All Questions
Tagged with convolution nt.number-theory
7 questions
2
votes
0
answers
139
views
Local Rankin-Selberg Zeta-function and Coates' p-adic L-Functions
$\DeclareMathOperator\Kern{Kern}\DeclareMathOperator\diag{diag}$
Let $F$ be a non-archimedean local field, $\mathcal{O}$ its ring of integers, $\mathfrak{p}$ its maximal ideal and $\pi$ a uniformizer.
...
- 748
5
votes
1
answer
612
views
Why does this convolution of the prime counting function $\pi$ look like a parabola?
In this previous question it is shown that the convolution of the prime counting function $\pi$ with itself, is related to the Goldbach conjecture:
$$\pi^*(n):=\sum_{k=0}^n \pi(k) \pi(n-k)$$
The ...
- 3,134
4
votes
1
answer
217
views
Why do convoluted convolved Fibonacci numbers pop up from this triangle?
Start with this triangle (OEIS A118981). This triangle is simple to generate with the following recurrence relation (though $T(0,0)$ ends up different from the OEIS version):
$$
T(0,0) = 2;T(1,0) = 1;...
- 194
3
votes
0
answers
238
views
Is there a closed form for the discrete convolution of $\sigma_1$ and $\sigma_2$?
I am trying to find a closed form for the following sum:
$$\sum_{k = 1}^{n-1}\sigma_1(k) \sigma_2(n-k)$$
where $\sigma_i$ is the sum of divisors and $\sigma_2$ is the sum of squares of divisors.
...
- 211
1
vote
0
answers
134
views
Convolution integral of series involving the non-trivial zeros of $\zeta(s)$
Let us consider the convolution $$f\left(x\right):=\int_{2}^{x-2}\sum_{\rho_{1}}\frac{u^{\rho_{1}}}{\rho_{1}}\sum_{\rho_{2}}\frac{\left(x-u\right)^{\rho_{2}}}{\rho_{2}}du,\,x>4$$ where $\rho_{i},\,...
- 219
5
votes
2
answers
556
views
Are there multiplicative functions which are not rational?
Vaidyanathaswamy calls an arithmetic function rational if it is the convolution of some finite collection of functions which are either completely multiplicative or inverse to a completely ...
- 9,114
4
votes
0
answers
181
views
Shifted convolution problem for Coefficients of automorphic forms
The shifted convolution problem for coefficients of modular forms is well studied and many estimates were established for the shifted convolution sums of Hecke eigenvalues. So, one may ask about the ...
- 1,257