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Discuss some applications of the generalized Riemann hypothesis to problems that are not at first directly about zeta or L-functions. For example, the Solovay-Strassen test leads to a polynomial-time …

Call a set $S$ of primes "Frobenian" if there is a finite Galois extension $K/\mathbf Q$ and a union of conjugacy classes $H$ in ${\rm Gal}(K/\mathbf Q)$ such that $S$ is equal to the set of primes $p …

Look at the list of examples of zeta-functions on Wikipedia, and not all of them are in number theory. Here are some specific applications of the idea of a zeta-function in other areas of mathematics …

See Edwards' book Riemann's Zeta Function. He introduces $J(x)$ on p. 22 (in fact Edwards created the notation $J(x)$, which Riemann had written as $f(x)$). On p. 26 he defines ${\rm Li}(x)$ to be $ …

I don't know why you are restricting the products to $p \geq 13$ or where the factor $5\sqrt{3}/12$ is coming from. I am going to ignore that and discuss the following product over all primes $p$: $$ …

Application 1: in the chapter on modular forms in Serre's Course in Arithmetic, he integrates the logarithmic derivative $f'/f$ of a modular form $f$ on ${\rm SL}_2(\mathbf Z)$ around a contour (Argu …

He means the function has an analytic continuation from the open half-plane ${\rm Re}(s) > 1$ to the closed half-plane ${\rm Re}(s) \geq 1$. By definition, to say a function is analytic on a closed s …

Here is a simpler solution to question 1. By the prime number theorem, the $n$th prime $p_n$ admits the asymptotic estimate $p_n \sim n\log n$. It follows for any real number $x>0$ that $p_{[nx]}/p_ …

Franz, I wrote a paper related to an analytic heuristic on Dirichlet series associated to these prime counting problems. See http://www.math.uconn.edu/~kconrad/articles/hlconst.pdf.

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, e …

As long as a result remains unproved it can be pure speculation about whether it is really hard or just nobody has found the right simple idea, though in this case it seems plausible that the desired …

If $a_n = 1$ when $n$ is squarefree and $a_n = 0$ when $n$ is not squarefree, then you are asking for estimates on $\sum_{n \leq x} na_n$ and $\sum_{n \leq x} (\log n)a_n$. It is classical that $\sum …

Theorem. Let $K$ and $L$ be finite Galois extensions of $\mathbf Q$. Set $F = K\cap L$. (1) If $F = \mathbf Q$, then for each conjugacy class $C$ in ${\rm Gal}(L/\mathbf Q)$ there are infinitely many …

The answer by so-called friend Don indicates why the existence of logarithmic density implies the existence of Dirichlet density, with the same value. Below is an argument explaining why the set of pr …

There is no need to use the subscript $4$ on the $L$-function: just write $L(s,\chi_4)$ and $L^*(s,\chi_4)$. The first nontrivial zero in the upper half of the critical strip is $1/2 + it$ where $t \a …