On Hecke Eigenvalues at
Piatetski-Shapiro Primes
Joint Work with Stephan Baier
Liangyi Zhao
Nanyang Technological University
arXiv:0808.1756
The mean-values of many arithmetic functions are more or less wellunderstood until we begin to average over sparse sequences. The
best known example of this is perhaps the following.
Consider the von Mangoldt function Λ(n), the usual weight with
which we count primes. Λ(n) = log p if n = pk for p ∈ P and k ∈ N
and Λ(n) = 0 otherwise.
We understand rather well, thanks to Dirichlet, the asymptotics for
X
Λ(an + b), for a, b ∈ Z, gcd(a, b) = 1.
n≤N
But the problem becomes hopelessly harder if we try to prove
(1)

Y 
2
Λ(n + 1) ∼
1 −
p>2
n≤N
X

−1
p 
 N.
p
1
In this spirit, asymptotic formulas are known for, due to Hooley and
Friedlander-Iwaniec, respectively
X
τ2(n2 + a) and
n≤N
X
τ3(n2 + m6),
n2+m6 ≤N
gcd(m,n)=1
where τk (n) is the number of ways to write n as the product of k
natural numbers.
The most celebrated results in this direction are due to FriedlanderIwaniec and Heath-Brown who developed, respectively, asymptotics
for
X
a2 +b4≤N
Λ(a2 + b4) and
X
a3 +2b3≤N
Λ(a3 + 2b3).
A classical result in this direction is the Piatetski-Shapiro prime
number theorem which states that
N
c
, as N → ∞
(2)
πc(N ) := #{n ≤ N : [n ] ∈ P} ∼
c log N
if c is a fixed number lying in the range 1 < c < 12/11 = 1.0909....
This c-range for which (2) holds has been widened by many authors, Heath-Brown, Kolesnik, Liu-Rivat. The most recent record
is due to Rivat and Sargos who proved that (2) holds in the range
1 < c < 2817/2426 = 1.16117....
This can be viewed as an intermediate problem between Dirichlet’s
theorem and the Hardy-Littlewood asymptotics in (1).
Lower bounds of the correct order of magnitude for πc(N ) have been
established for wider c-ranges.
It is conjectured that (2) holds for all non-integers c > 1.
D. Leitman and D. Wolke showed that (2) holds for almost all, with
respect to Lebesgue measure, c with 1 < c < 2.
Moreover, it was due to J.-M. Deshouillers that πc(N ) tends to infinity as N tends to infinity for almost all c > 1, with respect to
Lebesgue measure.
Certain extensions and generlizations of this type problem have been
considered by S. Baier.
Let f be a holomorphic cusp form of weight κ for the full modular
group. By λf (n) we denote the normalized n-th Fourier coefficient
of f , i.e.
f (z) =
∞
X
λf (n)n(κ−1)/2e(nz)
n=1
for ℑz > 0. The Ramanujan-Petersson conjecture, proved by Deligne,
states that for any fixed ε > 0, we have
|λf (n)| ≤ τ2(n) ≪ nε.(f ≪ g ⇐⇒ ∃c > 0, |f | ≤ cg.)
It is due to G. H. Hardy and S. Ramanujan, and A. Good, respectively that
X
n≤N
λf (n)e(αn) ≪ N
1/2
log(2N ), and
X
n≤N
λf (n) ≪ N 1/3+ε.
Recently, V. Blomer investigated the sum
X
λf (n2 + rn + s).
n≤N
Especially interesting is the distribution of Fourier coefficients of
cusp form at prime arguments. It is known that there exists a
constant C > 0, depending on the cusp form f , such that
(3)
X
p≤N
p
λf (p) ≪ N exp −C log N ,
where the implied constant depends on f . Under the GRH for modular L-functions, the majorant in (3) should be N 1/2+ε.
Ω-type statements have been conjectured and proved conditionally
by M. R. Murty and S. D. Adhikari for the sum in (3) for cusp forms
for the group Γ0(q).
A conjecture, known as the Sato-Tate conjecture for modular form,
describes more precisely the distribution of the values of λf (p) for
certain f ’s.
The speaker investigated special exponential sums with Fourier coefficients of cusp forms over primes, motivated by some surprising
heuristic due to Iwaniec, Luo and Sarnak.
Our result is a kind of hybrid of these two kinds of problems,
Piatetski-Shapiro primes and the value distribution of λf (n).
Theorem 1. Let 1 < c < 8/7 and λf be defined as above. P denotes
the set of primes. Then there exists a constant C > 0 depending
on f such that
(4)
X
c
p
λf ([n ]) ≪ N exp −C log N ,
n≤N
[nc]∈P
where the implied constant depends on c, C and f .
By (2) and 8/7 = 1.142... < 1.161... = 2817/2426 (recall the c-range
of Sargos and Rivat), the right-hand side of (4) is small compared
to the total number of Piatetski-Shapiro primes of the form [nc]
with n ≤ N if c lies in the range given in Theorem 1.
Analogous to the prime number theorem, it can be established that
N
, as N → ∞,
|λf (p)| ∼ cf
log
N
p≤N
X
2
where cf > 0 is some constant.
We conjecture an analogous result for Piatetski-Shapiro primes.
Conjecture 1. Under the assumptions of Theorem 1, there exists a
constant cf > 0 such that
X N
c 2
,
λf ([n ]) ∼ cf
c log N
n≤N
[nc]∈P
as N → ∞.
If Conjecture 1 holds, then by Ramanujan-Petersen conjecture
X
n≤N
[nc ]∈P
c τ2([n ]) λf ([n ]) ≥
c
X N
c 2
λf ([n ]) ∼ cf
c log N
n≤N
[nc]∈P
which is large compared to the right-hand side of (4). This implies the following conditional result on the oscillations of Fourier
coefficients of cusp forms at Piatetski-Shapiro primes.
Theorem 2. Assume that the conditions of Theorem 1 are satisfied
and Conjecture 1 holds. Then either ℜλf (p) or ℑλf (p) changes sign
infinitely often at primes of the form p = [nc], n ∈ N.
First, since Hecke eigenforms form a basis for the space of cusp
forms, it will suffice to prove Theorem 1 for Hecke eigenvalues.
The advantages of working with Hecke eigenvalues are that they
are multiplicative and real.
To detect Piatetski-Shapiro primes, we set γ = 1/c and note that
[nc] = m ⇐⇒ −(m + 1)γ < −n ≤ −mγ .
Hence
X
n≤N
c
c
Λ([n ])λ([n ]) ≈
X
([−mγ ] − [−(m + 1)γ ]) Λ(m)λ(m).
m≤N c
Set ψ(x) = x − [x] − 1/2. The above sum can be written as S1 − S2,
where S1 is a sum which does not involve [·] while S2 contains ψ(x).
S1 can be disposed using (3) and partial integration.
Treatment of S2 is considerably more complicated.
Note that ψ(x) is period with period 1. It can be well-approximated
using Fourier analytic means, a result due to J. D. Vaaler.
Moreover, the von Mangoldt function can be further decomposed
into so-called type I and type II sums using a Vaughan-type identity.
The type II sum, involving a kind of bi-linear form, is easier to
handle, as the treatment of this type of sums is generally insensitive
to λ(n). One can use the same techniques in the treatment of the
analogous sum in the classical Piatetski-Shapiro problem, a result
due to Heath-Brown.
The estimation of the type I sum makes up the core of the result.
After applying the above mentioned transformations, the relevant
sum becomes a triple sum, a double sum being introduced after applying the Vaughan-type identity and another after applying Vaaler’s
theorem.
For certain part (depending on the γ) of this triple sum, we use
estimates for some trilinear exponential sums with monomials, due
to Robert and Sargos, which is a refinement of a result of Fouvry
and Iwaniec. The under-lying ideas here are Weyl shift (A-process)
and the double large sieve. Again here, the method is completely
insensitive to the presence of λ.
The appearance of the Hecke eigenvalues requires us to introduce
some new ingredients.
For the remaining part of the sum, we shall use a method of M.
Jutila to transform exponential sums of the form
X
λf (n)e(g(n)), e(z) = exp(2πiz)
N1 <n≤N2
into other exponential sums involving Hecke eigenvalues of different
length.
Jutila’s formula may be viewed as an analogue of the B-process in
Weyl-van der Corput’s method in the theory of exponential sums.
The key under-lying idea of the formula is the Voronoi summation
formula.
The transformation formula of Jutila lies at the heart of our method.
Using this result, Jutila proved an estimate for “long” exponential
sums with Hecke eigenvalues which can be applied directly to our
sum. This simple application would give the c-range 1 < c < 9/8.
Note that 9/8 < 8/7.
We refine Jutila’s techniques, specific to our situation.
First we utilize Jutila’s transformation formula, after some further
splitting á la Jutila, to the inner-most of the triple sums, mentioned
before.
Collecting all terms, we arrive at a multi-linear exponential sum, a
quintuple sum. To estimate this, we refine the Jutila’s treatment
of “long” exponential sum and take advantage of the outer summations.
This leads to a spacing problem of certain point depending on the
summands of the outer summations. Resolving this and applying
the van der Corput lemma for classical exponential sums, we get
the final estimate.
The requirement c < 8/7 or γ > 7/8 appears so that the ranges in the
two methods over-lap. Recall that for certain ranges (depending on
γ) of the type II sum, we use one method (tri-linear sum estimate)
and use a different method (Jutila’s formula) in another range.
We believe that the c-range in Theorem 1 could be slightly widened
by modifying the methods in of Rivat and Sargos, who have the
best c-range for the classical Piatetski-Shapiro problem.
It would be highly desirable to prove Conjecture 1 for some c > 1,
making Theorem 2 unconditional for the same c-range. To this
end, we need estimates for exponential sums with squares of Hecke
eigenvalues (or more generally, with Fourier coefficients of RankinSelberg convolutions of cusp forms).
Finally, it would be interesting to generalize our result to cusp forms
of arbitrary level. One would need to work out a generalization of
Jutila’s method to arbitrary levels for this purpose.