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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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