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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.
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
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
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
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
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
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
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