Photonic neuromorphic computing is of particular interest due to its significant potential for ultrahigh computing speed and energy efficiency. The advantage of photonic computing hardware lies in its ultrawide bandwidth and parallel processing utilizing inherent parallelism. Here, we demonstrate a scalable on-chip photonic implementation of a simplified recurrent neural network, called a reservoir computer, using an integrated coherent linear photonic processor. In contrast to previous approaches, both the input and recurrent weights are encoded in the spatiotemporal domain by photonic linear processing, which enables scalable and ultrafast computing beyond the input electrical bandwidth. As the device can process multiple wavelength inputs over the telecom C-band simultaneously, we can use ultrawide optical bandwidth (~5 terahertz) as a computational resource. Experiments for the standard benchmarks showed good performance for chaotic time-series forecasting and image classification. The device is considered to be able to perform 21.12 tera multiplication–accumulation operations per second (MAC ∙ s−1) for each wavelength and can reach petascale computation speed on a single photonic chip by using wavelength division multiplexing. Our results are challenging for conventional Turing–von Neumann machines, and they confirm the great potential of photonic neuromorphic processing towards peta-scale neuromorphic super-computing on a photonic chip.