数理解析研究所講究録
第 1806 巻 2012 年 94-98

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Some Notes on the Signsof Kloosterman Sums
Wladimir de Azevedo Pribitkin
Let us consider the classical Kloosterman sum$A_{c}(m,n)= \sum_{a(md\mathcal{C}\},ad\equiv 1\mathring{(}mod c)}e^{2\pi i(\frac{ma+nd}{c})}.$

Here the numbers and are any integers, and the modulus is a positive integer.
$A_{c}(m, n)$ is a “genuine”
sum if $mn\neq 0,$ $A_{c}(m, 0)=A_{c}(0, m)$ is a
$A_{c}(0,0)$$m\neq0$. The signifi, andis simply Buler’s totient
Ramanujan sum ifcance of Kloosterman sums to the theory of modular forms dates back a century to an
astonishingly little-known work of Poincare [10]. In 1926 Kloosterman [4] publishedhis seminal paper regarding Ramanujan’s problem of representing sufficiently large
integers by quatemary quadratic forms. Since then these sums have surfaced with analmost unreasonable ubiquity throughout arithmetic.
It is plain that $A_{c}(m, n)=A_{c}(-m, -n)$ and hence $A_{c}(m, n)$ is real. As such, itis oscillatory for fixed integers
is natural to ask whether the sequenceand . That is, are there infinitely many such that $A_{c}(m, n)>0$ and infinitely
is positive. And it ismany such that $A_{c}(m, n)<0$ ? Obviously,$m\neq0$
because of the familiar formula
is oscillatory forclear that(see, for example, [2, p. 238])
$m$

$c$

$n$

$KlooSterm\mathfrak{W}$$\phi(c)$$\{A_{c}(m, n)\}_{c=1}^{\infty}$

$m$

$c$

$n$

$\{A_{c}(0,0)\}_{c=1}^{\infty}$$c$$\{A_{c}(m, 0)\}_{c=1}^{\infty}$$A_{c}(m, 0)= \mu(\tilde{c})\frac{\phi(c)}{\phi(\tilde{c})},$

is the Mobius function. (So as runs through just theandwheresquarefree numbers with at most 2 prime factors, the resulting subsequence is itselfoscillatory.) But what if $mn\neq 0$ ? On the one hand, Kuznetsov’s [6] estimate$\tilde{c}=\frac{c}{(c,m)}$$\mu(\cdot)$$c$$\sum_{1\leq c\leq x}\frac{A_{c}(m,n)}{c}=0_{m,n}(x^{1}6(\log x)^{1}3)$(1)Keywords: Kloosterman sums, oscillatory sequences, Dirichlet series, automorphic forms.
2010 Mathematics Subject $Cassif\iota cation$ ; Primary llF30, llF72, llL05.
This work was supported (in part) by The City University of New York PSC-CUNY Research Award
Program grant $\# 64565-0042)$ .
$($95has often been cited as evidence for the conclusion that cancellation occurs in sumsof Kloosterman sums. On the other hand, as apparently pointed out by Serre (seeSamak’s book [14, p. 33] as well as the survey paper by Michel [8, p. 247] , it isplausible that (1) may not be caused by $sign$ changes, but rather be a consequence of
all the sums $A_{c}(m, n)$ being uniformly small for fixed and . Nonetheless, Fouvry
and Michel [1] (see also [8]) established thatis indeed oscillatory. In
fact, they proved that as mns through the squarefree numbers with at most 23 prime
factors, the resulting subsequence is oscillatory. They achieved this result by using thetheory of automorphic forms coupled with sieve methods and techniques from -adic
cohomology. Subsequently, Sivak-Fischler replaced this number of prime factors by
22 in [16] and then further reduced it to 18 in [17]. And rather recently, Matomaki[7] brought this number down to 15.
Now, is it possible to prove merely thatis oscillatory in some simple
manner? Yes. By applying the spectral theory of automorphic forms to Selberg’s$)$

$m$

$n$

$\{A_{c}(1,1)\}_{c=1}^{\infty}$

$c$

$P$

$\{A_{c}(1,1)\}_{c=1}^{\infty}$

Kloosterman zeta-function$Z_{m,n}(s)= \sum_{c=1}^{\infty}\frac{A_{c}(m,n)}{c^{2s}},$it is not difficult to demonstrate the more general result thatis oscillatory for any fixed integers and such that $mn\neq 0$ . We sketch this in a few
broad strokes. Evidently, $Z_{m,n}(s)$ is holomorphic for $\sigma={\rm Re} s>i$ . Thanks tothe profound work of Selberg [15], it possesses a meromorphic continuation to thewhole -plane. What’s more, owing to the fact that the underlying group is,$Z_{m,n}(s)$ is holomorphic for $\sigma>0$ except for the presence of (infinitelymany) nonreal
poles in $0<\sigma\leq 1/2$ . By Landau’s Theorem (conceming the abscissa of convergence of Dirichlet series with nonnegative coefficients) this implies immediately thatis oscillatory. For further applications of Landau’s Theorem to establish the oscillatory behavior of certain sequences arising in number theory, see [5] aswell as [11] and [12]. And for more information conceming the analytic behavior of$Z_{m,n}(s)$ , please read [6] and [14, pp. 33-41].
But is it possible to prove thatis oscillatory in some elementaryway? Yes. The main purpose of these notes is to provide an extremely easy proof of
the following (more general) result.
$\{A_{c}(m, n)\}_{c=1}^{\infty}$

$m$

$n$

$s$

$SL(2, \mathbb{Z})$$\{A_{c}(m, n)\}_{c=1}^{\infty}$$\{A_{c}(1,1)\}_{c=1}^{\infty}$

Proposition Letsums$m$be any fixed nonzero integer. Then the sequence of Kloosterman
is oscillatory. Moreover the following hold.

$\{A_{c}(m, m)\}_{c=1}^{\infty}$$\cdot$For any fixed prime $p\equiv-1(mod 4|m|)$ , the subsequenceisalternating and $A_{p^{2}}(m, m)>0.$(ii) For any fixed odd integer $k>1$ , the subsequence $\{A_{p^{k}}(m, m)\}_{p\equiv\pm 1(mod 4|m|)}$ isoscillatory. $(Here, as usual_{J}p$ denotes prime)$Oi$$\{A_{p^{k}}(m, m)\}_{k=2}^{\infty}$$a$Proof Since $A_{c}(m, m)=A_{c}(|m|, |m|)$ , we may suppose that $m>0$ . To begin with,$\cdot$let be any odd prime such that $(p, m)=1$ and set $c=p^{k}$ , where $k\geq 2$ . By workof Sali\’e [13] from 1931 (for a modem reference, consult [3, p. 60]), we know that$p$$A_{c}(m, m)=2( \frac{m}{c})\sqrt{c}{\rm Re}(\epsilon_{c}e^{\underline{4\pi}_{\mathcal{C}}arrow;n})$

,

(2)

96
where$( \frac{m}{c})$is the Jacobi symbol and$\epsilon_{c}=\{\begin{array}{ll}1 if c\equiv 1(mod 4) ,i if c\equiv-1(mod 4) .\end{array}$

be any prime such that $p\equiv\pm 1(mod 4m)$ . By the Law of Quadratic, and soReciprocity, it follows easily that

Next, let$p$$( \frac{m}{p})=1$$(_{\overline{p}^{F}}m)=1.$

To prove (i), choose $p\equiv-1(mod 4m)$ . Since $p^{k}\equiv(-1)^{k}(mod 4)$ , we obtainfrom formula (2) that, for $k\geq 2,$$A_{p^{k}}(m, m)=\{\begin{array}{ll}2 p^{k/2}\cos(\frac{4\pi m}{p^{k}}) if k is even,-2p^{k/2}\sin(\frac{4\pi m}{p^{k}}) if k is odd.\end{array}$But for some positive integer , we surely have that
This implies (i).
To show (ii), take $p\equiv 1(mod 4m)$ . Because $p^{k}\equiv 1(mod 4)$ , we get from (2)that, for any $k\geq 2,$$j$$\frac{4\pi m}{p^{k}}=\frac{\pi(p+1)}{p^{k}j}\leq\frac{\pi(p+i)}{p^{2}}\leq\frac{4\pi}{9}.$$A_{p^{k}}(m, m)=2p^{k/2} \cos(\frac{4\pi m}{p^{k}})$.

As before, for some positive integer , we see that
Thus, for any fixed integer $k\geq 2,$ $\{A_{p^{k}}(m, m)\}_{p\equiv 1(mod 4|m|)}$ is positive. This fact
(in conjunction with $(i)$ ) establishes (ii).
$j$$\frac{4\pi m}{p^{k}}=\frac{\pi(p-1)}{p^{k}j}\leq\frac{\pi(p-1)}{p^{2}}\leq\frac{4\pi}{25}.$$\square$Remarks: (i) By Dirichlet’s Theorem there exist infinitely many primes such that$p\equiv 1(mod 4|m|)$ and also infinitely many primes
such that $p\equiv-1(mod 4|m|)$ .
(In fact, for such arithmetic progressions “Euclidean proofs” of infinitude are available. For more on such matters, see [9]. (ii) By exploiting Sali\’e’s formula (2), we
can readily deduce further results. For instance, let be any fixed nonzero integer
and suppose that is any fixed odd prime such that $(p, m)=1$ . Then it follows
is oscillatory if and only if $p \equiv-(\frac{|m|}{p})(mod 4)$ . What’s more,that
if satisfies this condition, thenis alternating, whereis given$p$$p$$)$$m$$p$$\{A_{p^{k}}(m, m)\}_{k=2}^{\infty}$$k_{0}$$\{\mathcal{A}_{p^{k}}(m, m)\}_{k>k_{0}}$$p$oostemansumsby
defmedasbefore. $(iii)BcorepropertiesofK1$$\max\{\frac{\log 8|m|}{w^{\log p}ithk}, i\}.Otherwise’ ifp\equiv(\frac{|m|}{yp})(mod 4),then\{A_{p^{k}}(m’ m)\}_{k>k_{0}}is$positive,$0$

where is any divisor ofsuch that$(\delta, c)=1$ . So the Proposition extends to sums such as $A_{c}(1,m^{2})=A_{c}(m^{2},1)$ .

$A_{c}(m, m)=A_{c}(_{T}^{m}, \delta m)=A_{c}(\delta m,mT)$ ,$m$$\delta$

We conclude by recording a significant special case of the Proposition.

Corollary The sequence of Kloosterman sums
over, the following hold.

$\{A_{c}(1,1)\}_{c=1}^{\infty}$

is oscillatory. More-$\cdot$(i) For any fixed prime $p\equiv-1(mod 4)$ , the subsequenceis alternating and $A_{p^{2}}(1,1)>0.$(ii) For any fxed odd integer $k>1$ , the subsequence $\{A_{p^{k}}(1,1)\}_{p>2}$ is oscillatory.
$(Here, as$ before,denotes prime.
$p$$a$$)$$\{A_{p^{k}}(1,1)\}_{k=2}^{\infty}$

97
Remarks: (i) For any fixed integer $k\geq 2,$ $\{A_{p^{k}}(1,1)\}_{p\equiv 1(mod 4)}$ is positive. (ii)rather challenging (and seemingly intractable) unsolved problem is to establish that$\{A_{p}(1,1)\}_{p>2}$ is oscillatory.

$A$

Acknowledgment: The author thanks Professor Takumi Noda for organizing a rathersplendid and higly successful conference at the Research Institute for Mathematical
Sciences at Kyoto University in the fall of 2011.

References[1] E. Fouvry and P. Michel, Sur le changement de signe des sommes de Kloosterman, Ann. of Math. (2) 165 (2007), pp. 675-715.
[2] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5thed, Oxford Univ. Press, Oxford, 1979.

[3] H. Iwamiec, Topics in Classical Automorphic Forms, Grad. Stud. Math. 17,Amer. Math. Soc., Providence, RI, 1997.

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[7] K. Matomaki, A note on signs of Kloosterman sums, Bull. Soc. Math. France139 (2011), pp. 287-295.
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[9] M. R. Murty and N. Thain, Primes in certain arithmetic progressions, Funct.
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[10] H. Poincar\’e Fonctions modulaires et fonctions fuchsiennes, Ann. Fac. Sci.
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$Z$. 34 (1931),[14] P. Samak, Some Applications of Modular Fonns, Cambridge Tracts in Math.

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[15] A. Selberg, On the estimation of Fourier coefficients of modular forms, Theory of Numbers (Cal. Tech., Pasadena, $CA$ , 1963), ed. A. L. Whiteman, Proc.
Sympos. Pure Math. 8, Amer. Math. Soc., Providence, $RI$ , 1965, pp. 1-15.

[16] J. Sivak-Fischler, Crible \’etrange et sommes de Kloosterman, Acta Arith. 128(2007), pp. 69-100.
[17] –, Crible asymptotique et sommes de Kloosteman, Bull. Soc. Math. France137 (2009), pp. 1-62.

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