WATT'S MEAN VALUE THEOREM AND CARMICHAEL NUMBERS

2008 ◽  
Vol 04 (02) ◽  
pp. 241-248 ◽  
Author(s):  
GLYN HARMAN
Keyword(s):  
Recent Work ◽  
Arithmetic Progression ◽  
Character Sums ◽  
Mean Value ◽  
Original Version ◽  
Mean Value Theorem ◽  
Carmichael Numbers ◽  
Short Intervals

It is shown that Watt's new mean value theorem on sums of character sums can be included in the method described in the author's recent work [6] to show that the number of Carmichael numbers up to x exceeds x⅓ for all large x. This is done by comparing the application of Watt's original version of his mean value theorem [8] to the problem of primes in short intervals [3] with the problem of finding "small" primes in an arithmetic progression.

scholarly journals A new proof of Halász’s theorem, and its consequences

2018 ◽  
Vol 155 (1) ◽  
pp. 126-163 ◽  
Author(s):  
Andrew Granville ◽  
Adam J. Harper ◽  
K. Soundararajan
Keyword(s):  
Current Literature ◽  
Arithmetic Progression ◽  
Upper Bound ◽  
Multiplicative Function ◽  
Mean Value ◽  
Arithmetic Progressions ◽  
Short Intervals ◽  
The Mean

Halász’s theorem gives an upper bound for the mean value of a multiplicative function$f$. The bound is sharp for general such$f$, and, in particular, it implies that a multiplicative function with$|f(n)|\leqslant 1$has either mean value$0$, or is ‘close to’$n^{it}$for some fixed$t$. The proofs in the current literature have certain features that are difficult to motivate and which are not particularly flexible. In this article we supply a different, more flexible, proof, which indicates how one might obtain asymptotics, and can be modified to treat short intervals and arithmetic progressions. We use these results to obtain new, arguably simpler, proofs that there are always primes in short intervals (Hoheisel’s theorem), and that there are always primes near to the start of an arithmetic progression (Linnik’s theorem).

scholarly journals A mean-value theorem for character sums.

1992 ◽  
Vol 39 (1) ◽  
pp. 153-159 ◽  
Author(s):  
J. Friedlander ◽  
H. Iwaniec
Keyword(s):  
Character Sums ◽  
Mean Value ◽  
Mean Value Theorem

scholarly journals INCOMPLETE KLOOSTERMAN SUMS AND MULTIPLICATIVE INVERSES IN SHORT INTERVALS

2012 ◽  
Vol 09 (02) ◽  
pp. 481-486 ◽  
Author(s):  
T. D. BROWNING ◽  
A. HAYNES
Keyword(s):  
Mean Value ◽  
Kloosterman Sums ◽  
Mean Value Theorem ◽  
Short Intervals ◽  
Mod P

We investigate the solubility of the congruence xy ≡ 1 ( mod p), where p is a prime and x, y are restricted to lie in suitable short intervals. Our work relies on a mean value theorem for incomplete Kloosterman sums.

The mean value theorem of the divisor problem for short intervals

1998 ◽  
Vol 71 (6) ◽  
pp. 445-453 ◽  
Author(s):  
Isao Kiuchi ◽  
Yoshio Tanigawa
Keyword(s):  
Mean Value ◽  
Divisor Problem ◽  
Mean Value Theorem ◽  
Short Intervals ◽  
The Mean

scholarly journals On Riemann—Liouville and Caputo Fractional Forward Difference Monotonicity Analysis

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1303
Author(s):  
Pshtiwan Othman Mohammed ◽  
Thabet Abdeljawad ◽  
Faraidun Kadir Hamasalh
Keyword(s):  
Time Scale ◽  
Initial Value Problems ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Discrete Fractional Calculus ◽  
Initial Value ◽  
Forward Difference ◽  
The Mean ◽  
Monotonicity Analysis ◽  
Monotonicity Results

Monotonicity analysis of delta fractional sums and differences of order υ∈(0,1] on the time scale hZ are presented in this study. For this analysis, two models of discrete fractional calculus, Riemann–Liouville and Caputo, are considered. There is a relationship between the delta Riemann–Liouville fractional h-difference and delta Caputo fractional h-differences, which we find in this study. Therefore, after we solve one, we can apply the same method to the other one due to their correlation. We show that y(z) is υ-increasing on Ma+υh,h, where the delta Riemann–Liouville fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to zero, and then, we can show that y(z) is υ-increasing on Ma+υh,h, where the delta Caputo fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to −1Γ(1−υ)(z−(a+υh))h(−υ)y(a+υh) for each z∈Ma+h,h. Conversely, if y(a+υh) is greater or equal to zero and y(z) is increasing on Ma+υh,h, we show that the delta Riemann–Liouville fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to zero, and consequently, we can show that the delta Caputo fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to −1Γ(1−υ)(z−(a+υh))h(−υ)y(a+υh) on Ma,h. Furthermore, we consider some related results for strictly increasing, decreasing, and strictly decreasing cases. Finally, the fractional forward difference initial value problems and their solutions are investigated to test the mean value theorem on the time scale hZ utilizing the monotonicity results.

scholarly journals On a hybrid version of the Vinogradov mean value theorem

2021 ◽  
Vol 163 (1) ◽  
pp. 1-17
Author(s):  
C. Chen ◽  
I. E. Shparlinski
Keyword(s):  
Mean Value ◽  
Mean Value Theorem

scholarly journals Arithmetic of higher-dimensional orbifolds and a mixed Waring problem

2021 ◽  
Author(s):  
Tim Browning ◽  
Shuntaro Yamagishi
Keyword(s):  
Circle Method ◽  
Rational Points ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Waring’S Problem ◽  
Waring Problem ◽  
Waring's Problem ◽  
Main Conjecture ◽  
Higher Dimensional ◽  
Thin Set

AbstractWe study the density of rational points on a higher-dimensional orbifold $$(\mathbb {P}^{n-1},\Delta )$$ ( P n - 1 , Δ ) when $$\Delta $$ Δ is a $$\mathbb {Q}$$ Q -divisor involving hyperplanes. This allows us to address a question of Tanimoto about whether the set of rational points on such an orbifold constitutes a thin set. Our approach relies on the Hardy–Littlewood circle method to first study an asymptotic version of Waring’s problem for mixed powers. In doing so we make crucial use of the recent resolution of the main conjecture in Vinogradov’s mean value theorem, due to Bourgain–Demeter–Guth and Wooley.

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