Waldspurger aim for maximum generality/encyclopedic-ness, ... Selberg's sketches were of an even more different time. My colleague D. Hejhal ...


15

Good question. I don't understand fully what's happening, but here is an idea. Let $f=\sum a_n q^n$ be a modular form of weight $k+1/2$, nebentypus $\chi$. Assuming $k \geq 1$, the Shimura correspondence attaches to $f$ a modular form of integral weight $2k$, $g = \sum b_n q^n$ such that $$L(g,s) = L(\chi',s-k+1) \sum \frac{a_{n^2}}{n^s},$$ where $\chi'(n)=\...


15

1) True. 2) True. 3) True. 4) They are equal in the case of "tame ramification", e.g. if the degree of $K(\mu_p)$ over $K$, $K$ the coefficient field of the motive, is greater than $d$. One just has to recall that for $V$ the $\ell$-adic Galois representation associated to the motive, $\dim V$ is the degree of the $L$-function, the local $L$-factor is ...


14

I think you know all this, but nevertheless... These two formulas are arguably incarnations of the general philosophy of Arakelov geometry according to which derivatives of zeta functions (regularized determinants) compute (or are computed by, depending on your perspective) arithmetic intersection numbers. See for instance the arithmetic Riemann-Roch ...


14

I think there is a misunderstanding on your side. Tate's thesis is not about special Artin $L$-functions, but about special automorphic $L$-functions. It is a reformulation of Hecke's ideas in a uniform adelic language. Of course the automorphic $L$-functions occurring in Tate's thesis (namely those associated to automorphic forms on ${\rm GL}_1$) agree with ...


14

The de Bruijn-Newman constant is nonnegative, as proved in this brand new preprint by Rodgers and Tao. It is also conjectured, but not proven yet, that $M(x)=O(\sqrt{x})$ is false, in which case it is actually equivalent to $\Lambda<0$. Time will tell.


14

The following theorem is due to Chris Skinner, in this 2014 paper. Let E/Q be an elliptic curve such that rank E(Q) = 1 and the Tate-Shafarevich group Sha(E / Q) is finite, and some other technical assumptions hold. Then $ord_{s = 1} L(E, s) = 1$, and in particular $L'(E, 1) \ne 0$. This is, as far as I know, the best one can do at the moment; if ...


14

Your restriction that all $a_n$ are integers is too restrictive to define $L$-functions in general (even most Dirichlet $L$-functions are not like that), and you left out the Euler product. Anyway, examples with the restrictions you imposed can be found among the zeta-functions of number fields. For a number field $F$, its zeta-function $\zeta_F(s)$ has ...


14

I believe you are correct and $b$ is zero, although I find it inexplicable why this is not better known (certainly I didn't know it before). Let's stick to a primitive Dirichlet character $\mod q$, but what follows should be applicable in general. If we take logarithmic derivatives, then $$ \frac{\Lambda^{\prime}}{\Lambda}(s) = b + \sum_{\rho} \frac{1}{s-...


13

EDIT. A colleague wrote to me to point out that my original answer to this question was actually completely wrong: I had confused "isobaric" representations with "pure" representations (which are not at all the same thing). The correct answer, according to my colleague, is this. Consider the one-dimensional representation $\sigma = |\cdot|^{1/2} \boxtimes|\...


13

We have that \[-\frac{L'}{L}(s,\mathrm{sym}^2 f) = \sum_{n = 1}^{\infty} \frac{\Lambda_{\mathrm{sym}^2 f}(n)}{n^s},\] where $\Lambda_{\mathrm{sym}^2 f}(n)$ is equal to $\lambda_f(p^2) \log p$ if $n = p$ with $p \nmid N$, is essentially a bounded multiple of $\log p$ if $n = p^k$, and vanishes otherwise. (There are some minor issues at $p \mid N$ that are no ...


13

1) What generalizations of the Kummer congruences are known? This is somewhat imprecise as a question and in particular, I would dispute a little your assertion that This is probably the same question as asking "what definitions of $p$-adic L functions are known?" because the existence of a $p$-adic $L$-function is a much stronger result than the ...


13

The right person to answer this question is probably Dinakar Rimakrishnan (a good working reference is his paper "Modularity of the Rankin–Selberg $L$-series, and multiplicity one for $\mathrm{SL}(2)$"), but since he doesn't seem to use MathOverflow, here is my understanding of this. In each case, I will first discuss the local theory, then the ...


12

I am not sure exactly of what is asked in this question, but I wish to correct what has been said in an answer because it is a little bit outdated. (Edited after Kevin's comment) For $n=2$, Artin's conjecture had been solved in the 80's by Langlands and Tunnel for representations of $G_{\mathbb Q}$ to $GL_2(\mathbb C)$ which have solvable image in $PGL_2(\...


12

Regarding the first of these conjectures, I believe it was first explicitly stated (in the more general setting of a relative extension $K/k$) in Artin's 1923 paper [Über die Zetafunktionen gewisser algebraischer Zahlkörper, Math. Ann. 89, pp. 147-156]. This of course is a very special case of the general Artin holomorphy conjecture, and it is easy in the ...


12

There is a K-theory reformulation of Zagier's theorem which explains both the appearance of the integrals $A(x)$ as well as the appearance of hyperbolic geometry. I will try to give a sketch what is behind the relation of the zeta-value with volumes of hyperbolic simplices. First, due to the work of Borel, there is a relation between zeta-values and ...


11

Here is an elementary proof, the basic idea for which is in Selberg's 1949 paper "An elementary proof of Dirichlet's theorem about primes in an arithmetic progression" (Ann. Math., vol 2, 1949, pp. 297-304). It is enough to prove the unboundedness $$ \sum_{\substack{ p < X \\ p \equiv 1 \mod{N}}} \frac{\log{p}}{p} \neq O_X(1). $$ This is just a ...


11

The Elliott-Halberstam conjecture is not known to follow from GRH. Even the weak version of EH (which is with $Q=x^{1/2+\epsilon}$ for any fixed $\epsilon>0$) does not follow from GRH. On the other hand, it is known that the Elliott-Halberstam conjecture almost implies the twin primes conjecture, i.e., it implies that there are infinitely many pairs of ...


11

These definitions are consistent, though it's not immediate. The conductor quantifies the extent to which $\pi$ is ramified. As an aside, I prefer to write $c(\pi)$ for the conductor exponent of $\pi$, which is a nonnegative integer, so that $\mathfrak{p}^{c(\pi)}$ is the conductor of $\pi$, and $q^{c(\pi)}$ is the absolute conductor of $\pi$, where $q = N(\...


11

Let me translate this into a problem purely about automorphic forms: Take a newform $f \in \mathcal{S}_k^{\ast}(q,\chi)$, and a primitive Dirichlet character $\psi$ modulo $q'$. Then there exists a newform $f \otimes \psi$ of weight $k$, level dividing $q {q'}^2$, and nebentypus induced by the primitive character inducing $\chi \psi^2$, such that whenever $(...


11

You may use the local method of Landau with some bounds for L(s,chi) (expressing L'/L in terms of the local zeros, the approximation being controlled by an upper bound for |L(s,chi)| in a slightly larger region). Then, the convexity bound, as given by Rademacher for instance, should give you about what you get here. I would say, this would remove the +15.1, ...


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