Let $f$ be multiplicative, with $|f(p)|\le A$ at primes and $\sum_{n\le x}|f(n)|^2\le A^2x$ for every $x\ge1$. If $|\alpha-a/q|\le q^{-2}$, $(a,q)=1$, and $3\le R\le q\le N/R$, we prove \[ \sum_{n\le N}f(n)\operatorname{e}(n\alpha) \ll_A \frac{N}{\log N} +\frac{N}{\sqrt R}\sqrt{\log\log(3R)} \] with effective implied constants. Montgomery and Vaughan proved this with second term $NR^{-1/2}(\log R)^{3/2}$, and, for $1$-bounded functions, Bachman replaced it by $NR^{-1/2}\sqrt{\log R\log\log R}$. We remove the factor $\sqrt{\log R}$ from Bachman's second term while retaining the original coefficient hypotheses of Montgomery and Vaughan. A more precise estimate records the distance from a rational number. The proof combines the Brun-Titchmarsh inequality on short intervals with maximal Fourier estimates derived from the Carleson-Hunt theorem; the local bounds permit arbitrary prime-dependent prefixes. We also prove sharpness of the square-root displacement dependence.
We construct finite Heath-Brown-type identities for the fractional powers $\zeta(s)^{\pm a/b}$ of the Riemann zeta-function, for every reduced fraction $a/b$ with $0<a/b<1$, from Newton's binomial series in the algebra of arithmetic functions, and we use them to prove the Vinogradov-quality bound $\sum_{n \le x} d_{\pm a/b}(n)e(n\alpha) \ll_{a,b} ( x q^{-1/2} + x^{4/5} + x^{1/2} q^{1/2} ) (\log 2x)^{C}$, for some constant $C=C(a,b)>0$, whenever $|\alpha - r/q| \le 1/q^2$ with $(r, q) = 1$. The bound carries no $x^{\varepsilon}$ loss, and the same machinery gives the endpoint case of the M\"obius function with an absolute constant. As applications we determine the major-arc expansion of $S_z(x, \alpha) = \sum_{n \le x} d_z(n)e(n\alpha)$ to arbitrary logarithmic precision for real $0<|z|<1$. For rational $z \in (-1,1)$, we determine the order of magnitude of $\sup_{\alpha} |S_z(x, \alpha)|$ and prove an asymptotic formula for the moments $\int_0^1 |S_z(x, \alpha)|^{s}\, d\alpha$ for every fixed real $s>2$. The minor-arc analysis avoids the theory of $L$-functions entirely, and all constants are effective except those inherited from the Siegel-Walfisz theorem.
Let $\mu$ be the M\"obius function and $e(t)=e^{2\pi it}$. We prove that if $N\ge2$, $\alpha\in\mathbb{R}$, $(a,q)=1$, and $|\alpha-a/q|\le q^{-2}$, then \[\bigg|\sum_{n\le N}\mu^2(n)e(\alpha n)\bigg|\ll\left(\frac Nq+q\right)(\log 2N)^5, \] with an absolute implied constant, and we deduce the corresponding estimate on the minor arcs of the Hardy--Littlewood dissection throughout the range $Q\le N^{1/2}$. The estimates of Schlage-Puchta [SP] and of Tolev [T] have the same dependence on $q$ and $Q$ but carry a factor $N^{\varepsilon}$. The proof uses Heath-Brown's square sieve with sieving primes confined to an interval $(P,2P]$, where $P$ may be as small as a multiple of $\log N$; a finite Fej\'er majorant in place of a truncated Fourier series; and, after completion of the character sums, a count of representations that exploits the restriction on the primes in place of the divisor function.
Nicolas Robles, Alexandru Zaharescu, Dirk Zeindler· 0 citations
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