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What is known about the order of $\zeta(1+it)$?

Iwaniec-Kowalski gives (pp. 226 citing a result of Vinogradov-Korobov)

$|\zeta(1+it)| \lesssim (\log t)^{2/3},$

and oppositely Titchmarsh gives (pp. 188 citing Bohr-Landau)

$|\zeta(1+it)| \gtrsim \log \log t$

for infinitely many values of $t$.

Is this the limit of our knowledge? Is it conditionally known (or even expected) unconditionally known that

$|\zeta(1+it)| = e^{o(\log \log t)}$?

[As David points out below, on RH the result in Titchmarsh's book is optimal.]

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+1, this is cool. What chapter of Iwaniec-Kowalski? –  Frank Thorne Mar 7 '11 at 5:12
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RH implies the optimality of Titchmarsh's result. –  David Hansen Mar 7 '11 at 5:20
    
I could refer you to Apostle's analytic number theory book chapter 13 which has a set of inequalities for zeta. –  Daniel Parry Mar 7 '11 at 5:20
    
Ah, I see now from chapter 14 of Titchmarsh's book (which to be fair I ought to have consulted) the result David mentions. @Frank, it's at the end of chapter 8. So, that morally answers my question at least; I'm going to edit it above to see if anyone happens to know an unconditional result, but thanks! –  Brad Rodgers Mar 7 '11 at 5:52

2 Answers 2

up vote 9 down vote accepted

The theorem quoted from I-K is due to, and should therefore be attributed to, Vinogradov-Korobov. It is the best known unconditional result on the line $\mathrm{Re}(s)=1$, and is directly related to the width of the zero-free region by a result of Landau (see around 3.10 in Titchmarsh), so that any improvement of either will give an improvement of the other, and then of the error term in the Prime Number Theorem.

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Thanks! No offense intended to Vinogradov or Korobov! –  Brad Rodgers Mar 7 '11 at 7:01

This a classical problem which has been worked on by Littlewood, Titchmarsh, Levinson, and more recently Granville & Soundararajan.

Littlewood showed that the Riemann Hypothesis (RH) implies

$$\limsup_{t\to \infty} \frac{|\zeta(1+it)|}{\log\log t} \leq 2e^\gamma$$ where $\gamma$ is Euler's constant. Titchmarsh and later Levinson showed (unconditionally) that $$\limsup_{t\to \infty} \frac{|\zeta(1+it)|}{\log\log t} \geq e^\gamma.$$

Hence, on RH, a factor of 2 remains unresolved. Littlewood conjectured that $$\limsup_{t\to \infty} \frac{|\zeta(1+it)|}{\log\log t} = e^\gamma$$ but states "there is perhaps no good reason for believing ... this hypothesis." A similar situation (a factor of 2 remaining unresolved) exits for large values of the function $$1/|\zeta(1+it)|$$ when $|t|\geq 1$, say.

I got this information from the introduction to Granville and Soundararajan's paper: http://arxiv.org/abs/math/0501232. It can also be found in the notes of Chapter 13 in Montgomery & Vaughan's book "Multiplicative Number Theory." I believe it is also discussed in Heath-Brown's notes in the end of one of the chapters of Titchmarsh's book on the zeta-function.

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