scholarly journals On the mean value of a sum analogous to character sums over short intervals

2008 ◽  
Vol 58 (3) ◽  
pp. 651-668
Author(s):  
Ren Ganglian ◽  
Zhang Wenpeng
Keyword(s):  
Character Sums ◽  
Mean Value ◽  
Short Intervals ◽  
The Mean

scholarly journals Mean Value of the General Dedekind Sums over Interval \({[1,\frac{q}{p})}\)

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2079
Author(s):  
Lei Liu ◽  
Zhefeng Xu
Keyword(s):  
Character Sums ◽  
Mean Value ◽  
Dedekind Sums ◽  
Hybrid Mean Value ◽  
Asymptotic Formulae ◽  
The Mean

Let q>2 be a prime, p be a given prime with p<q. The main purpose of this paper is using transforms, the hybrid mean value of Dirichlet L-functions with character sums and the related properties of character sums to study the mean value of the general Dedekind sums over interval [1,qp), and give some interesting asymptotic formulae.

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).

Multiplicative Functions in Short Intervals

1987 ◽  
Vol 39 (3) ◽  
pp. 646-672 ◽  
Author(s):  
Adolf Hildebrand
Keyword(s):  
Number Theory ◽  
Mean Value ◽  
Considerable Progress ◽  
Central Problem ◽  
Partial Sums ◽  
Multiplicative Functions ◽  
Short Intervals ◽  
Probabilistic Number ◽  
The Mean ◽  
Image Position

A central problem in probabilistic number theory is to evaluate asymptotically the partial sumsof multiplicative functions f and, in particular, to find conditions for the existence of the “mean value”1.1In the last two decades considerable progress has been made on this problem, and the results obtained are very satisfactory.

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.

Bombieri-type theorems of nonlinear Weyl sums over primes in short intervals

2018 ◽  
Vol 14 (10) ◽  
pp. 2571-2581
Author(s):  
Yanjun Yao
Keyword(s):  
Mean Value ◽  
Short Intervals ◽  
The Mean

In this paper, we consider the mean-value estimate for nonlinear Weyl sums over primes in short intervals and establish the related Bombieri-type theorems. These results have applications in additive problems with prime variables in short intervals.

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