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Results tagged with analytic-number-theory
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user 1384
On the blending of real/complex analysis with number theory. The study involves distribution of prime numbers and other problems and helps giving asymptotic estimates to these.
20
votes
3
answers
2k
views
Computing (on a computer) the first few (non-trivial) zeros of the zeta function of a number...
Let $M$ be the splitting field of
x^8 + 3*x^7 + 13*x^6 + 17*x^5 + 45*x^4 + 37*x^3 + 11*x^2 + 112*x + 108
over the rationals. If I've understood some tables correctly, the splitting field is (of cou …
- 41.4k
19
votes
1
answer
1k
views
constants in Gamma factors in functional equation for zeta functions.
Usually the Riemann zeta function $\zeta(s)$ gets multiplied by a "gamma factor" to give a function $\xi(s)$ satisfying a functional equation $\xi(s)=\xi(1-s)$. If I changed this gamma factor by a non …
- 41.4k
18
votes
Accepted
What can be said about this double sum?
I don't know how "classical" you find these values, but here's perhaps something.
Define $E=\sum_{m,n\in\mathbb{Z}}q^{m^2+n^2}$, which is known to be a weight 1 level 4 modular form. In fact $E$ is a …
- 41.4k
18
votes
2
answers
1k
views
Why isn't meromorphic continuation enough for converse theorems?
This is a very naive question which really does little more than highlight my ignorance of how converse theorems really work.
Take an algebraic gadget which should be conjecturally associated to an a …
- 41.4k
11
votes
0
answers
430
views
Growth of $n=n(k)$ for which there's a non-trivial solution to $x_1^k+\cdots+x_n^k=y^k$?
Walter Hayman just asked me the following question. What, if anything, is known about the growth of the function $n(k)$, where $k\geq1$ is an integer, and $n=n(k)\geq2$ is the smallest integer for whi …
- 41.4k
7
votes
Accepted
Effective detection of CM modular forms
If the form is CM then it will be isomorphic to a quadratic twist of itself. So I think what I'd do with a form which I suspect is or is not CM is to just twist by all the (finitely many ) possible qu …
- 41.4k
5
votes
Accepted
Infinite sets of primes of density 0
If you can prove any reasonable lower bound for the set of primes which are at most $x$ then it's trivial to find infinite sets of primes with density 0. For example using completely elementary method …
- 41.4k
2
votes
Sum of $\sum_{\substack{1<a<q \\(a,q)>1 \\ (a+1,q)>1}}1$
What you have written down is already a formula for calculating the sum, so really you need to be more precise about what the question is.
But here are some comments which give a simpler formula in t …
- 41.4k