On sums of two $k$-th powers: a mean-square bound over short intervals
نویسندگان
چکیده
منابع مشابه
Mean Value of Hardy Sums over Short Intervals
The main purpose of this paper is to study the mean value properties of certain Hardy sums over short intervals by using the mean value theorems of the Dirichlet L-functions, and to give two interesting asymptotic formulae.
متن کاملSums and Differences of Three k-th Powers
If k ≥ 2 is a positive integer the number of representations of a positive integer N as either x1 + x k 2 = N or x k 1 − x2 = N , with integers x1 and x2, is finite. Moreover it is easily shown to be Oε(N), for any ε > 0. It is known that if k = 2 or 3 then the number of representations is unbounded as N varies, but it is conjectured that the number of representations is bounded for k ≥ 4. Inde...
متن کاملOn Sums of Integrals of Powers of the Zeta-function in Short Intervals
The modified Mellin transform Zk(s) = ∫ ∞ 1 |ζ( 1 2 + ix)|x dx (k ∈ N) is used to obtain estimates for
متن کاملOn Powers as Sums of Two Cubes
In a paper of Kraus, it is proved that x 3 + y 3 = z p for p 17 has only trivial primitive solutions, provided that p satisses a relatively mild and easily tested condition. In this article we prove that the primitive solutions of x 3 + y 3 = z p with p = 4; 5; 7; 11; 13, correspond to rational points on hyperelliptic curves with Jaco-bians of relatively small rank. Consequently, Chabauty metho...
متن کاملSums of Two Squares – Pair Correlation & Distribution in Short Intervals
In this work we show that based on a conjecture for the pair correlation of integers representable as sums of two squares, which was first suggested by Connors and Keating and reformulated here, the second moment of the distribution of the number of representable integers in short intervals is Poissonian, where “short” means of length comparable to the mean spacing between sums of two squares. ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Functiones et Approximatio Commentarii Mathematici
سال: 2000
ISSN: 0208-6573
DOI: 10.7169/facm/1538186688