Fourth power mean of the general 4-dimensional Kloosterman sum mod p
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RESEARCH
Fourth power mean of the general 4-dimensional Kloosterman sum mod p Nilanjan Bag*
and Rupam Barman
* Correspondence:
[email protected] Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati, Assam 781039, India
Abstract In this article, we prove an asymptotic formula for the fourth power mean of a general 4-dimensional Kloosterman sum. We use a result of P. Deligne, which counts the number of Fp -points on the surface (x − 1)(y − 1)(z − 1)(1 − xyz) − uxyz = 0, u = 0, and then take an average of the error terms over u to prove the asymptotic formula. We also find the number of solutions of certain congruence equations mod p which are used to prove our main result. Keywords: The general s-dimensional Kloosterman sums, Dirichlet character, Asymptotic formula Mathematics Subject Classification: 11L05
1 Introduction and statement of results In 1926, to study certain positive definite integral quadratic forms, Kloosterman [6] introduced the exponential sum S(a, b; q) =
1≤x≤q (x,q)=1
e
ax + bx , q
where a, b and q are arbitrary integers with q ≥ 1. Here e is defined as e(y) = e2π iy and x denotes the multiplicative inverse of x mod q. Later this sum was known as a Kloosterman sum. Kloosterman had considerable interest in the order of magnitude of K (a, b; q). In his paper he proved that S(a, b; q) = O(q 3/4+ (a, q)1/4 )
(q −→ ∞),
for every positive . Indeed, he showed in [7] that any non trivial upper bound for S(a, b; q) gives a corresponding improvement of Hecke’s upper bound for the Fourier coefficients of certain cusp forms. Because of such connections to analytic number theory mathematicians were interested in finding the order of magnitude of S(a, b; q) and its arithmetic properties.
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© Springer Nature Switzerland AG 2020.
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N. Bag and R. Barman Res. Number Theory (2020)6:31
Page 2 of 15
Let q ≥ 3 be a positive integer. For any fixed integer s ≥ 1, the higher dimensional Kloosterman sum K (m, s; q) is defined by q q x1 + · · · + xs + mx1 · · · xs K (m, s; q) = ··· e q x1 =1
xs =1
and the general higher dimensional Kloosterman sum K (m, s, χ; q) is defined by q q x1 + · · · + xs + mx1 · · · xs K (m, s, χ; q) = ··· χ(x1 · · · xs ) × e , q x1 =1
xs =1
q
where
denotes the summation over all 1 ≤ x ≤ q such that gcd(x, q) = 1, m is
x=1
any integer and χ is a Dirichlet character mod q. Many authors studied the arithmetical properties of K (m, s; p), and obtained a series of interesting results. One of such results is due to Mordell [9]. For odd prime p, he got the following estimate |K (m, s; p)| p
s+1 2
.
Later Deligne [3] improved Mordell’s result and obtained the upper bound estimate s
|K (m, s; p)| ≤ (s + 1)p 2 .
(1)
Later on many other results are obtained, see for example ( [8,10–13]). It is well known that, for a principal character χ, √ K (m, 1, χ; p) = −2 p cos(θ(m)), where the angles θ(m) are equidistributed in [0, π] with respect to the Sato-Tate measure 2 2 π sin (θ)dθ. For example, see [5].
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