Infinitely many sign changes of the Liouville function on 𝑋²+𝐷

2021 ◽  
Author(s):  
Anitha Srinivasan
Keyword(s):  
Liouville Function ◽  
Sign Changes

scholarly journals Sign Changes of the Liouville Function on Quadratics

2013 ◽  
Vol 56 (2) ◽  
pp. 251-257
Author(s):  
Peter Borwein ◽  
Stephen K. K. Choi ◽  
Himadri Ganguli
Keyword(s):  
Prime Number ◽  
Prime Number Theorem ◽  
Liouville Function ◽  
Sign Changes ◽  
Integer Coefficients

AbstractLet λ(n) denote the Liouville function. Complementary to the prime number theorem, Chowla conjectured thatfor any polynomial f (x) with integer coefficients which is not of form bg(x)2.

scholarly journals BETWEEN THE PROBLEMS OF PÓLYA AND TURÁN

2012 ◽  
Vol 93 (1-2) ◽  
pp. 157-171 ◽  
Author(s):  
MICHAEL J. MOSSINGHOFF ◽  
TIMOTHY S. TRUDGIAN
Keyword(s):  
Zeta Function ◽  
Computational Methods ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Real Parameter ◽  
Liouville Function ◽  
Sign Changes ◽  
General Family ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

AbstractWe investigate the behaviour of the function $L_{\alpha }(x) = \sum _{n\leq x}\lambda (n)/n^{\alpha }$, where $\lambda (n)$ is the Liouville function and $\alpha $ is a real parameter. The case where $\alpha =0$ was investigated by Pólya; the case $\alpha =1$, by Turán. The question of the existence of sign changes in both of these cases is related to the Riemann hypothesis. Using both analytic and computational methods, we investigate similar problems for the more general family $L_{\alpha }(x)$, where $0\leq \alpha \leq 1$, and their relationship to the Riemann hypothesis and other properties of the zeros of the Riemann zeta function. The case where $\alpha =1/2$is of particular interest.

scholarly journals Sign changes in sums of the Liouville function

2008 ◽  
Vol 77 (263) ◽  
pp. 1681-1694 ◽  
Author(s):  
Peter Borwein ◽  
Ron Ferguson ◽  
Michael J. Mossinghoff
Keyword(s):  
Liouville Function ◽  
Sign Changes

scholarly journals Sign changes in the prime number theorem

2021 ◽  
Author(s):  
Thomas Morrill ◽  
Dave Platt ◽  
Tim Trudgian
Keyword(s):  
Prime Number ◽  
Prime Number Theorem ◽  
Sign Changes

scholarly journals Sarnak’s conjecture for sequences of almost quadratic word growth

2020 ◽  
pp. 1-56
Author(s):  
REDMOND MCNAMARA
Keyword(s):  
Box Dimension ◽  
Multiplicative Functions ◽  
Logarithmic Density ◽  
Liouville Function ◽  
Sign Patterns

Abstract We prove the logarithmic Sarnak conjecture for sequences of subquadratic word growth. In particular, we show that the Liouville function has at least quadratically many sign patterns. We deduce the main theorem from a variant which bounds the correlations between multiplicative functions and sequences with subquadratically many words which occur with positive logarithmic density. This allows us to actually prove that our multiplicative functions do not locally correlate with sequences of subquadratic word growth. We also prove a conditional result which shows that if the ( $\kappa -1$ )-Fourier uniformity conjecture holds then the Liouville function does not correlate with sequences with $O(n^{t-\varepsilon })$ many words of length n where $t = \kappa (\kappa +1)/2$ . We prove a variant of the $1$ -Fourier uniformity conjecture where the frequencies are restricted to any set of box dimension less than $1$ .

scholarly journals Ordering risk bounds in factor models

2018 ◽  
Vol 6 (1) ◽  
pp. 259-287 ◽  
Author(s):  
Jonathan Ansari ◽  
Ludger Rüschendorf
Keyword(s):  
Risk Factor ◽  
Systemic Risk ◽  
Factor Models ◽  
Worst Case ◽  
Sign Changes ◽  
Risk Factor Models ◽  
Risk Bounds ◽  
Derivatives Of ◽  
Dependence Structures

AbstractConditionally comonotonic risk vectors have been proved in [4] to yield worst case dependence structures maximizing the risk of the portfolio sum in partially specified risk factor models. In this paper we investigate the question how risk bounds depend on the specification of the pairwise copulas of the risk components Xiwith the systemic risk factor. As basic toolwe introduce a new ordering based on sign changes of the derivatives of copulas. This together with discretization by n-grids and the theory of supermodular transfers allows us to derive concrete ordering criteria for the maximal risks.

scholarly journals On the number of sign changes in the remainder-term of the prime-ideal theorem

1988 ◽  
Vol 56 (1) ◽  
pp. 185-197 ◽  
Author(s):  
J. Kaczorowski ◽  
W. Staś
Keyword(s):  
Prime Ideal ◽  
Remainder Term ◽  
Sign Changes ◽  
Prime Ideal Theorem

scholarly journals Extremal behaviour of ± 1-valued completely multiplicative functions in function fields

2021 ◽  
Vol 56 (1) ◽  
pp. 79-94
Author(s):  
Nikola Lelas ◽  
Keyword(s):  
Rational Function ◽  
Finite Fields ◽  
Function Fields ◽  
Mean Value ◽  
Multiplicative Functions ◽  
Irreducible Polynomials ◽  
Liouville Function ◽  
The Mean

We investigate the classical Pólya and Turán conjectures in the context of rational function fields over finite fields 𝔽q. Related to these two conjectures we investigate the sign of truncations of Dirichlet L-functions at point s=1 corresponding to quadratic characters over 𝔽q[t], prove a variant of a theorem of Landau for arbitrary sets of monic, irreducible polynomials over 𝔽q[t] and calculate the mean value of certain variants of the Liouville function over 𝔽q[t].

Sign changes of Fourier coefficients of cusp forms supported on prime power indices

2014 ◽  
Vol 10 (08) ◽  
pp. 1921-1927 ◽  
Author(s):  
Winfried Kohnen ◽  
Yves Martin
Keyword(s):  
Positive Integer ◽  
Prime Power ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Power Indices ◽  
Hecke Operator ◽  
Sign Changes ◽  
Integral Weight ◽  
Hecke Eigenform ◽  
Almost All

Let f be an even integral weight, normalized, cuspidal Hecke eigenform over SL2(ℤ) with Fourier coefficients a(n). Let j be a positive integer. We prove that for almost all primes p the sequence (a(pjn))n≥0 has infinitely many sign changes. We also obtain a similar result for any cusp form with real Fourier coefficients that provide the characteristic polynomial of some generalized Hecke operator is irreducible over ℚ.

Sign changes of the functionS(t) on short intervals

2005 ◽  
Vol 69 (4) ◽  
pp. 719-731
Author(s):  
M A Korolev
Keyword(s):  
Short Intervals ◽  
Sign Changes
Sign in / Sign up

Export Citation Format

Share Document