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On Goldbach numbers in short intervals
(Springer Science and Business Media Deutschland GmbH, 2026-09-01)
Assuming the Riemann Hypothesis, we prove that for all x⩾2, there exists at least one even integer within the interval (x,x+123log2x], that can be expressed as the sum of two primes. This result is an improvement over the recent work of Cully-Hugill and Dudek, who obtained the constant 9696 instead of 123.
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Conditional Estimates for the Logarithmic Derivative of Dirichlet L-Functions
(Elsevier BV, 2023-07-31)
Assuming the Generalized Riemann Hypothesis, we establish explicit bounds in the q-aspect for the logarithmic derivative L′/Lσ,χ of Dirichlet L-functions, where χ is a primitive character modulo q≥1030 and 1/2+1/loglogq≤σ≤1−1/loglogq. In addition, for σ=1 we improve upon the result by Ihara, Murty and Shimura (2009). Similar results for the logarithmic derivative of the Riemann zeta-function are given.

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