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Beyond the Riemann Hypothesis bounds: A pair-correlation approach to the least prime in arithmetic progression and the smallest quadratic non-residue
Neelam Kandhil, Alessandro Languasco, Pieter Moree
The Generalized Riemann Hypothesis (GRH) has long defined the expected bounds for the smallest prime in an arithmetic progression and the least quadratic non-residue. However, this…
Counting ideals in abelian number fields
Alessandro Languasco, Rashi Lunia, Pieter Moree
Already Dedekind and Weber considered the problem of counting integral ideals of norm at most in a given number field . Here we improve on the existing results in case $K/\m…
Euler-Kronecker constants of modular forms: beyond Dirichlet -series
Steven Charlton, Anna Medvedovsky, Pieter Moree
The Euler-Kronecker constants related to congruences of Fourier coefficients of modular forms that have been computed so far, involve logarithmic derivatives of Dirichlet -serie…
Pair Correlation of zeros of Dirichlet -Functions: A possible path towards the conjectures of Chowla, Elliott-Halberstam and Montgomery
Neelam Kandhil, Alessandro Languasco, Pieter Moree
Assuming the Generalized Riemann Hypothesis and a pair correlation conjecture for the zeros of Dirichlet -functions, we establish the truth of a conjecture of Montgomery (in its…
Relative class numbers and Euler-Kronecker constants of maximal real cyclotomic subfields
Neelam Kandhil, Alessandro Languasco, Pieter Moree +2
The Euler--Kronecker constant of a number field is the ratio of the constant and the residue of the Laurent series of the Dedekind zeta function at . We study the…
Euler constants from primes in arithmetic progression
Alessandro Languasco, Pieter Moree
Many Dirichlet series of number theoretic interest can be written as a product of generating series , with ranging…
The Kummer ratio of the relative class number for prime cyclotomic fields
Neelam Kandhil, Alessandro Languasco, Pieter Moree +2
Kummer's conjecture predicts the asymptotic growth of the relative class number of prime cyclotomic fields. We substantially improve the known bounds of Kummer's ratio under three…
The Brauer-Siegel ratio for prime cyclotomic fields
Neelam Kandhil, Alessandro Languasco, Pieter Moree
The Brauer-Siegel theorem concerns the size of the product of the class number and the regulator of a number field . We derive bounds for this product in case is a prime cyc…
On the discriminator of Lucas sequences. II
Matteo Ferrari, Florian Luca, Pieter Moree
The family of Shallit sequences consists of the Lucas sequences satisfying the recurrence with initial values and and w…
Sequences of integers generated by two fixed primes
Alessandro Languasco, Florian Luca, Pieter Moree +1
Let and be two distinct fixed prime numbers and the sequence of consecutive integers of the form with . Tijdeman gave a lower bou…
Values of the Euler phi-function not divisible by a given odd prime, and the distribution of Euler-Kronecker constants for cyclotomic fields
Kevin Ford, Florian Luca, Pieter Moree
For a fixed odd prime q we investigate the first and second order terms of the asymptotic series expansion for the number of n\le x such that q does not divide phi(n). Part of the…
Primes in a prescribed arithmetic progression dividing the sequence a^k+b^k
P. Moree, B. Sury
Given positive integers a,b,c and d such that c and d are coprime we show that the primes p=c(mod d)dividing a^k+b^k for some k>=1 have a natural density and explicitly compute thi…