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In a previous paper arXiv:2406.06294 [math.NT], the author proved the exact formulae for ranks of partitions modulo each prime $p\geq 5$. In this paper, for $p=5$ and $7$, we prove special vanishing properties of the Kloosterman sums appearing in the exact formulae. These vanishing properties imply a new proof of Dyson's rank conjectures. Specifically, we give a new proof of Ramanujan's congruences $p(5n+4)\equiv 0\pmod 5$ and $p(7n+5)\equiv 0\pmod 7$.

In 2009, Bringmann arXiv:0708.0691 [math.NT] used the circle method to prove an asymptotic formula for the Fourier coefficients of rank generating functions. In this paper, we prove that Bringmann's formula, when summing up to infinity and in the case of prime modulus, gives a Rademacher-type exact formula involving sums of vector-valued Kloosterman sums. As a corollary, in an upcoming paper, we will provide a new proof of Dyson's conjectures by showing that the certain Kloosterman sums vanish.

In the previous paper [Sun23], the author proved a uniform bound for sums of half-integral weight Kloosterman sums. This bound was applied to prove an exact formula for partitions of rank modulo 3. That uniform estimate provides a more precise bound for a certain class of multipliers compared to the 1983 result by Goldfeld and Sarnak and generalizes the 2009 result from Sarnak and Tsimerman to the half-integral weight case. However, the author only considered the case when the parameters satisfied $\tilde m\tilde n<0$. In this paper, we prove the same uniform bound when $\tilde m\tilde n>0$ for further applications.

The rise of the click farm business using Multi-purpose Messaging Mobile Apps (MMMAs) tempts cybercriminals to perpetrate crowdsourcing frauds that cause financial losses to click farm workers. In this paper, we propose a novel contrastive multi-view learning method named CMT for crowdsourcing fraud detection over the heterogeneous temporal graph (HTG) of MMMA. CMT captures both heterogeneity and dynamics of HTG and generates high-quality representations for crowdsourcing fraud detection in a self-supervised manner. We deploy CMT to detect crowdsourcing frauds on an industry-size HTG of a representative MMMA WeChat and it significantly outperforms other methods. CMT also shows promising results for fraud detection on a large-scale public financial HTG, indicating that it can be applied in other graph anomaly detection tasks. We provide our implementation at https://github.com/KDEGroup/CMT., Comment: Full technical report for our DASFAA 2024 paper: Crowdsourcing Fraud Detection over Heterogeneous Temporal MMMA Graph

Sums of Kloosterman sums have deep connections with the theory of modular forms, and their estimation has many important consequences. Kuznetsov used his famous trace formula and got a power-saving estimate with respect to $x$ with implied constants depending on $m$ and $n$. Recently, in 2009, Sarnak and Tsimerman obtained a bound uniformly in $x$, $m$ and $n$. The generalized Kloosterman sums are defined with multiplier systems and on congruence subgroups. Goldfeld and Sarnak bounded sums of them with main terms corresponding to exceptional eigenvalues of the hyperbolic Laplacian. Their error term is a power of $x$ with implied constants depending on all the other factors. In this paper, for a wide class of half-integral weight multiplier systems, we get the same bound with the error term uniformly in $x$, $m$ and $n$. Such uniform bounds have great applications. For the eta-multiplier, Ahlgren and Andersen obtained a uniform and power-saving bound with respect to $m$ and $n$, which resulted in a convergent error estimate on the Rademacher exact formula of the partition function $p(n)$. We also establish a Rademacher-type exact formula for the difference of partitions of rank modulo $3$, which allows us to apply our power-saving estimate to the tail of the formula for a convergent error bound.

Traces of singular moduli can be approximated by exponential sums of quadratic irrationals. Recently Andersen and Duke used theory of Maass forms to estimate generalized twisted traces with power-saving error bounds. We establish an asymptotic formula with effective error bounds for such traces. Our methods depend on an explicit bound for sums of Kloosterman sums on $\Gamma_0(4)$.

Nowadays, Multi-purpose Messaging Mobile App (MMMA) has become increasingly prevalent. MMMAs attract fraudsters and some cybercriminals provide support for frauds via black market accounts (BMAs). Compared to fraudsters, BMAs are not directly involved in frauds and are more difficult to detect. This paper illustrates our BMA detection system SGRL (Self-supervised Graph Representation Learning) used in WeChat, a representative MMMA with over a billion users. We tailor Graph Neural Network and Graph Self-supervised Learning in SGRL for BMA detection. The workflow of SGRL contains a pretraining phase that utilizes structural information, node attribute information and available human knowledge, and a lightweight detection phase. In offline experiments, SGRL outperforms state-of-the-art methods by 16.06%-58.17% on offline evaluation measures. We deploy SGRL in the online environment to detect BMAs on the billion-scale WeChat graph, and it exceeds the alternative by 7.27% on the online evaluation measure. In conclusion, SGRL can alleviate label reliance, generalize well to unseen data, and effectively detect BMAs in WeChat., Comment: WSDM 2023. This is the complete version containing the appendix

Regularization method and Bayesian inverse method are two dominating ways for solving inverse problems generated from various fields, e.g., seismic exploration and medical imaging. The two methods are related with each other by the MAP estimates of posterior probability distributions. Considering this connection, we construct a prior probability distribution with several hyper-parameters and provide the relevant Bayes' formula, then we propose a corresponding adaptive augmented regularization model (AARM). According to the measured data, the proposed AARM can adjust its form to various regularization models at each discrete point of the estimated function, which makes the characterization of local smooth properties of the estimated function possible. By proposing a modified Bregman iterative algorithm, we construct an alternate iterative algorithm to solve the AARM efficiently. In the end, we provide some numerical examples which clearly indicate that the proposed AARM can generates a favorable result for some examples compared with several Tikhonov and Total-Variation regularization models., Comment: 26 Pages