In the previous set of notes, we saw how zero-density theorems for the Riemann zeta function, when combined with the zero-free region of Vinogradov and Korobov, could be used to obtain prime number theorems in short intervals. It turns out that a more sophisticated version of this type of argument also works to obtain prime number theorems in arithmetic progressions, in particular establishing the celebrated theorem of Linnik:
In fact it is known that one can find a prime with , a result of Xylouris. For sake of comparison, recall from Exercise 65 of Notes 2 that the Siegel-Walfisz theorem gives this theorem with a bound of , and from Exercise 48 of Notes 2 one can obtain a bound of the form if one assumes the generalised Riemann hypothesis. The probabilistic random models from Supplement 4 suggest that one should in fact be able to take .
We will not aim to obtain the optimal exponents for Linnik’s theorem here, and follow the treatment in Chapter 18 of Iwaniec and Kowalski. We will in fact establish the following more quantitative result (a special case of a more powerful theorem of Gallagher), which splits into two cases, depending on whether there is an exceptional zero or not:
Assume that for some sufficiently large .
- (i) (No exceptional zero) If all the real zeroes of -functions of real characters of modulus are such that , then
for all and some absolute constant .
- (ii) (Exceptional zero) If there is a zero of an -function of a real character of modulus with for some sufficiently small , then
for all and some absolute constant .
The implied constants here are effective.
Note from the Landau-Page theorem (Exercise 54 from Notes 2) that at most one exceptional zero exists (if is small enough). A key point here is that the error term in the exceptional zero case is an improvement over the error term when no exceptional zero is present; this compensates for the potential reduction in the main term coming from the term. The splitting into cases depending on whether an exceptional zero exists or not turns out to be an essential technique in many advanced results in analytic number theory (though presumably such a splitting will one day become unnecessary, once the possibility of exceptional zeroes are finally eliminated for good).
Exercise 3 Assuming Theorem 2, and assuming for some sufficiently large absolute constant , establish the lower bound
when there is no exceptional zero, and
for any primitive residue class and any . This is (barely) consistent with the estimate (1). Any lowering of the coefficient in the Brun-Titchmarsh inequality (with reasonable error terms), in the regime when is a large power of , would then lead to at least some elimination of the exceptional zero case. However, this has not led to any progress on the Landau-Siegel zero problem (and may well be just a reformulation of that problem). (When is a relatively small power of , some improvements to Brun-Titchmarsh are possible that are not in contradiction with the presence of an exceptional zero; see this paper of Maynard for more discussion.)
for any non-principal character of modulus , where we assume for some large ; for the principal character one has the same formula with an additional term of on the right-hand side (as is easily deduced from Theorem 21 of Notes 2). Using the Fourier inversion formula
(see Theorem 69 of Notes 1), we thus have
when an exceptional zero is present.
Proposition 5 (Classical zero-free region) Let . Apart from a potential exceptional zero , all zeroes of -functions with of modulus and are such that
for some absolute constant .
Using this zero-free region, we have
where we recall that is the number of zeroes of any -function of a character of modulus with and (here we use conjugation symmetry to make non-negative, accepting a multiplicative factor of two).
In Exercise 25 of Notes 6, the grand density estimate
is proven. If one inserts this bound into the above expression, one obtains a bound for (2) which is of the form
Unfortunately this is off from what we need by a factor of (and would lead to a weak form of Linnik’s theorem in which was bounded by rather than by ). In the analogous problem for prime number theorems in short intervals, we could use the Vinogradov-Korobov zero-free region to compensate for this loss, but that region does not help here for the contribution of the low-lying zeroes with , which as mentioned before give the dominant contribution. Fortunately, it is possible to remove this logarithmic loss from the zero-density side of things:
The implied constants are effective.
We prove this estimate below the fold. The proof follows the methods of the previous section, but one inserts various sieve weights to restrict sums over natural numbers to essentially become sums over “almost primes”, as this turns out to remove the logarithmic losses. (More generally, the trick of restricting to almost primes by inserting suitable sieve weights is quite useful for avoiding any unnecessary losses of logarithmic factors in analytic number theory estimates.)
Now we turn to the case when there is an exceptional zero (3). The argument used to prove (2) applies here also, but does not gain the factor of in the exponent. To achieve this, we need an additional tool, a version of the Deuring-Heilbronn repulsion phenomenon due to Linnik:
In other words, the exceptional zero enlarges the classical zero-free region by a factor of . The implied constants are effective.
Exercise 10 Use Theorem 8 to give an alternate proof of (Tatuzawa’s version of) Siegel’s theorem (Theorem 62 of Notes 2). (Hint: if two characters have different moduli, then they can be made to have the same modulus by multiplying by suitable principal characters.)
Theorem 8 is proven by similar methods to that of Theorem 6, the basic idea being to insert a further weight of (in addition to the sieve weights), the point being that the exceptional zero causes this weight to be quite small on the average. There is a strengthening of Theorem 8 due to Bombieri that is along the lines of Theorem 6, obtaining the improvement
with effective implied constants for any and in the presence of an exceptional zero, where the prime in means that the exceptional zero is omitted (thus if ). Note that the upper bound on falls below one when for a sufficiently small , thus recovering Theorem 8. Bombieri’s theorem can be established by the methods in this set of notes, and will be given as an exercise to the reader.
Remark 11 There are a number of alternate ways to derive the results in this set of notes, for instance using the Turan power sums method which is based on studying derivatives such as
for and large , and performing various sorts of averaging in to attenuate the contribution of many of the zeroes . We will not develop this method here, but see for instance Chapter 9 of Montgomery’s book. See the text of Friedlander and Iwaniec for yet another approach based primarily on sieve-theoretic ideas.
Remark 12 When one optimises all the exponents, it turns out that the exponent in Linnik’s theorem is extremely good in the presence of an exceptional zero – indeed Friedlander and Iwaniec showed can even get a bound of the form for some , which is even stronger than one can obtain from GRH! There are other places in which exceptional zeroes can be used to obtain results stronger than what one can obtain even on the Riemann hypothesis; for instance, Heath-Brown used the hypothesis of an infinite sequence of Siegel zeroes to obtain the twin prime conejcture.
— 1. Log-free density estimate —
We now prove Theorem 6. We will make no attempt here to optimise the exponents in this theorem, and so will be quite wasteful in the choices of numerical exponents in the argument that follows in order to simplify the presentation.
(say); we may also assume that is larger than any specified absolute constant. We may then replace by in the estimate, thus we wish to show that
Observe that in the regime
the claim already follows from the non-log-free density estimate (4). Thus we may assume that
for some , and the claim is now to show that there are at most zeroes of -functions with , , and a character of modulus . We may assume that , since the case follows from the case (and also essentially follows from the classical zero-free region, in any event).
For minor technical reasons it is convenient to first dispose of the contribution of the principal character. In this case, the zeroes are the same as those of the Riemann zeta function. From the Vinogradov-Korobov zero-free region we conclude there are no zeroes with and . Thus we may restrict attention to non-principal characters .
Suppose we have a zero of a non-principal character of modulus with and . From equation (48) of Notes 2 we then have
(say) for all . One can of course obtain more efficient truncations than this, but as mentioned previously we are not trying to optimise the exponents. If one subtracts the term from the left-hand side, this already gives a zero-detecting polynomial, but it is not tractable to work with because it contains too many terms with small (and is also not concentrated on those that are almost prime). To fix this, we weight the previous Dirichlet polynomial by , where is an arithmetic function supported on to be chosen later obeying the bound . We expand
and hence by (7) and the upper bound on
Since , one sees from the divisor bound and the hypothesis that is large that
If we have , then we can extract the term and obtain a zero-detecting polynomial:
We now select the weights . There are a number of options here; we will use a variant of the “continuous Selberg sieve” from Section 2 of Notes 4. Fix a smooth function that equals on and is supported on ; we allow implied constants to depend on . For any , define
Observe from Möbius inversion that for all . The weight was used as an upper bound Selberg sieve in Notes 4.
We will need the following general bound:
Proof: Clearly, we can restrict to those numbers whose prime factors do not exceed , for some large absolute constant .
for some rapidly decreasing function , and thus the left-hand side of (10) may be written as
where we implicitly restrict to numbers whose prime factors do not exceed (note that this makes the integrand absolutely summable and integrable, so that Fubini’s theorem applies). We may factor this as
By the rapid decrease of , it thus suffices to show that
By Taylor expansion we can bound the left-hand side by
By Mertens theorem we can replace the constraint with . Since , it thus suffices to show that
But we can factor
and the claim follows from Mertens’ theorem.
We record a basic corollary of this estimate:
for any .
Proof: Writing , we can write the left-hand side of (12) as
Since is supported on and is bounded above by , the contribution of the error is which is acceptable. By Lemma 13 with , the contribution of the main term is , and the claim then follows from Mertens’ theorem.
Now we prove (13). Using Rankin’s trick, it suffices to show that
The left-hand side factorises as
From Lemma 13 with , we see that
(using Mertens theorem to control the error between and ) and the claim follows.
We will work primarily with the cutoff
the reason for the separate scales and will become clearer later. The function is supported on , equals at , and is bounded by , so from the previous discussion we thus have the zero-detector inequality
whenever with of modulus , , and . Our objective is to show that the number of such zeroes is .
We first control the number of zeroes that are very close together. From equation (48) of Notes 2 with (say), we see that
whenever , , and is non-principal of modulus ; also from equation (45) of Notes 2 we have
From Jensen’s theorem (Theorem 16 of Supplement 2), we conclude that for any given non-principal and any , there are at most zeroes of (counting multiplicity, of course) with and . To prove Theorem 6, it thus suffices by the usual covering argument to establish the bound
Note from the existing grand zero-density estimate in (4) that
We write (14) for the zeroes as
for all , where
and is a smooth function supported on which equals on . Note that the term in is .
We use the generalised Bessel inequality (Proposition 2 from Notes 3) with to conclude that
where are complex numbers with . (Strictly speaking, one needs to deal with the issue that the are not finitely supported, but there is enough absolute convergence here that this is a routine matter.) From Corollary 14 we have
(note how the logarithmic factors cancel, which is crucial to obtaining our “log-free” estimate) and so from (18), the inequality and symmetry it suffices to show that
for all .
Making the change of variables , this becomes
The integral is bounded by , and from two integration by parts it is also bounded by
On the other hand, for , the are -separated by hypothesis, and so
and the claim follows.
— 2. Consequences of an exceptional zero —
In preparation for proving Theorem 8, we investigate in this section the consequences of a Landau-Siegel zero, that is to say a real character of some modulus with a zero
for some with small. For minor technical reasons we will assume that is a multiple of , so that ; this condition can be easily established by multiplying by a principal character of modulus dividing . (We will not need to assume here that is primitive.)
Also, from the class number formula (equation (56) from Notes 2) we have
For the arguments below, one could also use the slightly weaker estimates in Exercise 67 of Notes 2 or Exercise 57 of Notes 3 and still obtain comparable results. We will however not rely on Siegel’s theorem (Theorem 62 of Notes 2) in order to keep all bounds effective.
We now refine this analysis. We begin with a complexified version of Exercise 58 from Notes 2:
for any . (Hint: use the Dirichlet hyperbola method and Exercise 44 from Notes 2.)
If is a non-principal character of modulus with , show that
For technical reasons it will be convenient to work with a completely multiplicative variant of the function . Define the arithmetic function to be the completely multiplicative function such that for all ; this is equal to at square-free numbers, but is a bit larger at other numbers. Observe that is non-negative, and has the factorisation
where is a multiplicative function that vanishes on primes and obeys the bounds
for all and primes . In particular is non-negative and for , since we assumed . From Euler products we see that
and also that
whenever with .
Taking Dirichlet series, we see that
whenever ; more generally, we have
for any character . Now we look at what happens inside the critical strip:
for any and .
for any and .
We record a nice corollary of these estimates due to Bombieri, which asserts that the exceptional zero forces to vanish (or equivalently, to become ) on most large primes:
Informally, Bombieri’s lemma asserts that for most primes between and . The exponent of here can be lowered substantially with a more careful analysis, but we will not do so here. For primes much larger than , becomes equidistributed; see Exercise 23 below.
Proof: Without loss of generality we may take to be a multiple of . We may assume that , as the claim follows from Mertens’ theorem otherwise; in particular .
By (28) for we have
Next, applying (30) with replaced by and subtracting, we have
As , we have by Taylor expansion. As before, the error term can be bounded by the main term and so
Since is non-negative and completely multiplicative, one has
and thus (since )
Since , we have , and the claim follows.
Exercise 18 With the hypotheses of Bombieri’s lemma, show that
for any natural number .
Now we can give a more precise version of (24):
for any with .
Observe that (24) is a corollary of the case of this proposition thanks to Mertens’ theorem and the trivial bounds . We thus see from this proposition and Bombieri’s lemma that the exceptional zero controls at primes larger than , but that is additionally sensitive to the values of at primes below this range. For an even more precise formula for , see this paper of Goldfeld and Schinzel, or Exercise 24 below.
Proof: By Bombieri’s lemma and Mertens’ theorem, it suffices to prove the asymptotic for .
We begin with the upper bound
Applying (26) with and we have
The left-hand side is non-negative and , so we conclude (using (25)) that
From Euler products and Mertens’ theorem we have
But from Lemma 17 and Mertens’ theorem we see that
and the claim follows.
Now we establish the matching lower bound
Applying (26) with and we have
For , we have , and thus by (25)
and the claim then follows from the preceding calculations.
Remark 20 One particularly striking consequence of an exceptional zero is that the spacing of zeroes of other -functions become extremely regular; roughly speaking, for most other characters whose conductor is somewhat (but not too much) larger than the conductor of , the zeroes of (at moderate height) mostly lie on the critical line and are spaced in approximate arithmetic progression; this “alternative hypothesis” is in contradiction to to the pair correlation conjecture discussed in Section 4 of Supplement 4. This phenomenon was first discovered by Montgomery and Weinberger and can roughly be explained as follows. By an approximate functional equation similar to Exercise 54 of Supplement 3, one can approximately write as the sum of plus times a gamma factor which oscillates like when . The smallness of on average for medium-sized (as suggested for instance by Bombieri’s lemma) suggests that these sums should be well approximated by much shorter sums, which oscillate quite slowly in . This gives an approximation to that is of the form for slowly varying , which can then be used to place the zeroes of this function in approximate arithmetic progression on the real line.
— 3. The Deuring-Heilbronn repulsion phenomenon —
We now prove Theorem 8. Let be such that there is an exceptional zero with small, associated to some quadratic character of modulus :
(say), since the claim is trivial otherwise. By multiplying by the principal character of modulus if necessary, we may assume as before that is a multiple of , so that we can utilise the multiplicative function from the previous section. By enlarging , we may assume as in Section 1 that
The task (35) is then equivalent to showing that
for some absolute constant .
We recall the sieve cutoffs
from Section 1, which were used in the zero detector. The main difference is that we will “twist” the polynomial by the completely multiplicative function :
Proposition 21 (Zero-detecting polynomial) Let the notation and assumptions be as above.
(say) for any . In particular, as is supported on and one has from the divisor bound, one has
for . By a similar calculation to before, we have
Using the estimates from the previous section, we can establish the following bound:
The point here is that the sieve weights and are morally restricting to almost primes, and that should be small on such numbers by Bombieri’s lemma. Assuming this proposition, we conclude that the left-hand sides of (40) or (41) are , and (39) follows.
We begin with the second bound (42), which we establish by quite crude estimates. By a Fourier expansion we can write
which factorises as
Bounding , we thus have
for any . Squaring and using Cauchy-Schwarz, we conclude that
for any . In particular, for , we have
and so we can bound the left-hand side of (43) by
which we bound by
By Mertens’ theorem we have
and the claim follows by taking large enough.
Now we establish (42). By dyadic decomposition it suffices to show that
for all . The left-hand side may be written as as
From the hyperbola method we see that
for any , and thus
The proof of Theorem 8 is now complete.
for all and some absolute constant . (Hint: use (3) and the explicit formula.) Roughly speaking, this exercise asserts that is equidistributed for primes with much larger than .