New answers tagged analytic-number-theory
8
votes
Is there a "convolution" of asymptotic growth?
$F(H)$ and $G(H)$ do not determine the asymptotic growth of the third display. Indeed, consider the following two functions from $\mathbb{Z}_{\geq 0}$ to $\mathbb{Z}_{\geq 0}$:
$$
f(x):=\begin{cases}0,...
- 89k
1
vote
On the explicit upper bound of $|\log\zeta(s)|$ near $\Re(s)=1$
Theorem 1 here shows that if $|t|\geq 3$ and $\frac{1}{2}\leq\sigma\leq 1$, then $|\zeta(\sigma+it)|\leq 76.2|t|^{4.45(1-\sigma)^{3/2}}(\log |t|)^{2/3}$. This is the best known explicit bound for $|\...
- 4,014
0
votes
Error term in França-LeClair approximation of zeta zeros
It is expected to have integer part correct. Even for a very large value.
Transcendental equations satisfied by the individual zeros of Riemann ζ, Dirichlet and modular L-functions Guilherme França ...
- 11
5
votes
Curious infinite product, convergence, connection to prime numbers
I believe that the doubly infinite product $\prod _{k=1} ^\infty \prod_{m=1} ^\infty \left( 1 - \frac {x^2} {k^2 m^2} \right)$ is absolutely convergent and equals $f(x) \frac {\sin^2 (\pi x)} {(\pi x)^...
- 121
10
votes
Accepted
Zeros of Dirichlet function $L(s,\chi_4)$
There is no need to use the subscript $4$ on the $L$-function: just write
$L(s,\chi_4)$ and $L^*(s,\chi_4)$. The first nontrivial zero in the upper half of the critical strip is $1/2 + it$ where
$t \...
- 45.7k
5
votes
Accepted
Positivity of partial Dirichlet series for a quadratic character?
Theorem 6.18 in the same chapter of Apostol shows, by partial summation, that
$$L_m(\chi) = L(1,\chi) + O_{\chi}(m^{-1})$$
where the implied constant is effective and depends on $\chi$. This can be ...
- 11k
1
vote
A sum related to the first moment of quadratic $L$-functions at $s=1$
If in doubt use the square-root cancellation heuristic.
You can think of this sum as
$$
\sum_{n < (\log x)^{100}} \varepsilon_n X_n
$$
where $\varepsilon_n$ is a random sign and $X_n$ is another ...
- 11
1
vote
Weak Siegel–Walfisz property
There are examples of real-valued functions $f$ for which this property fails even for bounded $q$. For instance, given a function $f$ let
$$s_x = \sum_{n \le x,\, n \equiv 1 \bmod 3}f(n),$$
$$ t_x = \...
- 11k
6
votes
Accepted
Inequalities for two functions related to the primorial function
Let $\theta(x)=\sum_{p\leq x}\log p$ be first Chebyshev function. Then we have $A(n)=e^{\theta(n)}$, and
$$B(n)=\prod_k A(n^{1/k})^k=e^{\sum_k k\theta(n^{1/k})}.$$
One can easily show that $\sum_k k\...
- 25.2k
2
votes
Inequalities for two functions related to the primorial function
Your $A(n)$ is usually denoted by $n\#$ https://en.wikipedia.org/wiki/Primorial and is related to the first Chebyshev function via
$$
\ln n\#=\theta(n)
$$
where
$$
\theta(x)=\sum_{p\leq x} \ln p.
$$
...
- 9,094
3
votes
Does there exist a sequence $(x,y) \in \mathbb{Z}^2$ such that $|\alpha x - y| \sqrt{x^2 + y^2}$ approaches a given real number?
First note that for all $(x,y)$ with $y/x$ sufficiently close to $\alpha$, we will have $\sqrt{x^2 + y^2}$ very close to $|x|\sqrt{1 + \alpha^2}$. So we might as well scale $c$ by $\sqrt{1 + \alpha^2}$...
- 2,831
0
votes
Exponential sum vs. exponential integral via Poisson summation
The connection between these summation formulas has been explored in The Summation Formulae of Euler–Maclaurin, Abel–Plana, Poisson.
The Euler-MacLaurin formula is the "partial summation" ...
- 158k
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