Results 61 to 70 of about 66,497 (169)

Fractional moments of L$L$‐functions and sums of two squares in short intervals

Mathematika, Volume 71, Issue 4, October 2025.
Abstract Let b(n)=1$b(n)=1$ if n$n$ is the sum of two perfect squares, and b(n)=0$b(n)=0$ otherwise. We study the variance of B(x)=∑n⩽xb(n)$B(x)=\sum _{n\leqslant x}b(n)$ in short intervals by relating the variance with the second moment of the generating function f(s)=∑n=1∞b(n)n−s$f(s)=\sum _{n=1}^{\infty } b(n)n^{-s}$ along Re(s)=1/2$\mathrm{Re}(s)=1/
Siegfred Baluyot, Steven M. Gonek
wiley   +1 more source

Counting primes with a given primitive root, uniformly

Mathematika, Volume 71, Issue 4, October 2025.
Abstract The celebrated Artin conjecture on primitive roots asserts that given any integer g$g$ that is neither −1$-1$ nor a perfect square, there is an explicit constant A(g)>0$A(g)>0$ such that the number Π(x;g)$\Pi (x;g)$ of primes p⩽x$p\leqslant x$ for which g$g$ is a primitive root is asymptotically A(g)π(x)$A(g)\pi (x)$ as x→∞$x\rightarrow \infty$
Kai (Steve) Fan, Paul Pollack
wiley   +1 more source

The Existence of Moving Spike Patterns in an Attractive Chemotaxis Model

Mathematical Methods in the Applied Sciences, Volume 48, Issue 14, Page 13505-13516, 30 September 2025.
ABSTRACT We prove rigorously the existence of moving spike patterns in an attractive chemotaxis model with small diffusion coefficient for the chemical. In the zero diffusion limit, ϵ→0$$ \epsilon \to 0 $$, we prove that the non‐monotone traveling wave solutions of the system with ϵ>0$$ \epsilon >0 $$ converge to those of the system with ϵ=0$$ \epsilon
Tong Li, Casey Stone
wiley   +1 more source

Eigenvalue Density, Li’s Positivity, and the Critical Strip [PDF]

We rewrite the zero-counting formula within the critical strip of the Riemann zeta function as a cumulative density distribution; this subsequently allows us to formally derive an integral expression for the Li coefficients associated with the Riemann xi-
He, Y., Jejjala, V., Minic, D.
core  

Riemann's Zeta Function: The alternating Xi-Function Xia(s)

, 2016
As well known, the study of Riemanns zeta function (s) involves the related entire function (s). A close relative of (s) is the alternating zeta function (s). Similar to (s), also (s) has a corresponding entire function _a (s). After establishing its definition and a related functional equation, formulas based on incomplete gamma ...
openaire   +2 more sources

A note on the Riemann ξ-function

Lietuvos matematikos rinkinys, 2000
We prove that for σ > 4.5, inf {ℜ(ξ'(σ + it)/ξ(σ + it) ) : - ∞ < t < ∞} = ξ'(σ)/ξ(σ).This improves the result obtained by Lagarias.  
openaire   +3 more sources

Zeros of derivatives of Riemann's ξ-function on the critical line

Journal of Number Theory, 1983
AbstractExplicit lower bounds for the proportion of zeros of the derivatives of Riemann's ξ-function on the critical line are given. In particular, it is shown that the proportion tends to one as the order of the derivative tends to infinity.
openaire   +4 more sources

Zeros of Jensen polynomials and asymptotics for the Riemann xi function

, 2020
Cormac O’Sullivan
openalex   +3 more sources

The integral representation for the Riemann Ξ-function

Journal of Number Theory, 1971
AbstractThe function Φ(u) in the integral representation for the Rieman Ξ-function is shown to be monotone decreasing for u ≥ 0, and an application made to show a function having a similar integral representation has no real zeros.
openaire   +2 more sources

Pair correlation of the zeros of the derivative of the Riemann $��$-function

, 2008
28 pages.
Farmer, David W., Gonek, Steven M.
openaire   +2 more sources