0
$\begingroup$

I learn from Montgomery & Vaughan's Multiplicative Number Theory I: Classical theory that there exists $c_1,c_2>0$ such that whenever $\sigma\ge1-c_1/\log|t|,|t|\ge4$ there is $$ |\log\zeta(\sigma+it)|\le\log\log|t|+c_2,\tag1 $$ As I want to bound $|\zeta(\sigma+it)|^k$ uniformly in $k$, I wonder whether there are literature that provides a version of (1) that has $c_1$ and $c_2$ explicitly calculated.

$\endgroup$

1 Answer 1

1
$\begingroup$

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 $|\zeta(\sigma+it)|$ when $\sigma\geq 1-e^{-1}(\log|t|)^{-2/3}(\log\log|t|)^{-1/3}$.

$\endgroup$
2
  • 2
    $\begingroup$ "As I want to bound $|\zeta(\sigma+it)|^k$..." $\endgroup$
    – 2734364041
    Jan 27, 2023 at 15:57
  • $\begingroup$ Oh, never mind. I didn't look at I asked before commenting😂 $\endgroup$
    – TravorLZH
    Jan 27, 2023 at 23:16

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.