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学者姓名：郗平
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Keyword ：
arcsine law character sum Dirichlet L-function equidistribution Kloosterman sum moment
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| GB/T 7714 | Xi, Ping . Moments and equidistributions of multiplicative analogues of Kloosterman sums [J]. | ACTA ARITHMETICA , 2022 , 206 (3) : 245-275 . |
| MLA | Xi, Ping . "Moments and equidistributions of multiplicative analogues of Kloosterman sums" . | ACTA ARITHMETICA 206 . 3 (2022) : 245-275 . |
| APA | Xi, Ping . Moments and equidistributions of multiplicative analogues of Kloosterman sums . | ACTA ARITHMETICA , 2022 , 206 (3) , 245-275 . |
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Abstract ：
We give a corrigendum to the previous paper [8] and recover the same quantitative statement: the Kloosterman sum changes sign infinitely often as the modulus runs over squarefree numbers with at most seven prime factors.
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| GB/T 7714 | Xi, Ping . Sign Changes of Kloosterman Sums with Almost Prime Moduli. II: Corrigendum [J]. | INTERNATIONAL MATHEMATICS RESEARCH NOTICES , 2022 , 2022 (1) : 556-574 . |
| MLA | Xi, Ping . "Sign Changes of Kloosterman Sums with Almost Prime Moduli. II: Corrigendum" . | INTERNATIONAL MATHEMATICS RESEARCH NOTICES 2022 . 1 (2022) : 556-574 . |
| APA | Xi, Ping . Sign Changes of Kloosterman Sums with Almost Prime Moduli. II: Corrigendum . | INTERNATIONAL MATHEMATICS RESEARCH NOTICES , 2022 , 2022 (1) , 556-574 . |
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Abstract ：
By elaborating a two-dimensional Selberg sieve with asymptotics and equidistributions of Kloosterman sums from l\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell $$\end{document}-adic cohomology, as well as a Bombieri-Vinogradov type mean value theorem for Kloosterman sums in arithmetic progressions, it is proved that for any given primitive Hecke-Maass cusp form of trivial nebentypus, the eigenvalue of the n-th Hecke operator does not coincide with the Kloosterman sum Kl(1,n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {Kl}(1,n)$$\end{document} for infinitely many squarefree n with at most 100 prime factors. This provides a partial negative answer to a problem of Katz on modular structures of Kloosterman sums.
Keyword ：
11F30 11L05 11N36 11T23
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| GB/T 7714 | Xi, Ping . When Kloosterman sums meet Hecke eigenvalues [J]. | INVENTIONES MATHEMATICAE , 2020 , 220 (1) : 61-127 . |
| MLA | Xi, Ping . "When Kloosterman sums meet Hecke eigenvalues" . | INVENTIONES MATHEMATICAE 220 . 1 (2020) : 61-127 . |
| APA | Xi, Ping . When Kloosterman sums meet Hecke eigenvalues . | INVENTIONES MATHEMATICAE , 2020 , 220 (1) , 61-127 . |
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Abstract ：
A classical problem of D. H. Lehmer suggests the study of distributions of elements of Z/pZ of opposite parity to the multiplicative inverse mod p. Zhang initiated this problem and found an asymptotic evaluation of the number of such elements. In this paper, an asymptotic formula for the fourth moment of the error term of Zhang is proved, from which one may see that Zhang's error term is optimal up to the logarithm factor. The method also applies to the case of arbitrary positive integral moments.
Keyword ：
D H Kloosterman sum Lehmer problem Moment
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| GB/T 7714 | Xi, Ping , Yi, Yuan . Kloosterman Sums and a Problem of D. H. Lehmer [J]. | CHINESE ANNALS OF MATHEMATICS SERIES B , 2020 , 41 (3) : 361-370 . |
| MLA | Xi, Ping 等. "Kloosterman Sums and a Problem of D. H. Lehmer" . | CHINESE ANNALS OF MATHEMATICS SERIES B 41 . 3 (2020) : 361-370 . |
| APA | Xi, Ping , Yi, Yuan . Kloosterman Sums and a Problem of D. H. Lehmer . | CHINESE ANNALS OF MATHEMATICS SERIES B , 2020 , 41 (3) , 361-370 . |
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Abstract ：
We prove that the Kloosterman sum S(1, 1; c) changes sign infinitely often as c runs over squarefree numbers with at most 7 prime factors, which improves the previous results of Fouvry and Michel, Sivak-Fischler, Matomaki, and the author.
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| GB/T 7714 | Xi, Ping . Sign Changes of Kloosterman Sums with Almost Prime Moduli. II [J]. | INTERNATIONAL MATHEMATICS RESEARCH NOTICES , 2018 , 2018 (4) : 1200-1227 . |
| MLA | Xi, Ping . "Sign Changes of Kloosterman Sums with Almost Prime Moduli. II" . | INTERNATIONAL MATHEMATICS RESEARCH NOTICES 2018 . 4 (2018) : 1200-1227 . |
| APA | Xi, Ping . Sign Changes of Kloosterman Sums with Almost Prime Moduli. II . | INTERNATIONAL MATHEMATICS RESEARCH NOTICES , 2018 , 2018 (4) , 1200-1227 . |
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Abstract ：
We prove that the exponent of distribution of tau 3 in arithmetic progressions can be as large as 1/2 + 1/34, provided that the moduli is squarefree and has only sufficiently small prime factors. The tools involve arithmetic exponent pairs for algebraic trace functions, as well as a double q-analogue of the van der Corput method for smooth bilinear forms.
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| GB/T 7714 | Xi, Ping . TERNARY DIVISOR FUNCTIONS IN ARITHMETIC PROGRESSIONS TO SMOOTH MODULI [J]. | MATHEMATIKA , 2018 , 64 (3) : 701-729 . |
| MLA | Xi, Ping . "TERNARY DIVISOR FUNCTIONS IN ARITHMETIC PROGRESSIONS TO SMOOTH MODULI" . | MATHEMATIKA 64 . 3 (2018) : 701-729 . |
| APA | Xi, Ping . TERNARY DIVISOR FUNCTIONS IN ARITHMETIC PROGRESSIONS TO SMOOTH MODULI . | MATHEMATIKA , 2018 , 64 (3) , 701-729 . |
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Abstract ：
Assume that alpha, beta are arbitrary arithmetic functions, and that gamma is an arithmetic function, which does not correlate with additive characters. By introducing a simple argument, we are able to give a general upper bound for the triple correlation & para;& para;Sigma(n is an element of(X,2X] alpha(n)beta(n - h)gamma(n + h)& para;& para;averaging over a short variable h. More precise estimates can be obtained by taking gamma to be Fourier coefficients of cusp forms, which improve and stream line the results of Lin and Singh. (C) 2017 Elsevier Inc. All rights reserved.)
Keyword ：
Circle method Cusp forms Fourier coefficients Triple correlation
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| GB/T 7714 | Lu, Guangshi , Xi, Ping . On triple correlations of Fourier coefficients of cusp forms [J]. | JOURNAL OF NUMBER THEORY , 2018 , 183 : 485-492 . |
| MLA | Lu, Guangshi 等. "On triple correlations of Fourier coefficients of cusp forms" . | JOURNAL OF NUMBER THEORY 183 (2018) : 485-492 . |
| APA | Lu, Guangshi , Xi, Ping . On triple correlations of Fourier coefficients of cusp forms . | JOURNAL OF NUMBER THEORY , 2018 , 183 , 485-492 . |
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Abstract ：
We consider a quadratic analogue of the classical Titchmarsh divisor problem and prove an upper bound for the average of tau(p(2) + 1) over p <= X. (C) 2017 Elsevier Inc. All rights reserved.
Keyword ：
Arithmetic exponent pairs Primes in arithmetic progressions q-Analogue of van der Corput method Titchmarsh divisor problem
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| GB/T 7714 | Xi, Ping . A quadratic analogue of Titchmarsh divisor problem [J]. | JOURNAL OF NUMBER THEORY , 2018 , 184 : 192-205 . |
| MLA | Xi, Ping . "A quadratic analogue of Titchmarsh divisor problem" . | JOURNAL OF NUMBER THEORY 184 (2018) : 192-205 . |
| APA | Xi, Ping . A quadratic analogue of Titchmarsh divisor problem . | JOURNAL OF NUMBER THEORY , 2018 , 184 , 192-205 . |
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Abstract ：
In this paper, we estimate the shifted convolution sum Sigma(n >= 1) lambda(1)(1, n)lambda(2)(n + h)V(n/x), where Visa smooth function with support in [1, 2], 1 <= vertical bar h vertical bar <= X, and lambda 1 (1, n) and lambda 2(n) are the n-th Fourier coefficients of SL(3, Z) and SL(2, Z) Hecke-Maass cusp forms, respectively. We prove an upper bound 0(X21/22+epsilon), updating a recent result of Munshi.
Keyword ：
exponential sums Fourier coefficients of cusp forms Shifted convolution sum
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| GB/T 7714 | Xi, Ping . A shifted convolution sum for GL(3) x GL(2) [J]. | FORUM MATHEMATICUM , 2018 , 30 (4) : 1013-1027 . |
| MLA | Xi, Ping . "A shifted convolution sum for GL(3) x GL(2)" . | FORUM MATHEMATICUM 30 . 4 (2018) : 1013-1027 . |
| APA | Xi, Ping . A shifted convolution sum for GL(3) x GL(2) . | FORUM MATHEMATICUM , 2018 , 30 (4) , 1013-1027 . |
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Keyword ：
Brun-Titchmarsh theorem greatest prime factor strong primes
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| GB/T 7714 | Xi, Ping . How strong can primes be [J]. | ACTA ARITHMETICA , 2017 , 179 (4) : 363-373 . |
| MLA | Xi, Ping . "How strong can primes be" . | ACTA ARITHMETICA 179 . 4 (2017) : 363-373 . |
| APA | Xi, Ping . How strong can primes be . | ACTA ARITHMETICA , 2017 , 179 (4) , 363-373 . |
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