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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,...
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 ...
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)^...
10 votes

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

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 ...
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 = \...
6 votes

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 $n\#$ 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" ...

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