scholarly journals Cotangent sums, quantum modular forms, and the generalized Riemann hypothesis

2018 ◽  
Vol 6 (1) ◽  
Author(s):  
John Lewis ◽  
Don Zagier
Keyword(s):  
Modular Forms ◽  
Riemann Hypothesis ◽  
Generalized Riemann Hypothesis ◽  
Quantum Modular Forms

Quadratic Non-Residues and Prime-Producing Polynomials

1989 ◽  
Vol 32 (4) ◽  
pp. 474-478 ◽  
Author(s):  
R. A. Mollin ◽  
H. C. Williams
Keyword(s):  
Riemann Hypothesis ◽  
Quadratic Polynomials ◽  
Generalized Riemann Hypothesis ◽  
Initial Values ◽  
Positive Discriminant

AbstractWe will be looking at quadratic polynomials having positive discriminant and having a long string of primes as initial values. We find conditions tantamount to this phenomenon involving another long string of primes for which the discriminant of the polynomial is a quadratic non-residue. Using the generalized Riemann hypothesis (GRH) we are able to determine all discriminants satisfying this connection.

scholarly journals Variant of the truncated Perron formula and primes in polynomial sets

2019 ◽  
Vol 16 (02) ◽  
pp. 309-323
Author(s):  
D. S. Ramana ◽  
O. Ramaré
Keyword(s):  
Dirichlet Series ◽  
Riemann Hypothesis ◽  
Partial Sums ◽  
Generalized Riemann Hypothesis ◽  
Perron’S Formula ◽  
Polynomial Formula

We show under the Generalized Riemann Hypothesis that for every non-constant integer-valued polynomial [Formula: see text], for every [Formula: see text], and almost every prime [Formula: see text] in [Formula: see text], the number of primes from the interval [Formula: see text] that are values of [Formula: see text] modulo [Formula: see text] is the expected one, provided [Formula: see text] is not more than [Formula: see text]. We obtain this via a variant of the classical truncated Perron’s formula for the partial sums of the coefficients of a Dirichlet series.

scholarly journals Quantum modular forms and plumbing graphs of 3-manifolds

2020 ◽  
Vol 170 ◽  
pp. 105145 ◽  
Author(s):  
Kathrin Bringmann ◽  
Karl Mahlburg ◽  
Antun Milas
Keyword(s):  
Modular Forms ◽  
Quantum Modular Forms

scholarly journals The André–Oort conjecture for Drinfeld modular varieties

2013 ◽  
Vol 149 (4) ◽  
pp. 507-567 ◽  
Author(s):  
Patrik Hubschmid
Keyword(s):  
Riemann Hypothesis ◽  
Classical Case ◽  
Base Field ◽  
Generalized Riemann Hypothesis ◽  
Fixed Set ◽  
Special Points ◽  
Modular Varieties

AbstractWe consider the analogue of the André–Oort conjecture for Drinfeld modular varieties which was formulated by Breuer. We prove this analogue for special points with separable reflex field over the base field by adapting methods which were used by Klingler and Yafaev to prove the André–Oort conjecture under the generalized Riemann hypothesis in the classical case. Our result extends results of Breuer showing the correctness of the analogue for special points lying in a curve and for special points having a certain behaviour at a fixed set of primes.

The distribution of quadratic residues and non-residues in the Goldwasser–Micali type of cryptosystem

2014 ◽  
Vol 8 (2) ◽  
Author(s):  
Benjamin Justus
Keyword(s):  
Riemann Hypothesis ◽  
Generalized Riemann Hypothesis ◽  
Quadratic Residues

Abstract.We provide unconditional results and conditional ones under the assumption of GRH (Generalized Riemann Hypothesis) on the distribution of quadratic residues and quadratic non-residues in

scholarly journals “Strange” combinatorial quantum modular forms

2017 ◽  
Vol 170 ◽  
pp. 315-346 ◽  
Author(s):  
Amanda Folsom ◽  
Caleb Ki ◽  
Yen Nhi Truong Vu ◽  
Bowen Yang
Keyword(s):  
Modular Forms ◽  
Quantum Modular Forms
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