ON THE NUMBER OF ZEROS OF THE FUNCTION ζ(s) ON "ALMOST ALL" SHORT INTERVALS OF THE CRITICAL LINE

1989 ◽  
Vol 32 (3) ◽  
pp. 475-499
Author(s):  
L V Kiselëva
Keyword(s):  
Critical Line ◽  
Short Intervals ◽  
Number Of Zeros ◽  
Almost All

scholarly journals Quantitative relations between short intervals and exceptional sets of cubic Waring-Goldbach problem

2017 ◽  
Vol 15 (1) ◽  
pp. 1517-1529
Author(s):  
Zhao Feng
Keyword(s):  
Short Intervals ◽  
Exceptional Sets ◽  
Goldbach Problem ◽  
Almost All

Abstract In this paper, we are able to prove that almost all integers n satisfying some necessary congruence conditions are the sum of j almost equal prime cubes with j = 7, 8, i.e., $\begin{array}{} N=p_1^3+ \ldots +p_j^3 \end{array} $ with $\begin{array}{} |p_i-(N/j)^{1/3}|\leq N^{1/3- \delta +\varepsilon} (1\leq i\leq j), \end{array} $ for some $\begin{array}{} 0 \lt \delta\leq\frac{1}{90}. \end{array} $ Furthermore, we give the quantitative relations between the length of short intervals and the size of exceptional sets.

Primes in short intervals

1997 ◽  
Vol 122 (2) ◽  
pp. 193-205 ◽  
Author(s):  
HONGZE LI
Keyword(s):  
Prime Number ◽  
Short Intervals ◽  
Almost All

In 1982, Glyn Harman [2] proved that for almost all n, the interval [n, n+n(1/10)+ε] contains a prime number. By this we mean that the set of n[les ]N for which the interval does not contain a prime has measure o(N) as n→+∞. It follows from Huxley's work [6] that if θ>1/6 then there will almost always be asymptotically nθ(log n)−1 primes in the interval [n, n+nθ]. In 1983, Glyn Harman [3] pointed that for almost all n, the interval [n, n+n(1/12)+ε] contains a prime number, and meantime Heath-Brown gave the outline of this result in [5]. The exponent was reduced to 1/13 by Jia [10], 2/27 by Li [12] and 1/14 by Jia [11], and meantime N. Watt [16] got the same result. In this paper we shall prove the following result.THEOREM. For almost all n, the intervalformula herecontains a prime number.

scholarly journals Primes in almost all short intervals

1998 ◽  
Vol 84 (3) ◽  
pp. 225-244 ◽  
Author(s):  
Alessandro Zaccagnini
Keyword(s):  
Short Intervals ◽  
Almost All

scholarly journals Moments of the Riemann zeta function on short intervals of the critical line

2021 ◽  
Vol 49 (6) ◽  
Author(s):  
Louis-Pierre Arguin ◽  
Frédéric Ouimet ◽  
Maksym Radziwiłł
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Critical Line ◽  
Short Intervals ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

scholarly journals AN ELECTROSTATIC DEPICTION OF THE VALIDITY OF THE RIEMANN HYPOTHESIS AND A FORMULA FOR THE NTH ZERO AT LARGE N

2013 ◽  
Vol 28 (30) ◽  
pp. 1350151 ◽  
Author(s):  
ANDRÉ LECLAIR
Keyword(s):  
Electric Field ◽  
Critical Line ◽  
Quantum Statistical Mechanics ◽  
Spatial Dimension ◽  
Transcendental Equation ◽  
Critical Strip ◽  
Number Of Zeros ◽  
Large N ◽  
Relativistic Gases ◽  
Counting Formula

We construct a vector field E from the real and imaginary parts of an entire function ξ(z) which arises in the quantum statistical mechanics of relativistic gases when the spatial dimension d is analytically continued into the complex z plane. This function is built from the Γ and Riemann ζ functions and is known to satisfy the functional identity ξ(z) = ξ(1-z). E satisfies the conditions for a static electric field. The structure of E in the critical strip is determined by its behavior near the Riemann zeros on the critical line ℜ(z) = 1/2, where each zero can be assigned a ⊕ or ⊖ parity, or vorticity, of a related pseudo-magnetic field. Using these properties, we show that a hypothetical Riemann zero that is off the critical line leads to a frustration of this "electric" field. We formulate this frustration more precisely in terms of the potential Φ satisfying E = -∇Φ and construct Φ explicitly. The main outcome of our analysis is a formula for the nth zero on the critical line for large n expressed as the solution of a simple transcendental equation. Riemann's counting formula for the number of zeros on the entire critical strip can be derived from this formula. Our result is much stronger than Riemann's counting formula, since it provides an estimate of the nth zero along the critical line. This provides a simple way to estimate very high zeros to very good accuracy, and we estimate the 10106-th one.

scholarly journals ON THE DISTRIBUTION OF -FREE NUMBERS AND NON-VANISHING FOURIER COEFFICIENTS OF CUSP FORMS

2012 ◽  
Vol 54 (2) ◽  
pp. 381-397 ◽  
Author(s):  
KAISA MATOMÄKI
Keyword(s):  
Fourier Coefficients ◽  
Cusp Forms ◽  
Short Intervals ◽  
Almost All

AbstractWe study properties of -free numbers, that is numbers that are not divisible by any member of a set . First we formulate the most-used procedure for finding them (in a given set of integers) as easy-to-apply propositions. Then we use the propositions to consider Diophantine properties of -free numbers and their distribution on almost all short intervals. Results on -free numbers have implications to non-vanishing Fourier coefficients of cusp forms, so this work also gives information about them.

Sign in / Sign up

Export Citation Format

Share Document