1 Introduction

This note is a companion to the forth-coming paper^[1]. For the details on the Kuznetsov trace formula, cf. the extended survey [2-6], etc.

The main objective is to prove Theorem 1.3 whence the reverse Kuznetsov trace formula, Theorem 2.4. The raison-d′être for this note is the following. There is no direct proof of Theorem 2.4 in [3], it is derived by reversing the Kuznetsov trace formula, which in turn is proved via Fay′s functional equation which comes from the theory of resolvent and is not easily accessible. In[5], the proof is given in the lines of Selberg-Kuznetsov but the outlook of the statement is different since the Neumann series is replaced by something else. This gap has been filled in the companion paper^[1] in a sketchy way with the aid of Meijer G-functions. We hereby provide new proofs of some formulas involving Bessel functions which are only sketched in [1], and in [3] and [5], proofs are not given or given in an inexplicit way.

As in these papers, we mainly follow the notation of Iwaniec^[3]. The notation for special functions are from [7]. For comparison′s sake, we state the trace formulas in the form of a Fuchsian group of the first kind and so the cusps are to be understood to mean ∞ when one thinks of the full modular group.

Let S(m, n; c) be the classical Kloosterman sum defined as

(1)

where the summation is over relatively prime residues a mod c with a denoting the multiplicative inverse of a.

In [3] the Whittaker function is defined by

(2)

which satisfies the symmetry condition.

(3)

(4)

Here λ_aj(n) and φ_ac(n, 1/2+it) are the Fourier coefficients of (a complete orthonormal system of) Maass forms ([8]) and the eigen packet of Eisenstein series in ([3]).Hence, in particular,

(5)

where σ indicates the sum-of-divisors function^[3].

On [3], the normalization is introduced, which we will use in this paper:

(6)

for n≠0.

We call any C²-function f(x) on [0, ∞] a test function if it satisfies the condition

(7)

The following theorem is stated as the Bruggeman-Kuznetsov formula on [3] (known as the Kuznetsov trace formula, cf.[9-11].

Theorem 1  (Kuznetsov [12]) Let be cusps of the Fuchsian group of the first kind Γ and mn≠0. Then for any test function f satisfying the condition (7), we have

(8)

where f^± are given on [3].

The Kuznetsov trace formula is a sort of the Poisson summation formula ([3]) which is equivalent to the functional equation (cf.[13]) and so Theorem 1 may be proved most naturally as a consequence of the functional equation in Theorem 2 (due to Fay [14]) for the Kloosterman sums zeta-function with Bessel function weight Z_s(m, n)[3].

(9)

where

(10)

Theorem 2  (Fay[14]) The series Z_s(m, n) has an analytic continuation over the whole s-plane and satisfies the functional equation

(11)

where P_s(m, n) is the residual function given by

(12)

Proof of Theorem 1 from Theorem 2 is given on [3].

On the other hand, the Kloosterman sums zeta-function L_n(m, n) is defined by

(13)

and studied by Selberg^[15] and later by Goldfeld and Sarnak^[16].In [1] the following was proved as an important corollary to Lemma 1.Our main purpose is to give a detailed proof of this theorem and thence of Theorem 7.

Theorem 3  For integers m, n > 0 and Re s > 1/4, we have

(14)

2 Kuznetsov formula reversed

As described on [3] and on [17], the reverse Kuznetsov trace formula is to be regarded as an expansion in J-Bessel functions due to Sears and Titchmarsh^[18] and in many literature this reversed form is referred to as the Kuznetsov trace formula^{[17, 19-21]}. In [1-2] we referred to it as the Kuznetsov sum formula, whence the title, since it gives an expression for the sum of Kloostermann sums with the test function in terms of eigen-values of the automorphic Laplacian. Since in the case mn < 0, the Kuznetsov trace formula is completely reversed as given on [3], we concentrate on the case m, n > 0.

Let f(x) be a continumous function of bounded variation on R⁺ such that

(15)

in particular f(x) may be an infinitely many times differentiable function with compact support. We follow [3] which gives the clearest exposition thereof.

Let f⁰ be the projection of f on the space spanned by odd indexed Bessel functions {J_2n+1|n≥0} and is given by the Neumann series

(16)

and

(17)

is the Neumann integral.In [21] Ven this formula is stated with a typo of 2ir which should be 2n+1.

On the other hand, let

(18)

and define the Titchmarsh integral T_f(t) by

(19)

Then define the continuous superposition of projections of f on B_2it by

(20)

Theorem 4  (Sears-Titchmarsh inversion) We have the Sears-Titchmarsh inversion

(21)

Also define the constant

(22)

Theorem 5^[3]  Let be cusps of the Fuchsian group of the first kind Γ and let m, n > 0 be integers. Then for any test function h satisfying the condition (7), we have

(23)

It seems that the corresponding formulas in [17] are incorrect in comparison with other refs.

Let denote the space of susp forms of weight k which is spanned by Poincaré series in contrast to formula (55)

(24)

where j is the denominator appearing in

(25)

so that

(26)

Let f_jk be orthogonal basis. Let

(27)

be the expansion of Poincaréseries with respect to this basis.The exact complement to Theorem 5 is

Theorem 6^[3]  Let be cusps of the Fuchsian group of the first kind Γ and let m, n > 0 be integers. Then for any test function h satisfying the condition (7), we have

(28)

where ψ_ajk(m) are the normalized Fourier coefficients

(29)

Adding Theorems 5 and 6 in view of the Sears-Titchmarsh inversion give the reversed Kuznetsov sum formula in contrast to Theorem 1.

Theorem 7^[3]  Let be cusps of the Fuchsian group Γ of the first kind and let m, n > 0 be integers. Then for any test function h satisfying the condition (7), we have

(30)

Remark 1  We remark that Theorem 4 coincides with [5]. Since in the latter, the Neumann series part ([5, (2.2.6)] is replaced by [5, (2.2.9)], it has a seemingly different outlook.

Using the Hankel transform, Theorem 4 may be clearly understood as a procedure corresponding to the mapping x↔1/x under which the intervals (0, 1) and (1, ∞) map each other.

The Hankel transform of order 0 with Re v > -1/2 is defined by

(31)

for y > 0 which has admits the inversion formula

(32)

Then the Neumann series (16) may be expressed as

(33)

Proof is given ^[3] and depends on the formula

(34)

Integrating over one has

(35)

Multiplying (36) by f(y)y^-1 and integrating in y, we obtain

(36)

By the Sears-Titchmarsh inversion (15) and the Hankel inversion (32), we conclude that

(37)

Hence we conclude that the Sears-Titchmarsh inversion is a counter part of the division of the real line into two parts, which then is responsible for the functional equation. Thus the Sears-Titchmarsh is a counterpart of the functional equation and this explains the reverse Kuznetsov formula is also deduced from the zeta-symmetry as the Kuznetsov formula has been most naturally deduced from Fay′s functional equation in Theorem 2.

We note the following correspondence between groups. The right half-plane is represented by the positive real axis, which is a multiplicative group, in view of analytic continuation. And the Spiegelung τ↔-1/τ, which is one of the generators of the modular group, corresponds to the inversion x↔1/x under which the two intervals (0, 1) and (1, ∞) maps into each other.

In the Riemann zeta-case, the variables are connected by

(38)

because of the presence of a simple pole at s = 1, the right-half plane is narrowed down to Re s > 1, the domain of absolute convergence, leaving the critical strip 0 < σ < 1 so mysterious (cf.22).

Proof of (36).We use two well-known formulas

(39)

and

(40)

Since

(36) follows on substituting above formulas.

3 Proof of Theorem 3

We shall prove Theorem 3 whence Theorem 7 by modifying Motohashi′s argument^[5] and under the assumption that

(41)

or

(42)

and

(43)

We appeal to

(44)

This is applied to the essential case appearing on [5, (2.4.13)]:

(45)

Another formula to be used is

(46)

This is applied in the special case

(47)

For the proof of Theorem 7 we need two more well-known formulas

(48)

and

(49)

In Motohashi [5], the integral

(50)

is computed in an indirect way and the result reads

(51)

Proof of (51).

A common procedure is to apply Euler′s formula and rewrite it as

(52)

Putting and factoring out and respectively, and applying (46), we further transform it into

(53)

by (48). Hence by (49)

(54)

Hence we arrive at (51), completing the proof.

The real analytic Poincaré series is defined on [23] which is referred to in[1], We use the one defined on [5]

(55)

The key idea for trace formulas is due to Selberg^{[15, 24]} and depends on two ways of expressing the inner product of two Poincaré series

The inner product of Poincaré series is computed in Motohashi^[5] for integers m, n > 0. Equating the Unfolding formula and the Parseval formula, we deduce the following theorem. For the proof cf.[1] and [2].

Lemma 1  (Unfolding=Parseval formula), [5 Lemma 2.2] Let { u_j } be as in (3). Then for s_j ′s satisfying Re s₂+α > Re s₁ > α+1/4 and Re s_j > 3/4, j=1, 2 we have

(56)

where [5]

(57)

Proof of Theorem 3.

In Lemma 1 we choose s₁=1, s₂=s with Re s > 1-α for and invoke ([5(2.4.3)])

(58)

(43) amounts to

(59)

(56) reduces to

(60)

Dividing both sides by Γ(1-s), we led to the form to which we may apply (58).

(61)

Dividing both sides by , we rewrite (61) as

(62)

Since

(62) leads to (14) completing the proof.

Proof of Theorem 7 We multiply (14) by f(x) in the form (42) interchange the order of integration and appeal to the known formulas for Bessel functions (46), (48), and (49) successively.