The accelerating pace of scientific publishing makes it increasingly difficult for researchers to stay current. We present Paper Espresso, an open-source platform that automatically discovers, summarizes, and analyzes trending arXiv papers. The system uses large language models (LLMs) to generate structured summaries with topical labels and keywords, and provides multi-granularity trend analysis at daily, weekly, and monthly scales through LLM-driven topic consolidation. Over 35 months of continuous deployment, Paper Espresso has processed over 13,300 papers and publicly released all structured metadata, revealing rich dynamics in the AI research landscape: a mid-2025 surge in reinforcement learning for LLM reasoning, non-saturating topic emergence (6,673 unique topics), and a positive correlation between topic novelty and community engagement (2.0x median upvotes for the most novel papers). A live demo is available at https://huggingface.co/spaces/Elfsong/Paper_Espresso.