Monatshefte für Mathematik

, Volume 122, Issue 3, pp 265–273

# Some remarks on Vinogradov's mean value theorem and Tarry's problem

• Trevor D. Wooley
Article

## Abstract

LetW(k, 2) denote the, least numbers for which the system of equations$$\sum\nolimits_{i = 1}^s {x_i^j = } \sum\nolimits_{i = 1}^s {y_i^j (1 \leqslant j \leqslant k)}$$ has a solution with$$\sum\nolimits_{i = 1}^s {x_i^{k + 1} \ne } \sum\nolimits_{i = 1}^s {y_i^{k + 1} }$$. We show that for largek one hasW(k, 2)≦1/2k2(logk+loglogk+O(1)), and moreover that whenK is large, one hasW(k, 2)≦1/2k(k+1)+1 for at least one valuek in the interval [K, K3/4+ε]. We show also that the leasts for which the expected asymptotic formula holds for the number of solutions of the above system of equations, inside a box, satisfiessk2(logk+O(loglogk).

## 1991 Mathematics Subject Classification

11D72 11P55 11L07

## Key words

Vinogradov's mean value theorem Tarry's problem exponential sums Hardy-Littlewood method

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