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Arithmetic problems in the theory of Dirichlet characters. (English. Russian original) Zbl 1230.11099
Russ. Math. Surv. 63, No. 4, 641-690 (2008); translation from Usp. Mat. Nauk 63, No. 4, 43-92 (2008).
The author’s abstract states: “A survey of results related to the distribution of values of Dirichlet characters in sparse sets of positive integers is presented.”
To be more detailed: The introduction gives fundamental properties of Dirichlet characters, the Vinogradov–Pólya estimate $$|\sum_{x\leq X} \chi(x)| \leq \sqrt{q}\cdot \log q$$ for [non-principal] $$\chi \bmod q$$, the Hasse–Weil estimate $$|\sum_{x\leq q} \chi(f(x))| \leq (n-1) \sqrt{q}$$ for polynomials of degree $$n$$, results of I. M. Vinogradov on character sums $$\sum_{p\leq X} \chi(p+a)$$ extended over primes, the estimates of Burgess (1957) and the author (1968), …, finally results on character sums over sparse sets, for example the Erdös–Shapiro theorem.
Chapter 1 gives estimates for complete character sums (Hasse–Weil, and, for example, the author’s estimate (1978)) $\sum_{1\leq \lambda\leq q}\sum_{1\leq\mu\leq q}\left| \sum_{1\leq x\leq X} \chi_1(\lambda + \mu\, x +a) \chi_2(\lambda + \mu\, x + b)\right|^{2r} \leq r^{2r} X^r q^2 + 6r^2 x^{2r}q,$ and Vinogradov’s sieve, which enables one to reduce a sum over primes to multiple sums).
Chapter 2 deals with sums over successive primes (for example a result of the author, 1970) $\left|\sum_{p\leq X} \chi(p+a) \right| \ll_\omega X\, q^{-\frac{\omega^2}{1024}},$ where $$0<\omega<\frac12, \;(a,q)=1$$, and $$q^{\frac12+\omega}\leq X\leq q^2$$), next with sums over primes in arithmetical progressions, e.g. (the author, 1971) $\left| \sum_{{p\leq X}\atop {p \equiv l \bmod Q}}\chi(p+a)\right| \ll \frac1Q\, X \cdot \Delta, \text{ where } \Delta= q^{-0.0004\omega^2},$ and the distribution of products of shifted primes in short intervals, for example (the author, 1970) $\#\{p\leq n_1,\, p^\prime\leq n_2,\; p(p^\prime +a)\equiv l \bmod q\} = \frac{\pi(n_1) \pi(n_2)}{q-1} + \mathcal O\Bigl( (n_1n_2)^{1+\epsilon} q^{-1-\gamma \omega^2}\Bigr)$ for a suitable $$\gamma$$) [in this review the exact assumptions needed in the results are not stated explicitly].
Chapter 3 estimates non-linear character sums over successive primes (the author, 1978), $\left| \sum_{p\leq X} \chi\Bigl( (p+a)(p+b) \Bigr) \right| \ll \pi(X) \, q^{-0.02\omega^2}$ for $$X\geq q^{\frac34 + \omega},$$ and sums over primes in arithmetical progressions (Karatsuba 1987), $\left| \sum_{{p\leq X}\atop {p\equiv l\bmod Q}} \chi\Bigl( (p+a)(p+b) \Bigr) \right| \ll \frac1Q \, X\, q^{-0.02 \omega^2},$ for $$q^3 \, q^{\frac34 + \omega}\leq X \leq Q^3 q^B$$. This result has consequences, for example, for the number of primes $$p\leq X,\; p\equiv l \bmod Q$$, such that $$p+a$$ and $$p+b$$ are both primitive roots modulo $$q$$.
Chapter 4 deals with weighted character sums:
The sum $$T(X)=\sum_{n\leq X} \tau_k(n) \chi(n+l)$$, where $$l\not\equiv 0 \bmod q$$, $$(l,q)=1$$ is estimated (for $$0<\omega \leq\frac14$$, $$q^{\frac12+\omega}\leq X\leq q^{10}$$) by $|T(X)| \ll X\, q^{-0.011 \omega^2} \cdot\exp\left( \frac{8k\log X}{\log\log X}\right).$ Next the sum $$\sum_{n\leq X} \tau_k(n) \left( \frac{(n+a)(n+b)}q\right)$$ (with the Legendre-symbol) is estimated similarly. Finally sums like $$\sum_{n\leq X} \tau_k(n) \cdot \chi\Bigl((n+a)(n+b)\Bigr)$$ or $$\sum_{x^2+y^2 \leq X} \chi_1(x^2+y^2+a)\, \chi_2(x^2+y^2+b)$$ are estimated.
Chapter 5 gives [non-trivial upper] estimates for character-sums like $$\sum_{p\leq N}\sum_{p^\prime \leq M} \chi(p+p^\prime)$$, and more general for sums $W_1 = \sum_{x\in X}\sum_{y\in Y} \psi_1(x)\, \psi_2(y) \cdot \chi(x+y+a),$ where $$\psi_1$$ and $$\psi_2$$ are arbitrary bounded complex-valued functions.
The last chapter 6 presents problems of the theory of characters, unsolved problems and conjectures.
The bibliography lists 99 papers.

##### MSC:
 11L40 Estimates on character sums
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