A Brun-Titchmarsh inequality for weighted sums over prime numbers

We prove explicit upper bounds for weighted sums over prime numbers in arithmetic progressions with slowly varying weight functions. The results generalize the well-known Brun-Titchmarsh inequality.Comment: 11 pages, to appear in Acta Arithmetic

Estimating π(x)\pi(x) and related functions under partial RH assumptions

The aim of this paper is to give a direct interpretation of the validity of the Riemann hypothesis up to a certain height $T$ in terms of the prime-counting function $\pi(x)$. This is done by proving the well-known explicit Schoenfeld bound on the RH to hold as long as $4.92 \sqrt{x/\log(x)} \leq T$. Similar statements are proven for the Riemann prime-counting function and the Chebyshov functions $\psi(x)$ and $\vartheta(x)$. Apart from that, we also improve some of the existing bounds of Chebyshov type for the function $\psi(x)$.Comment: 16 pages, final version, to appear in Math. Com

LACE: A light-weight, causal model for enhancing coded speech through adaptive convolutions

Classical speech coding uses low-complexity postfilters with zero lookahead to enhance the quality of coded speech, but their effectiveness is limited by their simplicity. Deep Neural Networks (DNNs) can be much more effective, but require high complexity and model size, or added delay. We propose a DNN model that generates classical filter kernels on a per-frame basis with a model of just 300~K parameters and 100~MFLOPS complexity, which is a practical complexity for desktop or mobile device CPUs. The lack of added delay allows it to be integrated into the Opus codec, and we demonstrate that it enables effective wideband encoding for bitrates down to 6 kb/s.Comment: 5 pages, accepted at WASPAA 202

Untersuchung der Primzahlzählfunktion und verwandter Funktionen

Die Verteilung der Primzahlen ist bekanntlich eng mit den Nullstellen der Riemannschen Zetafunktion verbunden. Insbesondere hat die Riemannsche Vermutung, nach welcher alle nichttrivialen Nullstellen den Realteil 1/2 besitzen sollen, starke Auswirkungen auf die Größe des Restglieds im Primzahlsatz, welcher besagt, dass die Primzahlzählfunktion zum Integrallogarithmus asymptotisch äquivalent ist. Die Nullstellen der Riemannschen Zetafunktion auf der kritischen Gerade lassen sich numerisch approximieren, und die Riemannsche Vermutung lässt sich auf diesem Wege zumindest partiell verifizieren. Der Beitrag dieser Arbeit besteht in der Anwendung derartiger Resultate zur Untersuchung der Primzahlzählfunktion und verwandter Funktionen. Dazu gehören hypothetische Abschätzungen, die auf der partiellen Gültigkeit der Riemannschen Vermutung bis zu einer gegebenen Höhe basieren, sowie numerische Fragen der Berechnung und der effizienten Abschätzung der Primzahlzählfunktion.Investigation of the prime-counting function and related functions The distribution of prime numbers is closely related to the distribution of non-trivial zeros of the Riemann zeta function. In particular, the Riemann hypothesis, which states that all non-trivial zeros have real part 1/2, implies a very tight bound for the remainder term in the prime number theorem. The zeros of the Riemann zeta function on the critical line can be approximated numerically, which can be turned into a partial verification of the Riemann hypothesis. This thesis concerns the problem of using such information to gain knowledge about the distribution of prime numbers. This includes hypothetical bounds for the remainder in the prime number theorem, which assume the correctness of the Riemann hypothesis up to a certain height, as well as efficient numerical algorithms for calculating and bounding the prime counting function

Framewise WaveGAN: High Speed Adversarial Vocoder in Time Domain with Very Low Computational Complexity

GAN vocoders are currently one of the state-of-the-art methods for building high-quality neural waveform generative models. However, most of their architectures require dozens of billion floating-point operations per second (GFLOPS) to generate speech waveforms in samplewise manner. This makes GAN vocoders still challenging to run on normal CPUs without accelerators or parallel computers. In this work, we propose a new architecture for GAN vocoders that mainly depends on recurrent and fully-connected networks to directly generate the time domain signal in framewise manner. This results in considerable reduction of the computational cost and enables very fast generation on both GPUs and low-complexity CPUs. Experimental results show that our Framewise WaveGAN vocoder achieves significantly higher quality than auto-regressive maximum-likelihood vocoders such as LPCNet at a very low complexity of 1.2 GFLOPS. This makes GAN vocoders more practical on edge and low-power devices.Comment: Submitted to ICASSP 2023, demo: https://ahmed-fau.github.io/fwgan_demo

NoLACE: Improving Low-Complexity Speech Codec Enhancement Through Adaptive Temporal Shaping

Speech codec enhancement methods are designed to remove distortions added by speech codecs. While classical methods are very low in complexity and add zero delay, their effectiveness is rather limited. Compared to that, DNN-based methods deliver higher quality but they are typically high in complexity and/or require delay. The recently proposed Linear Adaptive Coding Enhancer (LACE) addresses this problem by combining DNNs with classical long-term/short-term postfiltering resulting in a causal low-complexity model. A short-coming of the LACE model is, however, that quality quickly saturates when the model size is scaled up. To mitigate this problem, we propose a novel adatpive temporal shaping module that adds high temporal resolution to the LACE model resulting in the Non-Linear Adaptive Coding Enhancer (NoLACE). We adapt NoLACE to enhance the Opus codec and show that NoLACE significantly outperforms both the Opus baseline and an enlarged LACE model at 6, 9 and 12 kb/s. We also show that LACE and NoLACE are well-behaved when used with an ASR system.Comment: submitted to ICASSP 202