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In this article we unconditionally show that { μ ( n ) } \{\mu (n)\} is asymptotically orthogonal to { λ π ( n ) e ( n α ) } \{\lambda _{\pi }(n)e(n\alpha )\} where λ π ( n ) \lambda _{\pi }(n) is the n n -th Dirichlet coefficient of a GL ⁡ ( m ) \operatorname {GL}(m) cusp form π \pi . Our result does not assume any unproven hypothesis.

Let $\lambda _{\phi }(n)$ be the Fourier coefficients of a Hecke holomorphic or Hecke–Maass cusp form on $\textrm{SL}_{2}(\mathbb Z)$ and $f$ be any multiplicative function that satisfies two mild hypotheses. We establish a nontrivial upper bound for the correlation $\sum _{n \leq X}f(n)\lambda _{\phi }(n+h)$ uniformly in $0

Let π be an automorphic irreducible cuspidal representation of GL m {\operatorname{GL}_{m}} over ℚ {\mathbb{Q}} with unitary central character, and let λ π ⁢ ( n ) {\lambda_{\pi}(n)} be its n-th Dirichlet series coefficient. We study short sums of isotypic trace functions associated to some sheaves modulo primes q of bounded conductor, twisted by multiplicative functions λ π ⁢ ( n ) {\lambda_{\pi}(n)} and μ ⁢ ( n ) ⁢ λ π ⁢ ( n ) {\mu(n)\lambda_{\pi}(n)} . We are able to establish non-trivial bounds for these algebraic twisted sums with intervals of length of at least q 1 / 2 + ε {q^{1/2+\varepsilon}} for an arbitrary fixed ε > 0 {\varepsilon>0} .

With the rapid development of laser technology, the vision of attosecond world has been unfolded. Since the first isolated attosecond pulses produced by high-order harmonic generation in 2001, the time-resolved spectroscopy has entered the attosecond domain and greatly strengthened our understanding of ultrafast electron dynamics in atoms, molecules and condensed matter. Since it approaches the time scale of inner shell electron motion, attosecond pulses become one of the most important tools to explore the electron dynamics. Therefore, developing techniques of attosecond pulse generation is not only essential in optical research field but also imperiously demanded in atomic and molecular physics, which has attracted great attention of researchers worldwide. After years of continuous innovation and breakthroughs, the shortest pulse of 43 attosecond has been achieved. Comparing with the ultrafast laser pulse generation techniques in picosecond or femtosecond regions, the generation of attosecond pulses is rather complex and challenging. From high-order harmonic generation to single attosecond pulse characterization, several crucial technical details need to be considered and investigated carefully, while attosecond gating is a vital link in the chain of technics required by attosecond pulse generation. Up to now, various gating methods have been developed, such as amplitude gating, which, with spectral filtering, takes advantage of the continuous spectrum near the cut-off. In addition, new gating techniques, such as polarization gating, double optical gating, interference polarization gating, polarization-assisted amplitude gating, and phase-matched time gating, also have been proposed or demonstrated to generate isolated attosecond pulses. Recently, spatial gating, which takes advantage of the variation on the driving laser wave front to produce isolated attosecond pulses, has also been developed. Meanwhile, as the intensity at a femtosecond laser focal point now easily exceeds 1016 W/cm2, attosecond pulse generation via laser-plasma interaction has been investigated more intensively. Combined with atto-chirp compensation techniques, these methods are pushing the frontier of ultrafast optics toward shorter and stronger attosecond pulses, and achieving even better time resolution in a broader range of applications. In this article, starting with a general introduction of the principle of attosecond pulse generation, we review various gating techniques for attosecond pulse generation in gas targets, new development of attosecond pulses generated by laser-plasma interaction, as well as atto-chirp compensation for even shorter pulses, which are all crucial for the advancement of attosecond science.

Attosecond pulses have potential applications in the fields of physics, chemistry, biology, and medicine owing to the extremely high time resolution brought by their extremely short pulse widths. The generation and measurement technology of attosecond pulses provides a powerful means for the study of electron dynamical process in atoms. Since the date of its birth, the time width of an attosecond pulse has become shorter and shorter, making its measurement a challenging task. Although attosecond pulses are more and more widely used in the study of ultrafast dynamics and more interest is attracted into the field of attosecond science, it is undeniable that there are still difficulties and challenges for the measurement of attosecond pulses. This review is divided into two parts. The first part presents the development of technics for characterizing attosecond pulses, and briefly describes several experimental principles, setups and results for acquiring time-domain information of attosecond pulses, including RABITT (reconstruction of attosecond beating by interference of two-photon transitions) and laser-assisted lateral X-ray photoionization for measuring attosecond pulse train, attosecond streak camera for measuring single attosecond pulses, TIPTOE (tunneling ionization with a perturbation for the time-domain observation of an electric field) for measuring time-domain information of periodic pulses, as well as theoretical proposals such as attosecond cross correlation, asymmetric photoionization method, and SPIDER (attosecond spectral phase interferometry for direct electric-field reconstruction). The second part introduces the principle and structure of inversion algorithms for extracting time-domain information of attosecond pulses from spectrograms obtained with an attosecond streak camera, including FROG-CRAB (frequency resolved optical gating for complete reconstruction of attosecond bursts), PROOF (phase retrieval by omega oscillation filtering), VTGPA (Volkov transform generalized projections algorithm), PROBP-AC (phase retrieval of broadband pulses with autocorrelation) and CVGN (conditional variational generative network). We also evaluate the calculation time and accuracy of the algorithms. Finally, we summarize the experimental and theoretical challenges of the characterization of attosecond pulses, and look forward to the future development of attosecond pulse measurement.

The Bourgain-Sarnak-Ziegler (BSZ) criterion was initially established for an arbitrary multiplicative function a(n) satisfying |a(n)| ⩽ 1. Recently, Cafferata et al. (2020) showed that the condition |a(n)| ⩽ 1 can be replaced by a(p) 1 to some extent. In this paper, we formulate and prove a further extension of the BSZ criterion, in which the restriction a(p) ⩽ 1 is removed. As applications, we use it together with the analytic theory of automorphic L-functions to prove that there exist some cancellations in the sequence {λπ(n)e(nkα)}n⩽1 on GLm and the Mobius function is disjoint from this sequence.

We demonstrate extreme spectral broadening of third harmonic pulses from a 35-fs Ti:Sapphire laser using cross-phase modulation with the fundamental in Neon filled hollow core fibers. The potential for high energy sub-2-fs UV-pulses is explored.

Bi-level programming is widely used in processing various questions, but it cannot deal with the complex and fuzzy information contained in problems. In order to solve such problems better with intricate and vague information that can be efficiently handled by bifuzzy theory, a bifuzzy–bilevel programming model that sets the parameters to bifuzzy variables is proposed in this paper, which can process complex realistic data more accurately and improve the feasibility and validity of bi-level programming models. To ensure the solvability of the model, the equivalent form of the bifuzzy–bilevel programming model is obtained by utilizing the expected value operator. According to the linear and nonlinear characteristics of the model, the Karush–Kuhn–Tucker condition and particle swarm optimization algorithm are employed to handle the problem, respectively. Finally, by taking the distribution center location problem of the supplier as an example, the bifuzzy–bilevel programming model is applied in practice to balance highly intricate customer demands and corporate cost minimization, obtaining the feasible solution of functions at the upper and lower levels, and the bifuzzy information in the problem can also be processed well, which proves the effectiveness of the proposed methodology.

In this paper, we firstly generalize a result of Friedlander and Iwaniec. By analogy, let [Formula: see text] denote the [Formula: see text]th coefficient of [Formula: see text]-function [Formula: see text] associated to a Hecke–Maass form [Formula: see text] for [Formula: see text] and [Formula: see text] denote the number of representations of [Formula: see text] by the quadratic form [Formula: see text], we investigate the sum with [Formula: see text] [Formula: see text] under a certain assumption.

Microstructural evolutions during hot deformation and annealing of alumina-forming austenitic (AFA) steel were investigated using electron backscatter diffraction (EBSD). Tensile properties of deformed and as-received AFA steels were also tested. The yield strength and ultimate tensile stress of deformed AFA steel increase gradually with the increase in rolling reduction, while the fracture elongation decreases rapidly. Specifically, after hot-rolling with the reduction of 40%, the yield strength of the annealed specimen is high to 773.2 MPa, and the fracture elongation of this sample is about 43.6%. The excellent mechanical properties can be attributed to the smaller average grain, larger Schmid factor and coincident site lattice (CSL) boundaries. Such a rapid decline of the elongation of AFA steel occurs due to increasing low angle grain boundaries (LAGB, 15°).

Let F ( z ) be a Hecke–Maass form for S L ( 3 , Z ) (or S L ( 2 , Z ) ), A F ( n , 1 ) (or A F ( n ) ) be its Fourier coefficients, and L ( s , F ) the Godement–Jacquet L-function attached to F. And let μ F ( n ) be the coefficients of the reciprocal of L ( s , F ) in the absolutely convergent region ℜ s > 1 . In this paper we prove that μ F ( n ) is strongly orthogonal to 1-step nilsequence. As applications, we prove that the Mobius function μ ( n ) is strongly orthogonal to A F ( n , 1 ) e ( n α ) and A F ( n ) 2 e ( n α ) . More precisely, we establish ∑ n ≤ N μ ( n ) A F ( n , 1 ) e ( n α ) ≪ c N exp ⁡ ( − c log ⁡ N ) , and ∑ n ≤ N μ ( n ) A F ( n ) 2 e ( n α ) ≪ A N ( log ⁡ N ) − A . Moreover, we also show that μ F ( n ) is asymptotically orthogonal to any A C 0 ( d ) function, and all MMA and MMLS flows.

The flow behavior of alumina-forming austenitic steel was studied using axisymmetric hot compression on a Gleeble-3500 thermomechanical simulator. The temperature range was 900–1200 °C, and strain rate range was 0.1–100 s−1. The microstructures after deformation were investigated by electron backscattering diffraction (EBSD) and transmission electron microscopy (TEM). The deformation temperature and strain rate have a significant influence on the flow stress. A constitutive equation, describing the flow stress as a function of deformation temperature and strain rate, has been developed, and the hot deformation activation energy was confirmed as 579.4 kJ/mol. Dynamic recrystallization (DRX) progress had been finished after increasing hot deformation temperature to 1100 °C at a strain rate of 100 s−1, leading to the obvious transformation from low-angle grain boundaries (LAGBs) to high-angle grain boundaries (HAGBs), and a relatively stable fraction of HAGBs was obtained. At a strain rate of 100 s−1, the β-fiber at {011} transited to {112} (C orientation), and finally a recrystallized orientation of {100} formed after absolute DRX. GDRX is the primary DRX mechanism, but DDRX mechanism is dominant with the increase in deformation temperature at a high strain rate of 100 s−1.

We study the analogue of the Bombieri–Vinogradov theorem for $\operatorname{SL}_{m}(\mathbb{Z})$ Hecke–Maass form $F(z)$. In particular, for $\operatorname{SL}_{2}(\mathbb{Z})$ holomorphic or Maass Hecke eigenforms, symmetric-square lifts of holomorphic Hecke eigenforms on $\operatorname{SL}_{2}(\mathbb{Z})$, and $\operatorname{SL}_{3}(\mathbb{Z})$ Maass Hecke eigenforms under the Ramanujan conjecture, the levels of distribution are all equal to $1/2,$ which is as strong as the Bombieri–Vinogradov theorem. As an application, we study an automorphic version of Titchmarch’s divisor problem; namely for $a\neq 0,$$$\begin{eqnarray}\mathop{\sum }_{n\leqslant x}\unicode[STIX]{x1D6EC}(n)\unicode[STIX]{x1D70C}(n)d(n-a)\ll x\log \log x,\end{eqnarray}$$ where $\unicode[STIX]{x1D70C}(n)$ are Fourier coefficients $\unicode[STIX]{x1D706}_{f}(n)$ of a holomorphic Hecke eigenform $f$ for $\operatorname{SL}_{2}(\mathbb{Z})$ or Fourier coefficients $A_{F}(n,1)$ of its symmetric-square lift $F$. Further, as a consequence, we get an asymptotic formula $$\begin{eqnarray}\mathop{\sum }_{n\leqslant x}\unicode[STIX]{x1D6EC}(n)\unicode[STIX]{x1D706}_{f}^{2}(n)d(n-a)=E_{1}(a)x\log x+O(x\log \log x),\end{eqnarray}$$ where $E_{1}(a)$ is a constant depending on $a$. Moreover, we also consider the asymptotic orthogonality of the Möbius function against the arithmetic function $\unicode[STIX]{x1D70C}(n)d(n-a)$.

Let μ ( n ) be the Mobius function and x real. In this paper, we investigated the best possible estimates for the sum ∑ n ≤ x μ ( n ) e ( n k θ ) under the weak Generalized Riemann Hypothesis. A similar result also holds for the Liouville function λ ( n ) .

Microstructural evolution of modified 2.5Al alumina-forming austenitic steel after isothermal aging at 973 K was investigated by scanning electron microscope (SEM), and X-ray diffraction (XRD). Tensile and vicker's hardness tests were also carried out to study mechanical properties during isothermal aging. The average particle diameter of precipitates was statistically measured and the kinetic model was employed to describe the coarsening of precipitates. The results show that the precipitation sequence of Laves and NiAl phases was related to aging time, and precipitated after 20 h and 100 h in this work, respectively. The average particle diameter of precipitates continuously increased with aging time. And the coarsening rate of the precipitates decreased at 1000 h due to the exhaustion of Nb atoms in matrix. The precipitation of NiAl and Laves phases plays an important role in mechanical properties. The hardness and the work hardening rate continuously increased. In contrast, the elongation gradually decreased. The above results indicated that the effect of aging strengthening. The room-temperature yield strength firstly sluggish increased and then gradually decreased, finally increased significantly due to the increase in σgb, the decrease in σdis and σss, and the change in σppt.

In this paper, we establish a quantitative non-vanishing result on a class of twisted L-functions A(s,χ) of degree k, which satisfy some mild and standard assumptions. As a corollary, we show that for a positive proportion of characters in a specific set, the special values A(β,χ) are non-vanishing for ℜβ>1−1k. In particular, our argument holds for Rankin–Selberg L-functions and symmetric square L-functions on certain higher rank groups, and leads to some new non-trivial results for the first time.

Aiming at the problem of gradient disappearance and network degradation caused by using of recurrent neural networks in traditional question answering models, an automatic question answering model based on IMGRU-Seq2seq is proposed. First, based on the gated recurrent unit, the batch normalization technology and the rectified linear unit activation function are combined and the identity mapping is added to construct the IMGRU model; then, the text is represented by weighted word vectors through the TF-IDF method; finally, as the semantic extraction unit of the question answering model, the bidirectional IMGRU introduced attention mechanism and the beam search algorithm to realize automatic question and answer. The experimental results showed that the proposed method is 16.55% and 8.36% higher than the existing methods BLEU and ROUGE-L respectively.

We study the exponential sum involving multiplicative function f under milder conditions on the range of f, which generalizes the work of Montgomery and Vaughan. As an application, we prove cancellation in the sum of additively twisted coefficients of automorphic L-function on $$\text {GL}_m$$ $$(m\ge 4)$$ , uniformly in the additive character.

The Weapon Equipment Acquisition Information Network(WEAIN) is an important platform for opening and exchange the whole society in the field of weapon and equipment construction in China. It is a model of interaction between national defense construction and social resources, and plays an important role in gathering superior resources to participate in equipment construction. In order to describe the status and function of WEAIN website objectively, and maximize the connecting value of the bridge between the military and the society, the data composition and characteristics of WEAIN website are analyzed firstly. Then the data analysis requirements are put forward. Lastly, the multidimensional data analysis models and text analysis models of WEAIN website are established, which provide means support and model basis for mastering the effect of the website and the development trend of competitive equipment acquisition systematically and accurately.

Improving the generalization ability of ensemble learning is very important. This paper proposes a new ensemble learning method embedded with PCA-ReliefF Algorithm. It can efficiently remove the influence of noise features, and improve the classification accuracy of ensemble learning. Experiments on UCI dataset show that the method in this paper is feasible and effective, and provides a new research approach for feature classification in pattern recognition.

In this paper, we study some shifted convolution sums for higher rank groups. In particular, we establish an asymptotic formula for a GL ⁢ ( 4 ) × GL ⁢ ( 2 ) {\mathrm{GL}(4)\times\mathrm{GL}(2)} shifted convolution sum ∑ n ≤ x | λ f ⁢ ( n ) | 2 ⁢ r l ⁢ ( n + b ) , \sum_{n\leq x}\lvert\lambda_{f}(n)\rvert^{2}r_{l}(n+b), where λ f ⁢ ( n ) {\lambda_{f}(n)} are normalized Fourier coefficients of a Hecke holomorphic cusp form and r l ⁢ ( n ) {r_{l}(n)} denotes the number of representations of n by the quadratic form x 1 2 + ⋯ + x l 2 {x_{1}^{2}+\cdots+x_{l}^{2}} .

In this paper, we study shifted convolution sums for GL(2) and give an upper bound for $$\sum \nolimits _{n\ge 1}\lambda _f(n+b) r(n,Q)g(n)$$ , where g(n) is a smooth weight function. In particular, we get an upper bound for $$\sum \nolimits _{n\le x}\lambda _f(n+b)r_3(n)$$ , which improves the result in Lu et al. (Ramanujan J 40(1):C115–C133, 2016).

Let $$\mu (n)$$ be the Mobius function, $$\Lambda (n)$$ be the von Mangoldt function, and $$A_F(n, 1,\ldots ,1)$$ denote the nth coefficient of the Dirichlet series for L(s, F) associated to a Hecke–Maass form F for $$SL(m,\mathbb {Z})$$ . In this paper, as an appendix to our previous work, we firstly proved that for a Maass form for $$SL(3,\mathbb {Z})$$ , uniformly for all real numbers $$\theta $$ , the sequences $$\{\mu (n)\}$$ and $$\{A_F(n,1)e(n\theta )\}$$ are strong asymptotically orthogonal. Then as an analogue of Baker and Harman’s result on exponential sums formed with the Mobius function under the Generalized Riemann Hypothesis, we investigated the best possible estimates for the sum $$\sum _{n\le x}\Lambda (n) A_F(n, 1,\ldots ,1) e(n^k\theta )$$ under certain assumptions.

Metal-support interaction and corresponding interface regulation are of great importance for the preparation of well-defined, supported metal catalysts with outstanding performance. Herein a joint experimental-theoretical study has been conducted to investigate the role of polydopamine (PDA) as an interface facilitator for supported noble metal catalysts. The effects of PDA between various noble metals (Pt, Pd, Au, Ag) and a support (SBA-15) have been systematically investigated in the model reaction of 4-nitrophenol (4-NP) reduction by NaBH4. Based on the kinetic model of 4-NP reduction by NaBH4 and a set of density functional theory calculations on different 4-NP binding energies, a volcano-shaped relationship plot is obtained. The volcano-shaped dependence of the reaction rate constant on 4-NP adsorption energy is observed for supported noble metal catalysts with/without PDA. The general relationship indicates that PDA can effectively improve catalytic performance via adjusting 4-NP adsorption energy to a more suitable value. The results of charge density difference and Bader charge analysis illustrate the establishment of an ‘electron pool’ by PDA in the system to achieve controllable electronic charge redistribution for an optimized binding when the reactant molecule adsorbs on the metal nanoparticle.

Aiming at the evaluation of equipment competitive acquisition management, the paper proposes an evaluation method based on entropy theory. First, the concept of equipment competitive acquisition management entropy is defined. Second, the conceptual model, mathematical model, and evaluation index system of equipment competitive acquisition management entropy are constructed. Finally, taking policy and regulatory entropy as an example, the paper has calculated and analyzed its evaluation results to prove the effectiveness of the method.

Let π be a unitary cuspidal automorphic representation for GL m ⁢ ( 𝔸 ℚ ) {\mathrm{GL}_{m}(\mathbb{A}_{\mathbb{Q}})} , and let L ⁢ ( s , π ) {L(s,\pi)} be the automorphic L-function attached to π, which has a Dirichlet series expression in the half-plane ℜ ⁡ s > 1 {\Re s>1} , i.e. L ⁢ ( s , π ) = ∑ n = 1 ∞ λ π ⁢ ( n ) n s . L(s,\pi)=\sum_{n=1}^{\infty}\frac{\lambda_{\pi}(n)}{n^{s}}. In this paper we are interested in the upper bound of the fourth power moment of λ π ⁢ ( n ) {\lambda_{\pi}(n)} , i.e. ∑ n ≤ x λ π ⁢ ( n ) 4 {\sum_{n\leq x}\lambda_{\pi}(n)^{4}} . If m = 2 {m=2} , we are able to consider the sixteenth power moment of λ π ⁢ ( n ) {\lambda_{\pi}(n)} . As an application, we consider the lower bound of ∑ n ≤ x | λ π ⁢ ( n ) | {\sum_{n\leq x}\lvert\lambda_{\pi}(n)\rvert} , which improves previous results.

Let F(z) be a Hecke–Maass form for SL(m, ℤ) with m ⩽ 3, or be the symmetric power lift of a Hecke–Maass form for SL(2, ℤ) if m = 4, 5 and let AF(n, 1, . . ., 1) be the coefficients of L-function attached to F. We establish $$\sum_{q\leq Q}\max_{(a,q)=1}\max_{y\leq x}\left|\sum_{n\leq y \atop n\equiv a\bmod q}A_F(n,1, \dots, 1)\Lambda(n)\right| \ll x\log^{-A}x,$$ where Q = xϑ−ϵ with some ϑ > 0, the implied constant depends on F, A, ϵ.

The effects of the relative content of Al and Ti, and the addition of 2 at.% Cr or Mo to partly substitute Co on the microstructure and mechanical properties of Co–Ni–Al–Ti superalloy were investigated. The microstructures of five as-cast superalloys showed that γ'-(Co,Ni)3(Al,Ti) precipitates segregated on γ-Co matrix and exhibited dendritic morphology. After aging, nano-sized γ'-(Co,Ni)3(Al,Ti) precipitates uniformly precipitated on γ-Co matrix. Ni, Al, Ti and Mo atoms partitioned preferentially to γ′ phase, while Co and Cr atoms partitioned preferentially to γ matrix phase. The increase in Ti content, and the addition of 2 at.% Cr or Mo all increased the volume fraction and the average size of γ′ phase. The lattice misfit increased with the concentration of Ti and the addition of 2 at.% Mo, but decreased with the addition of 2 at.% Cr. With increasing aging time, the size of γ′ phase and lattice misfit increased, while γ′ phase volume fraction increased firstly, and then decreased after reaching the maximum at 100 h. The lattice misfit plays a key role on vicker's hardness in the time of 20 h–100 h aging, while γ′ phase volume fraction plays a decisive role in vicker's hardness after 100 h aging.

We utilized a set of fused silica thin plates to broaden the spectrum of 1 kHz, 30 fs Ti:sapphire amplified laser pulses to an octave. Following the compression by chirped mirror pairs, the generated few-cycle pulses were focused onto an argon filled gas cell. We detected high order harmonics corresponding to a train of 209 as pulses, characterized by the reconstruction of attosecond beating by interference of two-photon transition (RABITT) technique. Compared with the conventional attosecond pulse trains, the broad harmonics in such pulse trains cover more energy range, so it is more efficient in studying some typical cases, such as resonances, with frequency resolved RABITT. As the solid thin plates can support high power supercontinuum generation, it is feasible to tailor the spectrum to have different central wavelength and spectral width, which will make the RABITT source work in different applications.

We measured and stabilized the carrier-envelope phase with three different methods of supercontinuum generation by f-2f spectral interferometry, and recorded the CEP jitter of each method, finding that fused silica thin plates perfomed better than sapphire plate and hollow-core fiber.

Let $F(z)$ be a Hecke-Maass form for $SL(m,\mathbb{Z})$ and $A_F(n,1, \dots, 1)$ be the coefficients of $L$-function attached to $F.$ We study the cancellation of $A_F(n,1, \dots, 1)$ for twisted with a nonlinear exponential function at primes, namely the sum \begin{equation*} \sum_{n \leq N} \Lambda (n)A_F(n,1, \dots, 1)e ( \alpha n^\theta ), \end{equation*} where $0

Let \(\lambda _\phi (n)\) be the \(n\)-th Fourier coefficient of a doubly even and normalized Hecke–Maass cusp form for hyperbolic \(3\)-manifolds. In this paper, we investigate the behavior of summatory functions in the following (i) the \(j\)-th power sum of \(\lambda _\phi (n)\) $$\begin{aligned} \sum _{N(n)\le x}\lambda _\phi (n)^{j}, \end{aligned}$$ where \(j\le 8;\) (ii) the sum of \(\lambda _\phi (n)\) over the sparse sequence \({n^l}\) $$\begin{aligned} \sum _{N(n)\le x}\lambda _\phi (n^l), \end{aligned}$$ where \(l\le 4;\) (iii) the hybrid sum for \(\lambda _\phi (n)\) $$\begin{aligned} \sum _{N(n)\le x}\lambda _\phi (n^l)^{j}, \end{aligned}$$ where \(2\le l\le 4, j=2,\) or \(l=2, j=4.\)

Let ϕ ( z ) denote a holomorphic or Maass cusp form for the full modular group Γ = SL ( 2 , Z ) . And let λ Sym 2 ϕ ( n ) be the n -th coefficient of symmetric square L -function associated with ϕ ( z ) . We establish the uniform upper bound for the summatory function ∑ n ≤ x λ Sym 2 ϕ ( n ) , which improves the results of Ichihara [4] , Lu [10] , Sankaranarayanan [13] and Tang [14] .