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Results tagged with exponential-sums
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user 37555
The method of exponential sums is one of a few general methods enabling us to solve a wide range of miscellaneous problems from the theory of numbers and its applications. The strongest results have been obtained with the aid of this method. Therefore knowledge of the fundamentals of theory of exponential sums is necessary for studying modern number theory.
2
votes
Accepted
Exponential sum estimates similar to the one for $\sum_p (\log p) e(p \alpha)$, but for diff...
Bruedern, Granville, Perelli, Vaughan and Wooley, (Philos. Trans. Roy. Soc. London Ser. A, 356 (1998) 739 - 761) dealt with the sequence of $k$-free integers. Bruedern (in: Analytic Number Theory …
4
votes
On an observation which relates to the exponential sum $\sum_{n=1}^{[\sqrt{t/2\pi}]} n^{-\fr...
The assertion is generally believed to be true. In fact, much more is conjectured: For every $\epsilon>0$ we have $\left|\zeta(\frac{1}{2}+it)-\sum_{n\leq t^\epsilon} n^{-\frac{1}{2}-it}\right|= \math …
1
vote
Occurrence of simultaneous small remainders?
This is not always true. Suppose that $c_1+c_2=c_0$. Then $pc_1+pc_2\equiv pc_0\pmod{n}$. Hence if $pc_1\bmod n$, $pc_2\bmod n$ are in $[0, n^r]$, then $pc_0\bmod n$ is in $[0, 2n^r]$, which falls out …