Research article

On triple correlation sums of Fourier coefficients of cusp forms

  • Received: 17 June 2022 Revised: 02 August 2022 Accepted: 12 August 2022 Published: 01 September 2022
  • MSC : 11F67, 11F66

  • Let p be a prime. In this paper, we study the sum

    m1n1anλg(m)λf(m+pn)U(mX)V(nH)

    for any newforms gBk(1) (or Bλ(1)) and fBk(p) (or Bλ(p)), with the aim of determining the explicit dependence on the level, where a={anC} is an arbitrary complex sequence. As a result, we prove a uniform bound with respect to the level parameter p, and present that this type of sum is non-trivial for any given H,X2.

    Citation: Fei Hou, Bin Chen. On triple correlation sums of Fourier coefficients of cusp forms[J]. AIMS Mathematics, 2022, 7(10): 19359-19371. doi: 10.3934/math.20221063

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  • Let p be a prime. In this paper, we study the sum

    m1n1anλg(m)λf(m+pn)U(mX)V(nH)

    for any newforms gBk(1) (or Bλ(1)) and fBk(p) (or Bλ(p)), with the aim of determining the explicit dependence on the level, where a={anC} is an arbitrary complex sequence. As a result, we prove a uniform bound with respect to the level parameter p, and present that this type of sum is non-trivial for any given H,X2.



    A basic but important problem in number theory is the triple correlation sums problem, which concerns the non-trivial bounds for

    hHnXa(n)b(l1n+l2h)c(h)orhHnXa(n)b(n+l1h)c(n+l2h).

    Here, a(n),b(n) and c(n) are three arithmetic functions, H,X2 and l1,l2Z. These type of sums play the vital roles of their own in many topics, such as the moments of L-functions (or zeta-functions), subconvexity, the Gauss circle problem and the Quantum Unique Ergodicity (QUE) conjecture, etc (see for instance [1,5,7,8,9,11,12,13,15,19] and the references therein). In the case of all the arithmetic functions being the divisor functions, Browning [4] found that

    hH,nXa(n)b(n+h)c(n+2h)=118Υ(h)p(11p)2(1+2p)HXlog3X+o(HXlog3X)

    for certain function Υ(h), provided that HX3/4+ε. It is remarkable that Blomer [2] used the spectral decomposition for partially smoothed triple correlation sums to prove that

    h1nXW(hH)τ(n)a(n+h)τ(n+2h)=HˆW(1)nXa(n)d1S(2n,0;d)d2×(logn+2γ2logd)2+O((H2X+HX14+XH+XH)a2).

    Here, the gamma constant γ0.57721, W is a bump function supported on [1/2,5/2], satisfying that xjW(j)(x)1 for any jN+; while, ˆW is the Mellin transform of W and

    a2=nX|a2(n)|

    is the 2-norm. Notice that, the parameter H is reduced to HX1/3+ε. Let k,k2N+. Let f1Bk(1) and f2Bk(1) be two Hecke newforms on GL2, with λf1 and λf2 being their n-th Hecke eigenvalues, respectively (see §2 for definitions). Subsequently, Lin [20] claimed that

    h1nXW(hH)λf1(n)a(n+h)λf2(n+2h)Xε(XH+XH)a2,

    which is non-trivial, provided that HX2/3+ε. Recently, Singh [28], however, were able to attain that, for any f1,f2,f3Bk(1) (or Bλ(1)) and some constant η>0,

    h1n1W1(hH)W2(nX)λf1(n)λf2(n+h)λf3(n+2h)X1η+εH,

    where W1,W2 are any bump functions compactly supported on [1/2,5/2] with bounded derivatives. Here, for any NN+, Bλ(N) denotes the collection of the primitive newforms of Laplacian eigenvalue λ on Γ0(N) (see §2 for backgrounds). Until now, the best result is due to Lü-Xi [21,22] who achieved that

    h1nXW(hH)a(n)b(n+h)λf1(n+2h)XεΔ1(X,H)a2b2,

    which allows one to take HX2/5+ε. Here, the definition of Δ1(X,H) can be referred to [22,Theorem 3.1]. More recently, Hulse et al. [14] successfully attained

    h1n1λg1(n)λg2(h)λg3(2nh)exp(hHnX)Xκ1+ϑ+12+εHκ12ϑ+12+ε, (1.1)

    where ϑ<7/64 denotes the currently best record for the Generalized Ramanujan Conjecture. Here, λg1(n) (resp. λg1(n) and λg1(n)) denote the n-th non-normalized coefficients of holomorphic cusp forms g1 (resp. g2 and g3), each of weight κ2 and level M2.

    In the present paper, we aim to consider the level aspect for the triple correlation sums. It is noticeable that, just lately, Munshi [26] obtained that, whenever X1/3+εpX, one has the inequality

    n1λf(n)λf(n+pm)p14X34+ε (1.2)

    for any fBk(p) and fixed integer m such that |m|X/p. In this paper, we would like to go further obtaining the following quantitive estimate.

    Theorem 1.1. Let X,H2, and p be a prime such that pX. Let U,V be two smooth weight functions supported [1/2,5/2] with bounded derivatives. Then, for any sequence a={anC} and any newforms gBk(1) (or Bλ(1)) and fBk(p) (or Bλ(p)), we have

    m1n1anλg(m)λf(m+pn)U(mX)V(nH)Xεmax(XHp,X)a2, (1.3)

    where the implied constant depends only on the weight k (or the spectral parameter λ) and ε.

    Remark 1.2. Our main result (1.3) is non-trivial for any given parameters X and H; particularly, for any automorphic cusp form π of any rank N, N2, with λπ(n) being its n-th normoailzed Fourier coefficient, we find

    m1n1λπ(n)λg(m)λf(m+pn)U(mX)V(nH)Xεmax(HXp,XH)

    by the Rankin-Selberg's bound, which says that nX|λπ(n)|2π,εX1+ε (see for instance [6,Remark 12.1.8]).

    Remark 1.3. The merits that comes from (1.3) is that the implied constant does not depend on the level parameter anymore. One may verify that our result (1.3), however, is exhibited to be a strengthened upper-bound whenever HX/p, compared with an application of Munshi's estimate (1.2). Indeed, Munshi's estimate implies the upper-bound p1/4X3/4+εHa2 for the triple sum above. Moreover, one might save roughly a magnitude of H in the interesting case of pHX; in the average sense, the main result improves upon the estimate due to Munshi. One, on the other hand, wanders whether or not the non-trivial bounds for the scenarios where the cusp forms f,g being of higher rank could be achieved; we shall plan to investigates this problem in the future work.

    Notations. Throughout the paper, ε always denotes an arbitrarily small positive constant. nX means that X<n2X. For any integers m,n, (m,n) means the great common divisor of m,n. Finally, μ(n) denotes the Möbius function of n.

    For any k2N+ and NN+, let us denote by Sk(N) the vector space of the normalized holomorphic cusp forms on Γ0(N) of weight k and trivial nebentypus. Whenever fSk(N), one has

    f(z)=n1λf(n)nk12e(nz)

    for Im(z)>0. We also denote by Sλ(N) the vector space of the normalized Maaß forms on Γ0(N) of weight 0, Laplacian eigenvalue λ=1/4+r2 (rR) and trivial nebentypus. For any fSλ(N), there exists the following Fourier expansion

    f(z)=2|y|n0λf(n)Kir(2π|ny|)e(nx),

    where z=x+iy. The set of the primitive forms Bk(N) (or Bλ(N)) consists of common eigenfunctions of all the Hecke operators Tn for any n1. Regarding the individual bounds for λf(n), we have

    λf(n)(nN)ε, (2.1)

    whenever fBk(N) (or Bλ(N)).

    We will need the Voronoi summation formula in the analysis; see [18,Theorem A.4].

    Lemma 2.1. Let k,N and the form f be as before. For any a,qN+ such that (a,q)=1, we set N2:=N/(N,q). Let h be a bump function of bounded derivatives. Then, there exists a constant ϱ of modulus one and a newform fBk(N) (or Bλ(N)) such that

    n1λf(n)e(anq)h(nX)=2πϱqN2n1λf(n)e(¯aN2nq)H(nXq2N2;h)+2πϱqN2n1λf(n)e(¯aN2nq)H(nXq2N2;h),

    where

    H(x;h)=0h(ξ)Jf(4πxξ)dξ,andH(x;h)=0h(ξ)Kf(4πxξ)dξ.

    Here, if f is holomorphic

    Jf(x)=2πikJk1(x),Kf(x)=0;

    while, if f is a Maaß form

    Jf(x)=πsin(πir)(J2ir(x)J2ir(x)),Kf(x)=4cosh(πr)K2ir(x).

    For any sR, one may write

    Jk1(s)=s12(F+k(s)e(x)+Fk(s)e(s)) (2.2)

    for some smooth functions F± satisfying that

    sjF±k(j)(s)k,js(1+s)32

    for any j0; the resource might be referred to [29,Section 6.5] if s<1 and [29,Section 3.4] if s1. One thus sees that, for H, the n-variable is essentially truncated at nq2N2/X1ε, by repeated integration by parts. Furthermore, notice that, by Appendix of [3],

    K2ir(s)r,ε{s12exp(s),s>1+π|r|,(1+|r|)ε,0<s1+π|r|; (2.3)

    one will find the n-variable enjoys the analogous truncation range for H with that for H.

    Now, let us recall the following Wilton-type bound; the resource, however, may be referred to [17], together with [10,27].

    Lemma 2.2. Let X2 and W be a smooth function, compactly supported on [1/2,5/2] such that xjW(j)(x)1 for any jN+. For any newform fBk(N) (or Bλ(N)) and αR, we thus have

    n1λf(n)e(nα)W(nX)XN13+ε, (2.4)

    where the implied -constant depends merely on k (or λ) and ε.

    As a variant of the circle method, the δ-symbol method plays a focal role in number theory. We will now briefly recall a version of the circle method; see for instance [16,Chapter 20].

    Lemma 2.3. Fix X,Q1. For any nX, one has

    δ(n)=1QqQ1qamodq(a,q)=1e(anq)Rg(q,τ)e(nτqQ)dτ,

    where

    g(q,τ)=1+h(q,τ)withh(q,τ)=O(1qQ(|τ|+qQ))A,τjjτjg(q,τ)logQmin(Qq,1|τ|),

    and g(q,τ)|τ|A for any sufficiently large A. In particular, the effective range ofthe τ-integral is [Xε,Xε].

    We will have a need of the following lemmas which will be applied in §3.

    Lemma 2.4. Let Q2. Let F(x,y) be a smooth bump functionsupported on [1/2,5/2]×[1/2,5/2], which satisfies that

    XiYjixijyjF(xX,yY)i,j1

    for any integers i,jN+ and any X,Y1. Then, for any cZ, sequence a={anC} and newform fBk(1) (or Bλ(1)), there holds that

    q1n1anS(n,c;q)F(nX,qQ)Xε(XQ+1Q2>XQ2)a2, (2.5)

    where the symbol 1P equals 1 if the assertion P is true, and 0 otherwise.

    Proof. First, via the Cauchy-Schwarz inequality, we might evaluate the double sum as

    (q1,q21n1S(n,c;q1)¯S(n,c;q1)F(nX,q1Q)F(nX,q2Q))12a2.

    It thus follows from the Weil bound that the non-generic terms q1=q2 shall contribute a upper-bound X1+εQ2 to the parentheses above, which gives the term XQa2 on the RHS of (2.5); while, for the generic terms q1q2, if one writes q1=ˆq1δ, q2=ˆq2δ with δ=(q1,q2) satisfying that (δ,ˆq1)=1, Poisson summation formula with the modulus ˆq1ˆq2δ thus might produce the following bound for the triple sum that

    δQˆq1,ˆq2Q/δsup0<|l|ˆq1ˆq2δ/X1ε|αmodˆq1ˆq2δS(α,c;ˆq1δ)¯S(α,c;ˆq2δ)e(αlˆq1ˆq2δ)|.

    Notice, here, the inner-most sum vanishes, if l=0, and it is necessary that Q2>X as well. At this point, on applying Chinese remainder theorem, the sum over α turns out to be

    ˆq1ˆq2δe(c¯δlˆq2ˆq1)smodδe(¯aˆq1sδˆq1c¯(ˆq2s+l)ˆq2δ)

    with ¯δδ1modˆq1, ¯ll1modˆq1, ¯ˆq1ˆq11modδ and ¯ˆq2s+l(ˆq2s+l)1modˆq2δ; trivially evaluating everything thus exactly leads to the term Q2a2 in (2.5).

    Lemma 2.5. Let the parameters X,Q,c, the form f and the sequence a={anC} be as in Lemma 2.4. Let W(x,y,z) be a smooth bump functionsupported on [1/2,5/2]×[1/2,5/2]×[1/2,5/2], with the partial derivatives satisfying

    XiYjZkixijyjkzkW(xX,yY,zZ)i,j,k1

    for ever integers i,j,kN+ and any X,Y,Z1. There thus holds that

    q1n1anm1λf(m)mS(mnp,c;q)W(mX,nH,qQ)Xε(HQ+Q2)a2. (2.6)

    Proof. To show the lemma, the initial procedure is to invoke the Cauchy-Schwarz inequality; we are thus led to evaluating

    q1,q21n1m1,m21λf(m1)¯λf(m2)m1m2S(m1np,c;q1)¯S(m2np,c;q2)W(m1X,nH,q1Q)W(m2X,nH,q2Q). (2.7)

    (1) First, let us begin with considering the generic terms q1=q2=q, say. In this moment, Poisson is applicable, which yields an alternative form for (2.7):

    Hl1ZqQ1qm1,m21λf(m1)¯λf(m2)m1m2Y0(l1,m1,m2,p,c;q)I0(l1,m1,m2),

    where

    Y0(l,m1,m2,p,c;q)=αmodqS(m1αp,c;q)¯S(m2αp,c;q)e(αlq),

    and

    I0(l,m1,m2)=RW(m1X,ξ,qQ)W(m2X,ξ,qQ)e(lHξq)dξ.

    Notice that the exponential sum modulo q asymptotically equals

    qe(m1¯plq)αmodqe((m1m2)α+n¯α¯pln¯αq),

    where we have employed the relation involving Ramanujan sum that

    S(n,0;q)=ab=qμ(a)βmodqe(βnb). (2.8)

    Upon combining with Lemma 2.2, one thus sees that the zero-frequency shall contribute a bound by HQXε for any ε>0. While, on the other hand, if Q>H, one may find the non-zero frequencies will be indispensable to contribute a magnitude to (2.7). It can be demonstrated that, in this situation, the contribution, however, is estimated as Q2Xε; this gives totally a quantity by

    HQXε+1Q>HQ2Xε. (2.9)

    (2) Now, we are left with the non-generic case where q1q2. One writes q1=q1h, q2=q2h, with (q1,q2)=h and (q1,q2)=1. Notice that h is co-prime with one of factors q1,q2; without loss of generality, one assumes that (h,q1)=1. The expression in (2.7) thus becomes

    hQq1,q2Q/hn1m1,m21λf(m1)¯λf(m2)m1m2S(m1np,c;q1h)ׯS(m2np,c;q2h)W(m1X,nH,q1hQ)W(m2X,nH,q2hQ). (2.10)

    Upon exploiting the Poisson twice, we thus arrive at

    Hl2ZhQq1,q2Q/h1q1q2hm1,m21λf(m1)¯λf(m2)m1m2Y(l2,m1,m2,p,c;q1,q2,h)I(l2,m1,m2,q1,q2h),

    where the exponential sum Y and the resulting integral I are defined as

    Y(l,m1,m2,p,c;q1,q2,h)=αmodq1q2hS(m1αp,c;q1h)¯S(m2αp,c;q2h)e(αlq1q2h),

    and

    I(l,m1,m2,q1,q2,h)=RW(m1X,ξ,q1hQ)W(m2X,ξ,q2hQ)e(lHξq1q2h)dξ.

    Here, one finds that the zero-frequency l2=0 does exist anymore. Indeed, upon recalling that (h,q1)=1, one writes α=q1¯q1x+q2h¯q2y, with xmodq2h and ymodq1 such that (x,q2h) and (y,q1)=1; applying Chinese remainder theorem and (2.8), the sum over α thus essentially turns out to be

    q1q2he(m1¯q2pl+nq2p¯hlq1+m2¯q1plq2h)αmodhe((m1m2)¯q1q2αnq1¯q2α¯pl+nq2¯q1¯αq2h).

    Via Lemma 2.2, it thus follows that the display (2.10) is dominated by εXεQ4, upon opening the Kloosterman sum above. This, together with (2.9), shows the desired estimates in the parentheses of (2.6).

    In this part, let us focus on the proof of Theorem 1.1. We shall first manage to separate the variables n,m by applying Lemma 2.3; in this paper, we shall employ a vital trick, that is, the 'conductor lowering mechanism' (see [23,24] or the survey [25]). One may see that actually there holds the following

    δ(n)=1pQqQ1qamodqp(a,q)=1e(anqp)Rg(q,τ)e(nτqQp)dτ; (3.1)

    while, the parameter Q shall be taken as Q=X/p. Now, for three smooth functions U,V,R, supported [1/2,5/2] with bounded derivatives, we shall detect the shift l=m+pn via (3.1), which yields an alternative form for the triple sum in (1.3) as follows

    S(X,p,H)=XHpQRqQg(q,τ)qγmodpq(γ,q)=1l1alle(γlq)Vτ(lH)m1λg(m)m×e(mγpq)Uτ(mX)n1λf(n)ne(nγpq)Rτ(nX)dτ, (3.2)

    where

    Uτ(m)=U(m)e(mXτpqQ),Rτ(n)=R(n)e(nXτpqQ),Vτ(u)=V(u)e(uHτqQ).

    We shall proceed to distinguish whether (γ,p)=1 or not in the analysis, so that we are led to three parts, i.e., the non-degenerate term SNonde., the degenerate term SDeg. and the error term SErr., which are respectively given by

    SNonde.(X,p,H)=XHpQRqQ(q,p)=1g(q,τ)qγmodpql1alle(lγpq)Vτ(lH)×m1λg(m)me(mγpq)Uτ(mX)n1λf(n)ne(nγq)Rτ(nX)dτ, (3.3)
    SDeg.(X,p,H)=XHpQRqQ(q,p)=1g(q,τ)qγmodql1alue(lpγq)Vτ(lH)×m1λπ(1,m)me(mγq)Uτ(mX)n1λf(n)ne(nγq)Rτ(nX)dτ, (3.4)

    and

    SErr.(X,p,H)=XHpQRqQp|qg(q,τ)qγmodpql1alle(lγpq)Vτ(lH)×m1λg(m)me(mγpq)Uτ(mX)n1λf(n)ne(nγq)Rτ(nX)dτ. (3.5)

    One might see that, here, it suffices to consider SNonde.; the same argument works for SErr. which serves as a noisy term and for which we save more. We shall now now begin with SNonde.; the analysis of the term SDeg. will be postponed to the end of this paper.

    In this part, let us concentrate on the analysis of SNonde.. For any ι,ν,υ,ρR, write

    Wτ(ι,ν,υ,ρ)=Uτ(ι)Rτ(ν)Vτ(υ)ηQ(ρQ),

    where ηQ is a smooth bump function supported on [1Q/2,5Q/2], satisfying that η(j)Q(x)1 for any jN+. One might find that the quantity we are focusing on is the following

    XHpQsupτXεsupQQq1(q,p)=1g(q,τ)qγmodpqu1alle(lγq)m1λg(m)m×e(mγpq)n1λf(n)ne(nγpq)Wτ(mX,nX,lH,qQ). (3.6)

    We intend to invoking the Voronoi formula, Lemma 2.1; we thus arrive at

    XHpQsupτXεsupQQq1(q,p)=1g(q,τ)ql1allm1λg(m)mn1λf(n)n×S(nm,pl;pq){^Wτ(mXpQ2,nXpQ2,lH,qQ)+^Wτ(mXpQ2,nXpQ2,lH,qQ)}, (3.7)

    where, for any ,{,}, each integral ^W,τ is defined as

    ^W,τ(ι,ν,υ,ρ)=ηQ(ρQ)Vτ(υ)H(Q2ιq2;Rτ)H(Q2νq2;Uτ).

    It is remarkable that, here, from (2.2) and (2.3), we have the identical crude estimate that ^W,τXε for any ε>0; in this sense, one sees that it suffices to deal simply with ^W,τ, upon noticing that the argument of the other terms (i.e., ^W,τ, ^W,τ and ^W,τ) can follow similarly with it. One, however, on the other hand, sees that the inner-most sum modulo pq can be converted into

    pS(¯p(nm),l;q)

    with nmmodp. Now, if one writes n=m+pk with kXε, we find that (3.6) is no more than

    XHQsupτXεsupQQmpXεkXελf(m)λf(pk+m)m(pk+m)×q1(q,p)=1g(q,τ)ql1allS(l,k;q)^W,τ(mXpQ2,nXpQ2,lH,qQ).

    At this point, an application of Lemma 2.3 shows that the RHS of the expression above is bounded by

    X1+εQsupτXεsupQQ(H+1Q2>HQ)a2Xεmax(XHp,X)a2, (3.8)

    upon recalling the value of Q.

    Now, let us have a look at the multiple-sum SDeg.. One might verify that SDeg. is of the form

    XHpQsupτXεsupQQq1(q,p)=1g(q,τ)qγmodql1alue(lpγq)m1λg(m)m×e(mγq)n1λf(n)ne(nγq)Wτ(nX,mX,uH,qQ) (3.9)

    with Wτ being as before. We will now proceed by appealing to the Voronoi formula, Lemma 2.1, again to transform the sums over n into the dualized form, so that we infer that the expression above should be controlled by

    XHpQsupτXεsupQQsupnpQ2X1+ε|q1(q,p)=1g(q,τ)ql1allm1λg(m)m=×S(mlp,n;q){~Wτ(mX,nXpQ2,lH,qQ)+~Wτ(mX,nXpQ2,lH,qQ)}|,

    where, for {,}, each integral transform ~Wτ is given by

    ~Wτ(ι,ν,υ,ρ)=ηQ(ρQ)Vτ(υ)Uτ(ι)H(Q2νq2;Rτ).

    Via Lemma 2.5, one thus deduces

    SDeg.(X,p,H)X1+εpQsupτXεsupQQ(H+Q)a2Xε(XHp+Xp)a2.

    This leads to the estimates we would like to prove in Theorem 1.1, upon combining with (3.8).

    In this paper, we investigate the triple correlations sums of Fourier coefficient of newforms on GL2, with the levels aspects being explicitly determined; our method is flexible enough to deal with the Maaß new forms. It is also remarkable that more recently, the authors are able to establish a sharp bound in the scenario where one of the froms f,g in (1.3) is a Maaß cuspidal form on GL3 (not necessarily self-dual) with the trivial level.

    This research was funded by National Natural Science Foundation of China (grant numbers 413618022 and 61402335).

    The authors declare that they have no conflicts of interest.



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