Under the Generalized Density Hypothesis, the prime number theorem holds in shorter intervals than the classic bounds for arithmetic progressions with moduli up to log powers of x.
Huxley, Large values of Dirichlet polynomials, III, Acta Arithmetica, 26 (1975), 435--444
2 Pith papers cite this work. Polarity classification is still indexing.
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Proves asymptotic for the number of ternary Goldbach representations of large odd N with one prime bounded by U = N^{4/49} exp(log^{2/3+ε}N) unconditionally or log^{4+ε}N under GRH.
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Refinements for primes in short arithmetic progressions
Under the Generalized Density Hypothesis, the prime number theorem holds in shorter intervals than the classic bounds for arithmetic progressions with moduli up to log powers of x.
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On the Goldbach problem with restricted primes
Proves asymptotic for the number of ternary Goldbach representations of large odd N with one prime bounded by U = N^{4/49} exp(log^{2/3+ε}N) unconditionally or log^{4+ε}N under GRH.