# The Riemann hypothesis and Hilbert's tenth problem

@article{Chowla1966TheRH,
title={The Riemann hypothesis and Hilbert's tenth problem},
journal={American Mathematical Monthly},
year={1966},
volume={73},
pages={906}
}
• S. Chowla
• Published 1 October 1966
• Mathematics
• American Mathematical Monthly
134 Citations
On the random Chowla conjecture
• Mathematics
• 2022
We show that for a Steinhaus random multiplicative function f : N → D and any polynomial P (x) ∈ Z[x] of degP ≥ 2 which is not of the form w(x+ c) for some w ∈ Z, c ∈ Q, we have 1 √ x ∑ n≤x f(P (n))
The Hardy–Littlewood–Chowla conjecture in the presence of a Siegel zero
• Mathematics
Journal of the London Mathematical Society
• 2022
Assuming that Siegel zeros exist, we prove a hybrid version of the Chowla and Hardy–Littlewood prime tuples conjectures. Thus, for an infinite sequence of natural numbers x, and any distinct integers
Sums of two squares are strongly biased towards quadratic residues
Chebyshev famously observed empirically that more often than not, there are more primes of the form 3 mod 4 up to x than of the form 1 mod 4. This was conﬁrmed theoretically much later by Rubinstein
Averages of the M\"obius function on shifted primes
It is a folklore conjecture that the M\"obius function exhibits cancellation on shifted primes; that is, $\sum_{p\le X}\mu(p+h) \ = \ o(\pi(X))$ as $X\to\infty$ for any fixed shift $h>0$. We prove
VALUE PATTERNS OF MULTIPLICATIVE FUNCTIONS AND RELATED SEQUENCES
• Mathematics
Forum of Mathematics, Sigma
• 2019
We study the existence of various sign and value patterns in sequences defined by multiplicative functions or related objects. For any set $A$ whose indicator function is ‘approximately
Correlation of arithmetic functions over $$\mathbb {F}_q[T]$$
• Mathematics
Mathematische Annalen
• 2019
For a fixed polynomial $\Delta$, we study the number of polynomials $f$ of degree $n$ over $\mathbb F_q$ such that $f$ and $f+\Delta$ are both irreducible, an $\mathbb F_q[T]$-analogue of the twin
THE LOGARITHMICALLY AVERAGED CHOWLA AND ELLIOTT CONJECTURES FOR TWO-POINT CORRELATIONS
• T. Tao
• Mathematics
Forum of Mathematics, Pi
• 2016
Let $\unicode[STIX]{x1D706}$ denote the Liouville function. The Chowla conjecture, in the two-point correlation case, asserts that \begin{eqnarray}\mathop{\sum }_{n\leqslant
An averaged form of Chowla's conjecture
• Mathematics
• 2015
Let $\lambda$ denote the Liouville function. A well known conjecture of Chowla asserts that for any distinct natural numbers $h_1,\dots,h_k$, one has $\sum_{1 \leq n \leq X} \lambda(n+h_1) \dotsm A dynamical point of view on the set of B-free integers • Mathematics • 2013 We extend the study of the square-free flow, recently introduced by Sarnak, to the more general context of B-free integers, that is to say integers with no factor in a given family B of pairwise On the Hardy–Littlewood–Chowla conjecture on average • Mathematics Forum of Mathematics, Sigma • 2022 Abstract There has been recent interest in a hybrid form of the celebrated conjectures of Hardy–Littlewood and of Chowla. We prove that for any$k,\ell \ge 1$and distinct integers$h_2,\ldots