# E. Kowalski's blog

## Fouvry 60

We are currently enjoying in Marseille the warmth and delights of a French Mediterranean Bouillabaisse while celebrating analytic number theory and the achievements of É. Fouvry, on the occasion of his 60th birthday.

I think everyone who has been in contact with any of his papers has immense respect for his scientific work. All those of us who have been fortunate enough to talk with him beyond purely scientific matters will also attest to his exemplary intellectual honesty, rectitude, generosity and — also important to my mind — to his sense of humor.

In analytic number theory, we play day to day in a wild down-to-earth jungle. We also all know that somewhere there is a Garden of Eden, where the Riemann Hypothesis roams free, and we hope to go there one day. Fewer know that there is a place even beyond, a Nirvana where even the Riemann Hypothesis is but a shadow of a deeper truth. And fewer still are those who have set foot in this special place. É. Fouvry did, and he was among the very first ones, if not the very first; and more people have walked on the moon than been there.

A few years ago, I wrote a nominating letter for Étienne’s application to the Institut Universitaire de France. There is one sentence that I wrote which still seems to me to summarize best my feelings about this part of his work: Rarely in history was so much owed by so many arithmeticians to so few. This is even truer today than it was then. Reader, if you care at all about prime numbers, recall that without É. Fouvry and very few others (two of whom are with us in Marseille), you might well never have known that the gaps between successive primes do not grow to infinity.

June 20th, 2013 at 7:37 pm

## Yet another remark on the Friedlander-Iwaniec sum

one comment

I just remembered a point I had intended to make concerning the exponential sum of Friedlander and Iwaniec that is crucial in Zhang’s work on gaps between primes, but which slipped my mind. This may present the argument in my note with Fouvry and Michel in a more enlightening way, although it does not simplify the proof (I’ve now added this as a remark in the PDF file.)

We view the sum as
$S(\alpha,\beta)=\sum_{x\in \mathbf{F}_p} a_1(x)\overline{a_2(x)}$
where $p$ is a prime and the coefficients $a_1(x)$ and $a_2(x)$ are values of Kloosterman sums:
$a_1(x)=K_2(x),\quad\quad a_2(x)=K_2(\beta x/(x+\alpha))$
for some parameters $\alpha$ and $\beta$, non-zero elements of $\mathbf{F}_p$, putting
$K_2(x)=-\frac{1}{\sqrt{p}}\sum_y e\Bigl(\frac{xy+1/y}{p}\Bigr).$

Now the point is that, from the “automorphic” view of trace functions (as discussed in one of my previous posts), both $a_1(x)$ and $a_2(x)$ can be seen as the Hecke eigenvalues, at the prime $T-x$, of a cusp form on $GL_2(\mathbf{F}_p(T))$, i.e., of the analogue of a classical cusp form, living however over a function field instead of $\mathbf{Q}$ — so the polynomial ring $\mathbf{F}_p[T]$ plays its role of cousin of $\mathbf{Z}$. This result (the existence of these cusp forms) is by no means obvious, but it follows from Deligne’s construction of Kloosterman sheaves and Drinfeld’s proof of the Langlands correspondance for $GL(2)$ over function fields.

In any case, if one admits this, the sum above is clearly an exact analogue of a sum of Hecke eigenvalues at primes of a Rankin-Selberg $L$-function associated to two cusp forms! If we know the Riemann Hypothesis for this Rankin-Selberg $L$-function, we can then very classically establish a bound
$S(a,b)\ll \sqrt{p},$
in full analogy with conditional results for Rankin-Selberg $L$-functions over number fields, provided the two cusp forms are not the same (otherwise there is a pole with a larger contribution). But here the two cusp forms are not the same simply because their “conductor” (in the arithmetic sense: in fact, just the location of ramified primes matters, i.e., the analogue of the divisors of the level of a primitive cusp form) are not equal!

Finally, the implied constant in applying the Riemann Hypothesis is uniformly bounded in terms of the conductor of the two cusp forms, and these are bounded independently of the prime and parameters, “explaining” the Birch-Bombieri result.

I emphasize again that this is not really a good way of writing a proof, because showing that the Rankin-Selberg $L$-functions over function fields satisfy GRH is harder than proving Deligne’s Riemann Hypothesis in the more geometric language of sheaves and their trace functions, simply because the argument reduces to Deligne’s result (this was, I think, first done by Lafforgue, though maybe Drinfeld had proved it for $GL(2)$). But this interpretation should make the argument very readable and natural to an analytic number theorist.

June 13th, 2013 at 10:17 pm

Posted in Mathematics

## Bounded gaps between primes: some grittier details

Because I was teaching a course on prime numbers this semester and had just finished a chapter on Vinogradov’s method when his paper appeared, I promptly switched my plans for the last classes in order to present some aspects of Yitang Zhang’s theorem on bounded gaps between primes. In addition, one of the speakers of this week’s conference celebrating 25 years of the “Zahlentheorie Seminar” of ETH had to cancel at short notice, and I replaced her and gave yesterday another survey-style talk. The notes for the latter (such as they are…) can be downloaded in scanned form.

My insights to Zhang’s work remain clearly superficial, but here are some remarks going a bit beyond what I mentioned in the previous post, coming after these lectures, and some discussions with Ph. Michel and P. Nelson.

(1) The most delicate estimates seem to be those for the “Type III” sums. These concern the “good” distribution in invertible residue classes of an arithmetic function $f(n)$ for integers $N, modulo a "large modulus" $q$, where $f(n)$ is of the very special type
$f(n)=\sum_{m_1n_1n_2n_3=n}\alpha(m_1),$
and the variables $m_1$, $n_1$, $n_2$ and $n_3$ are, roughly, of size $M_1$, $N_i$ with $M_1N_1N_2N_3=N$, and (crucially) $q$ is a bit larger than $N^{1/2}$: one needs to handle these for $q$ up to $N^{1/2+\delta}$ for some $\delta>0$ in order to obtain bounded gaps between primes.

The lengths $M_1$ and $N_i$ are constrained in various ways, and the most critical case seems to be when $M_1\approx q^{1/8}$, $N_i\approx q^{5/8}$ (the $\approx$ means that one must be able to go a bit beyond such a case, since $N$ is a bit beyond $q^2$).

(2) Another point is that Zhang manages to bound these sums for each individual residue class $a\pmod q$ (coprime to $q$). In other words, denoting
$\Delta_f(N,q,a)=\sum_{N
he proves individual bounds for $\Delta_f(N,q,a)$, instead of average bounds over $q$ (as in the other main part of this argument).

Also, he does not need to use the variable $m_1$ at all (but since the $\alpha(m_1)$ are mostly unknown coefficients, and the sum is rather short, exploiting it does not seem easy). Hence the result looks enormously like controlling the distribution in residue classes of the ternary divisor function. This is exactly the question that Friedlander and Iwaniec had studied in the famous paper where they proved that the exponent of distribution is at least $1/2+1/230$, but their argument is not quite sufficient for Zhang's purpose.

(3) One of the last tricks is Zhang's second use of the structure of the moduli $q$ that are involved in his argument: these were chosen to have only prime factors $\leq z=N^{\delta}$ for some small positive $\delta$), and Zhang exploits the "granular" (or friable) structure of such moduli in order to obtain flexibility in the possibility of factoring them as $q=rs$ with $r$ of size determined up to a factor at most $z$. This is particularly important for the “other” sums (it gives bilinear structure and makes it possible to use the dispersion method of Linnik, as already done by Fouvry-Iwaniec and Bombieri-Friedlander-Iwaniec in their works on primes in arithmetic progressions). For the "type III" case, it does not seem to be so much of the essence, but Zhang needs to gain a very small amount compared with Friedlander-Iwaniec, and does so by factoring $q=rs$ with $r$ rather small. He then gains from $r$ a factor $r^{1/2}$ which is essential, by exploiting the fact that a Ramanujan sum modulo $r$ is bounded (so he gets more than square-root cancellation from $r$…)! This is an extremely special situation, and right now, it is what seems the most “miraculous” about this proof (at least to me).

It is for the contribution of the complementary divisor $s$ that Zhang manages to position himself into applying the estimate of Birch and Bombieri for the exponential sums which Friedlander-Iwaniec had also encountered.

(4) This use of Deligne’s work is also very delicate: one can not relax the requirement of square-root cancellation, except by very tiny amounts. For instance, obtaining a bound of size $p^2$ for the three-variable sum modulo $p$ is useless; in fact, the bound $p^2$ can be considered here as the trivial estimate, since the sum can be written as an average of one-variable Kloosterman sums. With Zhang’s parameters, one needs an estimate of for the sum which is no larger than $p^{3/2+1/2000}$ (or so) in order to get the desired gain. However, as I explained in the last five minutes of my talk today (and as is explained in this note with Fouvry and Michel) the Birch-Bombieri bound is very well understood from a conceptual point of view.

(5) I was very curious, when first looking at the paper, to see how Zhang would handle the residue classes in the Goldston, Pintz, Yıldırım method, since the most uniform results on primes in arithmetic progressions (those of Fouvry-Iwaniec and Bombieri-Friedlander-Iwaniec) are constrained to use (essentially) a single residue class. What happens is that Zhang detects these classes by inserting a factor corresponding to their characteristic function, and by avoiding the Kloostermania approaches that rely on spectral theory of automorphic forms. The important properties of these residue classes are their multiplicative structure (coming from the Chinese Remainder Theorem) and the fact that, on average over moduli, there are not too many of them (the average is bounded by a power of $\log N$). In particular, his use of the dispersion method is in fact closer in spirit to some of its earliest uses by Fouvry and Iwaniec (for integers without small prime factors instead of primes), which also involved, in the final steps, the classical Weil bound for Kloosterman sums instead of (seemingly stronger) results on sums of Kloosterman sums.

June 4th, 2013 at 8:39 am

Posted in ETH,Mathematics

## Q.E.D, H.E.Q.D.I, H.E.Q.D.P, Q.D.I, Q.F.I, and all that

Today’s “Word of the day” from the Oxford English Dictionary is quod erat demonstrandum (which prompts the question of the largest number of words in an OED “word”, but let us leave that aside for now).

The full entry shows latin translators of Euclid toiling mightily to catch just the right sizzle of the Greek “ὅπερ ἔδει δεῖξαι” (which, we are told, means literally “what needed to be [shown] has been shown”), just like Joyce or Wodehouse trying to find the perfect mot juste for their latest novel.

So were tried, in turn, hoc est quad demonstrare intendimus (“This is what we intended to demonstrate”, too pedestrian); hoc est quod demonstrare proposimum (“This is what we proposed to demonstrate”, too egocentric); quod demonstrare intendimus (“What we intended to demonstrate”; dashing, but reeks of false modesty); quod fuit propositum (“Which was proposed”, yes, it was proposed, but is it proved?) — and who knows how many other versions which go unmentioned.

I personally think that the Greek sounds and reads much better; this suggests introducing the abbreviation “ὅἔδ”, or the transliteration “OED” if needed, instead of “QED” in the more refined mathematical literature…

May 29th, 2013 at 12:15 pm

Posted in Language,Mathematics

## À la…

A few months ago, I wondered what could be the largest cluster of foreign words in the Oxford English Dictionary, citing the examples of femme something-or-other and sympathique and company. It turns out that there is a much larger one! Here is the à la cluster:

à la 1579
à la bonne heure 1750
à la broche 1806
à la brochette 1821
à la carte 1816
à la crème 1741
à la fourchette 1817
à la Française 1589
à la modality 1753
à la mode 1637
à-la-modeness 1669
à la mort 1536
à la page 1930
à la roi 1852
à la royale 1853
à la Russe 1775
à la Turquie 1676

That’s no less than 18 items (the date on the right is the first OED citation). It’s interesting that so many have to do with food, and even more that three or four are basically synonyms of “in fashion” (this is what à la page basically means). I have to admit to being partial to à-la-modeness for its translanguage qualities, although I don’t know if I will be able to use it intelligently anytime soon (though one never knows; after all, I did manage to sneak ptarmigan in a recent paper…)

May 27th, 2013 at 8:38 pm