Structure of large quadratic character sums

In this article, we study the distribution of large quadratic character sums. Based on the recent work of Lamzouri~\cite{La2022}, we obtain the structure results of quadratic characters with large character sums.Comment: 15 page

Distribution of Dirichlet LL-functions

In this article, we study the distribution of values of Dirichlet $L$-functions, the distribution of values of the random models for Dirichlet $L$-functions, and the discrepancy between these two kinds of distributions. For each question, we consider the cases of \frac12<\RE s<1 and \RE s=1 separately.Comment: 24 page

Large zeta sums

In this article, we investigate the behaviour of values of zeta sums $\sum_{n\le x}n^{it}$ when $t$ is large. We show some asymptotic behaviour and Omega results of zeta sums, which are analogous to previous results of large character sums $\sum_{n\le x}\chi(n)$.Comment: 11 pages

A Survey of Large Language Models

Language is essentially a complex, intricate system of human expressions governed by grammatical rules. It poses a significant challenge to develop capable AI algorithms for comprehending and grasping a language. As a major approach, language modeling has been widely studied for language understanding and generation in the past two decades, evolving from statistical language models to neural language models. Recently, pre-trained language models (PLMs) have been proposed by pre-training Transformer models over large-scale corpora, showing strong capabilities in solving various NLP tasks. Since researchers have found that model scaling can lead to performance improvement, they further study the scaling effect by increasing the model size to an even larger size. Interestingly, when the parameter scale exceeds a certain level, these enlarged language models not only achieve a significant performance improvement but also show some special abilities that are not present in small-scale language models. To discriminate the difference in parameter scale, the research community has coined the term large language models (LLM) for the PLMs of significant size. Recently, the research on LLMs has been largely advanced by both academia and industry, and a remarkable progress is the launch of ChatGPT, which has attracted widespread attention from society. The technical evolution of LLMs has been making an important impact on the entire AI community, which would revolutionize the way how we develop and use AI algorithms. In this survey, we review the recent advances of LLMs by introducing the background, key findings, and mainstream techniques. In particular, we focus on four major aspects of LLMs, namely pre-training, adaptation tuning, utilization, and capacity evaluation. Besides, we also summarize the available resources for developing LLMs and discuss the remaining issues for future directions.Comment: ongoing work; 51 page

Real-time Monitoring for the Next Core-Collapse Supernova in JUNO

Core-collapse supernova (CCSN) is one of the most energetic astrophysical events in the Universe. The early and prompt detection of neutrinos before (pre-SN) and during the SN burst is a unique opportunity to realize the multi-messenger observation of the CCSN events. In this work, we describe the monitoring concept and present the sensitivity of the system to the pre-SN and SN neutrinos at the Jiangmen Underground Neutrino Observatory (JUNO), which is a 20 kton liquid scintillator detector under construction in South China. The real-time monitoring system is designed with both the prompt monitors on the electronic board and online monitors at the data acquisition stage, in order to ensure both the alert speed and alert coverage of progenitor stars. By assuming a false alert rate of 1 per year, this monitoring system can be sensitive to the pre-SN neutrinos up to the distance of about 1.6 (0.9) kpc and SN neutrinos up to about 370 (360) kpc for a progenitor mass of 30$M_{\odot}$ for the case of normal (inverted) mass ordering. The pointing ability of the CCSN is evaluated by using the accumulated event anisotropy of the inverse beta decay interactions from pre-SN or SN neutrinos, which, along with the early alert, can play important roles for the followup multi-messenger observations of the next Galactic or nearby extragalactic CCSN.Comment: 24 pages, 9 figure

Distribution de valeurs de la fonction zeta de Riemann

L’étude de la distribution de la valeur de la fonction zêta de Riemann DOLzeta(s)DOL remonte au début du XXe siècle lorsque Bohr a montré que pour tout DOLzinC^*DOL et DOLvarepsilon>0DOL,il existe une infinité de DOLsDOL avec DOL1frac{1}{2}DOL. Jusqu’à présent, un certain nombre de théories ont été développées. Sur la ligne critique, le théorème de la limite centrale de Selberg indique que le logarithme de Riemannfonction zeta DOLlog|zeta(frac12+i t)|DOL se comporte comme un complexeVariable aléatoire gaussienne de moyenne 0 et variance DOLfrac12log_2TDOL comme DOLTtoinftyDOL, où DOLtDOL varie en DOL[T,2T]DOL. Sur la ligne 1, Granville et Soundararajan ont établi la distribution de DOL|zeta(1+i t)|DOL, qui est asymptotiquement une fonction à double exposant. Dans la bande critique DOLfrac120DOL, il existe arbitrairement grand DOLtDOL tel que DOLlog|zeta(sigma+i t)|ge(log t)^{1-sigma-varepsilon}DOL. En 1977, Montgomery cite{Mon77} a montré que DOLlog|zeta(sigma+i t)|DOL peut être supérieur à DOLc(sigma)(log t)^{1-sigma}/(log_2t)^sigmaDOL pour une constante DOLc(sigma)DOL. Il a également conclu qu ’il s’agit du maximum de l’ordre de DOLlog|zeta(sigma+i t)|DOL jusqu’à DOLc(sigma)DOL. Ainsi, tous les derniers éléments de ce problème se concentrent sur l’obtention de valeurs plus grandes de DOLc(sigma)DOL. En 2011, Lamzouri cite{La2011} a donné une valeur conjecturale de DOLc(sigma)DOL. En 2018, Bondarenko et Seip ont examiné les cas de DOLsigmasearrowfrac12DOL et DOLsigmanearrow1DOL. Nous étudions également le premier cas et obtenons une amélioration du résultat de Bondarenko-Seip.The study of the value distribution of the Riemann zeta function DOLLARzeta(s)DOLLAR can date back to the early twentieth century when Bohr showed that for any DOLLARzinC^*DOLLAR and DOLLARvarepsilon>0DOLLAR,there are infinitely many DOLLARsDOLLAR's with DOLLAR1frac{1}{2}DOLLAR. Till now, quite a few of theory has been developed. On the critical line, Selberg's Central Limit Theorem states that the logarithm of the Riemannzeta function DOLLARlog|zeta(frac12+i t)|DOLLAR behaves like a complexGaussian random variable of mean 0 and variance DOLLARfrac12log_2TDOLLAR as DOLLARTtoinftyDOLLAR, where DOLLARtDOLLAR varies in DOLLAR[T,2T]DOLLAR. On the 1-line, Granville and Soundararajan established the distribution of DOLLAR|zeta(1+i t)|DOLLAR, which is asymptotically a double-exponent function. In the critical strip DOLLARfrac120DOLLAR there exists arbitrarily large DOLLARtDOLLAR such that DOLLARlog|zeta(sigma+i t)|ge(log t)^{1-sigma-varepsilon}DOLLAR. In 1977, Montgomery {Mon77} showed that DOLLARlog|zeta(sigma+i t)|DOLLAR can be larger than DOLLARc(sigma)(log t)^{1-sigma}/(log_2t)^sigmaDOLLAR for some constant DOLLARc(sigma)DOLLAR. He also conjected that this is the maximum of the order of DOLLARlog|zeta(sigma+i t)|DOLLAR up to DOLLARc(sigma)DOLLAR. So all the later improments for this problem focus on getting larger values of DOLLARc(sigma)DOLLAR. In 2011, Lamzouri {La2011} gave a conjectural value of DOLLARc(sigma)DOLLAR. In 2018, Bondarenko and Seip {BS18} considered the cases of DOLLARsigmasearrowfrac12DOLLAR and DOLLARsigmanearrow1DOLLAR. We also study the first case and get an improvement of the result of Bondarenko-Seip

Distribution de valeurs de la fonction zeta de Riemann

The study of the value distribution of the Riemann zeta function DOLLARzeta(s)DOLLAR can date back to the early twentieth century when Bohr showed that for any DOLLARzinC^*DOLLAR and DOLLARvarepsilon>0DOLLAR,there are infinitely many DOLLARsDOLLAR's with DOLLAR1frac{1}{2}DOLLAR. Till now, quite a few of theory has been developed. On the critical line, Selberg's Central Limit Theorem states that the logarithm of the Riemannzeta function DOLLARlog|zeta(frac12+i t)|DOLLAR behaves like a complexGaussian random variable of mean 0 and variance DOLLARfrac12log_2TDOLLAR as DOLLARTtoinftyDOLLAR, where DOLLARtDOLLAR varies in DOLLAR[T,2T]DOLLAR. On the 1-line, Granville and Soundararajan established the distribution of DOLLAR|zeta(1+i t)|DOLLAR, which is asymptotically a double-exponent function. In the critical strip DOLLARfrac120DOLLAR there exists arbitrarily large DOLLARtDOLLAR such that DOLLARlog|zeta(sigma+i t)|ge(log t)^{1-sigma-varepsilon}DOLLAR. In 1977, Montgomery {Mon77} showed that DOLLARlog|zeta(sigma+i t)|DOLLAR can be larger than DOLLARc(sigma)(log t)^{1-sigma}/(log_2t)^sigmaDOLLAR for some constant DOLLARc(sigma)DOLLAR. He also conjected that this is the maximum of the order of DOLLARlog|zeta(sigma+i t)|DOLLAR up to DOLLARc(sigma)DOLLAR. So all the later improments for this problem focus on getting larger values of DOLLARc(sigma)DOLLAR. In 2011, Lamzouri {La2011} gave a conjectural value of DOLLARc(sigma)DOLLAR. In 2018, Bondarenko and Seip {BS18} considered the cases of DOLLARsigmasearrowfrac12DOLLAR and DOLLARsigmanearrow1DOLLAR. We also study the first case and get an improvement of the result of Bondarenko-Seip.L’étude de la distribution de la valeur de la fonction zêta de Riemann DOLzeta(s)DOL remonte au début du XXe siècle lorsque Bohr a montré que pour tout DOLzinC^*DOL et DOLvarepsilon>0DOL,il existe une infinité de DOLsDOL avec DOL1frac{1}{2}DOL. Jusqu’à présent, un certain nombre de théories ont été développées. Sur la ligne critique, le théorème de la limite centrale de Selberg indique que le logarithme de Riemannfonction zeta DOLlog|zeta(frac12+i t)|DOL se comporte comme un complexeVariable aléatoire gaussienne de moyenne 0 et variance DOLfrac12log_2TDOL comme DOLTtoinftyDOL, où DOLtDOL varie en DOL[T,2T]DOL. Sur la ligne 1, Granville et Soundararajan ont établi la distribution de DOL|zeta(1+i t)|DOL, qui est asymptotiquement une fonction à double exposant. Dans la bande critique DOLfrac120DOL, il existe arbitrairement grand DOLtDOL tel que DOLlog|zeta(sigma+i t)|ge(log t)^{1-sigma-varepsilon}DOL. En 1977, Montgomery cite{Mon77} a montré que DOLlog|zeta(sigma+i t)|DOL peut être supérieur à DOLc(sigma)(log t)^{1-sigma}/(log_2t)^sigmaDOL pour une constante DOLc(sigma)DOL. Il a également conclu qu ’il s’agit du maximum de l’ordre de DOLlog|zeta(sigma+i t)|DOL jusqu’à DOLc(sigma)DOL. Ainsi, tous les derniers éléments de ce problème se concentrent sur l’obtention de valeurs plus grandes de DOLc(sigma)DOL. En 2011, Lamzouri cite{La2011} a donné une valeur conjecturale de DOLc(sigma)DOL. En 2018, Bondarenko et Seip ont examiné les cas de DOLsigmasearrowfrac12DOL et DOLsigmanearrow1DOL. Nous étudions également le premier cas et obtenons une amélioration du résultat de Bondarenko-Seip

Distinguishable Colorimetric Biosensor for Diagnosis of Prostate Cancer Bone Metastases

Abstract Castration‐resistant prostate cancer (PCa) causes severe bone metastasis (BM), which significantly increases mortality in men with PCa. Imaging tests and radiometric scanning require long analysis times, expensive equipment, specialized personnel, and a slow turnaround. New visualization technologies are expected to solve the above problems. Nonetheless, existing visualization techniques barely meet the urgency for precise diagnosis because the human eyes cannot recognize and capture even slight variations in visual information. By using dye differentiated superposition enhancement colorimetric biosensors, an effective method to diagnose prostate cancer bone metastases (PCa‐BM) with excellent accuracy for naked‐eye quantitative detection of alkaline phosphatase (ALP) is developed. The biomarker ALP specific hydrolytic product ascorbic acid can be detected by rhodamine derivatives (Rd) as gold nanobipyramids (Au NBPs) are deposited and grown. Color‐recombining enhancement effects between Rd and Au NBPs significantly improved abundance. The 150 U L−1 threshold between normal and abnormal can be identified by color. And with color enhancement effect and double signal response, the ALP index is visually measured to diagnose PCa‐BM and provide handy treatment recommendations. Additionally, the proposed colorimetric sensing strategy can be used to diagnose other diseases