The zero-free region of the Riemann zeta function

In the previous post about the zeta function the Vinogradov–Korobov zero-free region was stated, together with what it tells us about the error term involved in using {\text{Li}(x)} to approximate {\pi(x)}. In this post it is examined how this zero-free region, together with various other zero-free regions can be derived.

The Zero-Free Regions

In the last post it was stated that it was known that only the trivial zeros of the zeta function lie outside of the strip {0 < \Re(s) < 1}. Inside this strip lie an infinite number of zeros which we refer to as the non-trivial zeros. It has been possible to produce some further restrictions on the region in which we know the non-trivial zeros lie, these results are known as the zero-free regions.

The symmetry of the location of zeros in the critical strip around both the critical line and around the real axis make it sufficient to consider zeros in the region {0.5 \leq \beta \leq 1} and {t \geq 0}. The symmetries of the zero set can then be used to derive similar results in the other three segments that the critical strip is divided into by the lines {\Re(s)=\frac{1}{2}}, and {\Im(s)=0}. For sufficiently large t the following inequalities hold for some constant c (which will be different for the different inequalities below):

  •  The de la Vallée Poussin zero-free region: {1-\beta \geq \frac{c}{\log~t}} [6].
  •  The Littlewood zero-free region: {1-\beta \geq \frac{c\log\log~t}{\log~t}} [6].
  •  The Vinogradov-Korobov zero-free region, {1-\beta \geq \frac{c}{(\log~t)^{2/3}(\log\log~t)^{1/3}}} [6].

It should be noted that as {t \rightarrow \infty} the width of these zero-free regions within the critical strip tends to zero. Hence asymptotically they are equivalent to the result that {\zeta(s) \neq 0} for {\Re(s)=1}, however the improvement for bounded t is significant, and has been used to improve bounds on the error term involved in approximating {\pi(x)} by the log integral function.

It has been possible to calculate explicit versions of these constants in some cases, of course these constants are probably improvable, certainly in the case the Riemann Hypothesis holds they are significantly improvable. An explicit version of the de la Vallée Poussin zero-free region given by Kadiri [4], is that for {t \geq 2}, the inequality holds with c=\frac{1}{5.69693}. An explicit version of the Vinogradov-Korobov zero free region given by Ford [1], is that the inequality holds for {t \geq 3} with c=\frac{1}{57.54}.

Some Elementary Properties of \zeta(s)

In this section is some background work to the previous post’s announcements about the zeta function. This section will sketch a proof for the functional equation for the zeta function.

In Chapter II of Titchmarsh [6], no fewer than seven proofs of this equation can be found. Below a proof is outlined that uses the properties of theta functions (expect to hear more about theta functions on this blog soon — they have very interesting links to the Heisenberg group, complex tori and complex abelian varieties). The complete form of this proof can be found in either Titchmarsh [6] or Mumford [5].

However for the purposes of this derivation we only need to know the definition of the zeta function by:

{\theta(z,t)= \sum_{n\in \mathbb{Z}}{exp(\pi in^{2}\tau+2\pi inz)}}.

This converges for all complex numbers {z \in \mathbb{C}} and {\tau \in \mathbb{H} = {z \in \mathbb{C}: \Im(z) \geq 0}}.

This proof uses a link between the Mellin transform of the theta function and the zeta function. The Mellin transformation is an integral transform defined by

{Mf(s)=\int_{0}^{\infty}x^{s-1}f(x)dx}.

This transform can be inverted (subject to certain boundedness conditions for f of Mf, these are outlined in the Mellin Inversion Theorem), using the inverse Mellin transform defined below (note that c must be appropriately chosen — the restrictions on c depend on how \phi satisfies the boundedness conditions):

{M^{-1}\phi(x)=\frac{1}{2\pi i}\int^{c+i\infty}_{c-i\infty}x^{-s}\phi{s}ds}.

We now take the Mellin transform of \theta(0, ix)-1 to gain

{M(\theta(0, ix)-1)(\frac{1}{2}s) = 2\int^{\infty}_{0}(\sum_{n=1}^{\infty}{e^{-\pi n^{2}x}})\frac{dx}{x}}

This converges due to certain boundedness properties that the theta function satisfies — hence we can interchange integration and summation, and we then substitute y=\pi n^{2}x to obtain

{M(\theta(0, ix)-1)(\frac{1}{2}s) = 2\pi^{-s/2}\zeta(s)\Gamma(\frac{s}{2})}.

We can now use properties of the theta function to derive properties of the zeta function.  Specifically we have that \theta(0, \frac{i}{x})=x^{1/2}\theta(0, ix), we can use this to rearrange the Mellin transform to give

2\pi^{-s/2}\zeta(s)\Gamma(\frac{s}{2})=\int^{\infty}_{0}(\theta(0,ix)-1)x^{s/2}\frac{dx}{x}= \frac{-2}{s}-\frac{2}{1-s}+\int_{1}^{\infty}(\theta(0, ix)-1)(x^{s/2}+x^{(1-s)/2})\frac{dx}{x}

The above function on the left hand side is invariant under the substitution s \rightarrow 1-s, this invariance then gives us the functional equation.

We note that the functional equation is not ideal for actual estimation (or calculation) of the value of the zeta function at a point. This is because it relates the value of \zeta(s) to the value of the zeta function at another point, rather than expressing the value of \zeta(s) solely in terms of other functions.

Estimates of \zeta(s)

In order to find information out about the zeros of \zeta(s) within the critical strip we need some information about \zeta(s) withing this strip.  Specifically we use approximations of the value of \zeta(s).  In order to do this we use a class of results known as the approximate functional equations which relate the value of \zeta(s) within the critical strip to the values of partial sums of the series \sum_{n=1}^{\infty}{n^{-s}}, that is to say sums of the form \sum_{n \leq x}n^{-s}.

A well-known example of approximate functional equations is

\zeta(s)=\sum_{n\leq x}n^{-s}+\chi(s)\sum_{n\leq y}{n^{s-1}}+O(x^{-\sigma})+O(t^{1/2-\sigma}y^{\sigma-1}) for 0<\sigma<1, 2\pi xy=t, and x,y> h>0 for some positive constant h

where {\chi(s)=2^{s}\pi^{s-1}\Gamma(1-s)\sin(\pi s/2)}. There also exist explicit versions which give upper bounds for the implied constants within the big-O terms of the above expression. If we wish to gain a zero-free region with explicit constants then an explicit version of the above estimate (or other similar estimates) will have to be used. For details on the derivation of these approximation formulae see Titchmarsh [6] or Ivic [2].

We now need to develop bounds for the sums {\sum_{n\leq x}{n^{-s}} = \sum_{n\leq x}e^{-s\log n}}, in order to use some approximation formula to estimate \zeta(s).

Estimating exponential sums is often necessary in number theory and mathematics more generally. Different bounds on the value of these partial sums can be used to give different zero-free regions. For example the using Van der Corput method to approximate the partial sum can be used to give the Littlewood zero-free region, while using the Vinogradov–Korobov method gives the Vinogradov–Korobov zero-free region. The bounds on exponential sums generally can then be related to the specific sum above. For more details about these estimation processes see chapter 8 of [3] on which the below sections is based.

An exponential sum is a sum of the form {\sum_{n\leq x}{e^{2\pi if(n)}}}, where f maps the natural numbers into the real numbers, and x is some positive number. Clearly the individual terms e^{2\pi if(x)} will be somewhere on the unit circle. Hence without additional data on the function f(n), we cannot gain an upper bound better than \sum_{n\leq x}{e^{2\pi if(n)}} \leq x. However if we have additional data about f it may be possible to gain a better bound.

For example in the Van der Corput method we use bounds on the n^{\text{th}} derivative of f in order to gain a better bound on the exponential sum. Suppose that we have a bound on the f^{(n)}(x) for some function f \in C^{n} (i.e. it has continuous k^{\text{th}} derivatives for 1 \leq k \leq n), then we can relate this to a exponential sum with bounded (n-1)-th derivative. Let g(x)=f(x+r)-f(x) for some number r, then g^{(n-1)}(x)=f^{(n-1)}(x+r)-f^{(n-1)}(x)=rf^{((n)}(\eta) for some \eta \in (x, x+r), thereby giving bounds on the (n-1)-th derivative of g. If we can estimate the sum \sum_{n\leq x}{e^{2\pi ig(n)}}=\sum_{n\leq x}{e^{2\pi i(f(x+r)-f(x)))}}, it is possible to then relate this estimate to the sum \sum_{n\leq x}{e^{2\pi if(n)}} \leq x. We can use this link to reduce the estimation problem to where we have bounds on the second derivative of f. In this case we can use integral approximations to the series summation to gain an estimate for the sum, with the bounds on the second derivative of f allowing the error term in this approximation to be estimated.

In contrast to the above method the Vinogradov–Korobov method relates the sums to the number of solutions to certain Diophantine equations. We start by shifting the summation by q: \sum_{a<n<b}{e^{2\pi if(n)}}=\sum_{a-q<n<b-q}{e^{2\pi if(n+q)}}. We then approximate f(n+q) by using a Taylor polynomial (F) for f around n, which has a known error bound. We do this for multiple different q values, specifically the set q\in Q for a certain set Q, and essentially average the upper bounds for the different values of q. In order to do this we need to estimate S=\sum_{q\in Q}{e^{2\pi iF(q)}} for F(q) a polynomial.

We choose Q={xy|1\leq x,y\leq P}, with each element appearing with the multiplicity of how many times it can be represented in such a form. Thus S:=\sum_{q\in Q}{e^{2\pi iF(q)}}=\sum_{1\leq x\leq P}\sum_{1\leq y\leq P}e^{2\pi iF(xy)}.

Rather than estimating S directly we instead estimate some power S^{l}. By rearranging and using Hölder’s inequality we gain

S^{l}\leq P^{l-1}\sum_{x}|\sum_{y}e^{2\pi iF(sy)}|^{l}
=P^{l-1}\sum_{\lambda_{1},...\lambda_{k}}\nu(\lambda_{1},...\lambda_{k})\sum_{x}{\zeta(x)e^{2\pi i(\alpha_{1}\lambda_{1}x+...+\alpha_{k}\lambda_{k}x^{k})}}
where \zeta_{x} is some complex number of modulus one for each x, and \nu(\lambda_{1},...\lambda_{k}) is the number of integer solutions to the equations \sum_{i=1}^{n}y_{i}^{n}=\lambda_{n} for 1\leq n\leq k and 1\leq y_{i}\leq P, and the \alpha_{i}‘s are the coefficients to the Taylor Polynomial.
Thus
|S|^{2l^{2}}\leq P^{2l(l-1)}I_{l,k}(P)^{2(l-1)}J_{k,l}(P)Z_{l,k}(P)
where
_{l,k}(P)=\sum_{\lambda_{1},...,\lambda_{k}}\nu(\lambda_{1},...,\lambda_{k})
J(l,k)(P)=\sum_{\lambda_{1},...,\lambda_{k}}\nu(\lambda_{1},...,\lambda_{k})^{2}
Z(l,k)=\sum_{\lambda_{1},...,\lambda_{k}}{|\sum_{x}\zeta_{x}e^{2\pi i (\alpha_{1}\lambda_{1}x+...+\alpha_{k}\lambda_{k}x^{k})}|^{2l}}
It turns out that it is possible to estimate Z_{l,k}(P) in terms of J_{l,k}(P).
We can interpret J_{l,k}(P) as the number of integer solutions to the equations
y_{1}^{n}+...+y_{l}^{n}-y_{1+1}^{n}-...-y_{2l}^{n}=0 for 1\leq n\leq k, and 1\leq y_{j} \leq P for 1\leq j\leq 2l.
Bounds on the number of solutions to this Diophantine equation (and those defining I(l,k)(P)) can then be traced back through the above process to give abound on exponential sums, which can then be used to give the following bound on \zeta(s):
For s=\sigma+it, t\geq2, \frac{1}{2}\leq \sigma\leq 1: \zeta(s)=O(t^{\sigma(1-\sigma)^{3/2}}(\log(t))^{2/3}.

The bound on \zeta(s) that will be used to prove the de-la-Vallée  Poussin zero free region, is that \zeta(s)=O(\log(t)) in the region 1-\frac{A}{\log(t)}\leq \sigma \leq 1 for any positive constant A, and sufficiently large t[6].

Deriving the Zero Free Regions

We now use the approximations of \zeta(s) to show the existence of a zero-free region.  We will link the zeros of the zeta function to its magnitude by use of the Cauchy Integral formula.  Consider a circle in the upper plane of radius r, centre s_{0}.  We then remove the singularities of \zeta(s) within the concentric circle of radius \frac{r}{2} by defining g(s)=\zeta(s)\prod({s-\rho})^{-1} where the product is over the zeros within the smaller circle (counting multiplicity).  We can then use bounds on \zeta(s) to gain bounds on g(s), which can in turn be used to show that h(s)=\log(\frac{s}{s_{0}}) is regular and has bounded real part.  Thus we can use the Borel-Carathéodery Theorem to bound h(s).  Then we use the regularity of h(s) and the Cauchy Integral formula to use the bound on h(s) to gain a bound for |h'(s)|=|\frac{\zeta'(s)}{\zeta(s)}-\sum_{rho}{\frac{1}{s-\rho}}|, where the summation is over all zeros \rho in the circle of radius \frac{r}{2} centred at s_{0}.

We now choose functions \theta(t) and \phi(t) satisfying:
1)\phi(t) and 1/\theta(t) are increasing functions for t\geq 0, with \theta(t)\leq 1, and \phi(t) \rightarrow \infty as t\rightarrow\infty.
2)\frac{\phi(t)}{\theta(t)}=o(e^{\phi(t)})
3)\zeta(s)=O(^{\phi(t)}) as t\rightarrow\infty in the region 1-\theta(t)\leq \sigma\leq 2.
We also suppose there is zero of the form \beta + i\gamma in the upper half plane, and we choose 1+e^{-\phi(2\gamma+1)}\leq \sigma_{0}\leq 2.

We can use the above with circles of radius r=\theta(2\gamma+1) to gain, and  B\frac{\phi(2\gamma+1)}{\theta(2\gamma+1)}-\frac{1}{\sigma_{0}-\beta}, if \beta \geq \sigma_{0}-\frac{r}{2}, as an upper bound for \Re(\frac{\zeta'(\sigma+i\gamma)}{\zeta(\sigma+i\gamma)}), and B\frac{\phi(2\gamma+1)}{\theta(2\gamma+1)} as an upper bound for \Re(\frac{\zeta'(\sigma+2i\gamma)}{\zeta(\sigma+2i\gamma)}) (for some constant B).   If \beta < \sigma_{0}-\frac{r}{2} the inequality derived in this section follows from this inequality for sufficiently large t.

Consider the trigonometric polynomial 3+4cos(\theta)+cos(2\theta)= =2(1+2cos(\theta)+cos^{2}(\theta))=2(1+cos(theta))^{2} \geq 0.  We can now gain an inequality involving the zeta function by:

-3\frac{\zeta'(\sigma)}{\zeta(\sigma)}-4\Re(\frac{\zeta'(\sigma+i\gamma)}{\zeta(\sigma+i\gamma)})-\Re(\frac{\zeta'(\sigma+2i\gamma)}{\zeta(\sigma+2i\gamma)})=\sum_{p,m}\frac{\log(p)}{p^{m\sigma}(3+4\cos(m\gamma\log(p))+2\cos(2m\gamma\log(p)))}. Note that the identity for \frac{\zeta'(s)}{\zeta(s)} used comes from logarithmic differentiation of the Euler product formula for \zeta(s).

Due to the pole of \frac{\zeta'(s)}{\zeta(s)} at s=1 we have that as for \sigma_{0} close to one, we can use \frac{a}{\sigma_{0}-1} for some a, as an upper bound on \frac{\zeta'(s)}{\zeta(s)}.

Putting the estimates for \frac{\zeta'(s)}{\zeta(s)}, \Re(\frac{\zeta'(\sigma+2i\gamma)}{\zeta(\sigma+2i\gamma)}) and \Re(\frac{\zeta'(\sigma+i\gamma)}{\zeta(\sigma+i\gamma)}) into the inequality coming from the trigonometric inequality, with a suitable choices of a, gives the below theorem[6].

Theorem 1:  Let \phi(t) and \theta(t) be two functions such that:
1)\phi(t) and 1/\theta(t) are increasing functions for t\geq 0, with \theta(t)\leq 1, and \phi(t) \rightarrow \infty as t\rightarrow\infty.
2)\frac{\phi(t)}{\theta(t)}=o(e^{\phi(t)})
3)\zeta(s)=O(^{\phi(t)}) as t\rightarrow\infty in the region 1-\theta(t)\leq \sigma\leq 2.

Then for sufficiently large t, there are no zeros in the region \sigma \geq 1-A\frac{\theta(2t+1)}{\phi(2t+1)}, for some positive constant A[6].

Using the bounds for \zeta(s) from the previous section, we can gain the zero-free regions stated in the first section of this post.

We can see that in the above theorem we require; a bound on the growth of \zeta(s) within a certain region close to the line \Re(s)=1, note that due to the symmetry of the set of zeros discussed in the previous blog post it is only necessary to consider the part of the critical strip with \Im(s)\geq 0 and \frac{1}{2} \leq Re(s) \leq 1.  The bounds on the exponential sums in the above section allowed us to gain, via the approximate functional equations, bounds on \zeta(s), within a certain area.  When we substitute these into Theorem 1 above these give us the zero-free regions.

If we want known constants, as opposed to unknown constants in the zero-free regions the implicit constants in the above derivations must all be replaced by known constants.  Over last summer I did this for the Littlewood zero-free region, however unfortunately the resulting constants were of a magnitude such that they did not improve upon the explicit version of the Vinogradov-Korobov zero free region given by Ford [1].

Conclusion

While the above method has given us many of the zero-free regions, including many of the most powerful zero-free regions, there are some other methods of deriving such results.  It should be noted that all the zero-free regions currently existing have their width asymptotically approach zero when t approaches infinity, and this will be the case for all regions derived using Theorem 1.  In the notes to chapter 3 of [6] some other methods of deriving zero-free regions are looked at, however none give improvements on the Vinogradov-Korobov region.

The next and final post in this short series will look at how some of the theory in this post and in the previous post, can be extended to more general cases, in particular that of Dirichlet L-functions (of which the Riemann zeta function is a single example).

References
[1] K. Ford, Zero-free regions for the Riemann zeta function, in Number theory
for the millennium, II (Urbana, IL, 2000), A K Peters, Natick, MA, 2002,
pp. 25-56.
[2] A. Ivic, The Riemann Zeta-Function: Theory and Applications, Wiley, New
York, 1985.
[3] H. Iwaniec and E. Kowalski, Analytic number theory, no. v. 53 in American
Mathematical Society Colloquium publications, American Mathematical
Society, 2004.
[4] H. Kadiri, Une region explicite sans zeros pour la fonction  de Riemann,
Acta Arith., 117 (2005), pp. 303-339.
[5] D. Mumford, Tata Lectures on Theta, Progress in Mathematics Series,
Birkhäuser, 1994.
[6] E. Titchmarsh, The Theory of the Riemann Zeta-Function, Oxford University
Press, New York, 2nd ed., 1986.

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