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Let $t:{\mathbb F_p} \to C$ be a complex valued function on ${\mathbb F_p}$. A classical problem in analytic number theory is bounding the maximum
$$M(t): = \mathop {\max }\limits_{0 \le H < p} \left| {{1 \over {\sqrt p }}\sum\limits_{0 \le n < H} {t(n)} } \right|$$
of the absolute value of the incomplete sums $(1/\sqrt p )\sum\nolimits_{0 \le n < H} {t(n)} $. In this very general context one of the most important results is the Pólya–Vinogradov bound
where $\hat t:{\mathbb F_p} \to \mathbb C$ is the normalized Fourier transform of t. In this paper we provide a lower bound for certain incomplete Kloosterman sums, namely we prove that for any $\varepsilon > 0$ there exists a large subset of $a \in \mathbb F_p^ \times $ such that for $${\rm{k}}{1_{a,1,p}}:x \mapsto e((ax + \bar x)/p)$$ we have
For any prime
p, let
$y(p)$
denote the smallest integer y such that every reduced residue class (mod p) is represented by the product of some subset of
$\{1,\dots ,y\}$
. It is easy to see that
$y(p)$
is at least as large as the smallest quadratic nonresidue (mod p); we prove that
$y(p) \ll _\varepsilon p^{1/(4 \sqrt e)+\varepsilon }$
, thus strengthening Burgess’ classical result. This result is of intermediate strength between two other results, namely Burthe’s proof that the multiplicative group (mod p) is generated by the integers up to
$O_\varepsilon (p^{1/(4 \sqrt e)+\varepsilon })$
, and Munsch and Shparlinski’s result that every reduced residue class (mod p) is represented by the product of some subset of the primes up to
$O_\varepsilon (p^{1/(4 \sqrt e)+\varepsilon })$
. Unlike the latter result, our proof is elementary and similar in structure to Burgess’ proof for the least quadratic nonresidue.
In this paper, we obtain a variation of the Pólya–Vinogradov inequality with the sum restricted to a certain height. Assume $\chi $ to be a primitive character modulo q, $ \epsilon>0$ and $N\le q^{1-\gamma }$, with $0\le \gamma \le 1/3$. We prove that
In this paper, we prove a one level density result for the low-lying zeros of quadratic Hecke L-functions of imaginary quadratic number fields of class number 1. As a corollary, we deduce, essentially, that at least
$(19-\cot (1/4))/16 = 94.27\ldots \%$
of the L-functions under consideration do not vanish at 1/2.
We consider autocorrelation functions for supersymmetric quantum mechanical systems (consisting of a fermion and a boson) confined in trigonometric Pöschl–Teller partner potentials. We study the limit of rescaled autocorrelation functions (at random time) as the localization of the initial state goes to infinity. The limiting distribution can be described using pairs of Jacobi theta functions on a suitably defined homogeneous space, as a corollary of the work of Cellarosi and Marklof. A construction by Contreras-Astorga and Fernández provides large classes of Pöschl-Teller partner potentials to which our analysis applies.
We prove effective equidistribution of primitive rational points and of primitive rational points defined by monomials along long horocycle orbits in products of the torus and the modular surface. This answers a question posed in joint work by the first and the last named author with Shahar Mozes and Uri Shapira. Under certain congruence conditions we prove the joint equidistribution of conjugate rational points in the 2-torus and the modular surface.
Recently E. Bombieri and N. M. Katz (2010) demonstrated that several well-known results about the distribution of values of linear recurrence sequences lead to interesting statements for Frobenius traces of algebraic curves. Here we continue this line of study and establish the Möbius randomness law quantitatively for the normalised form of Frobenius traces.
We prove that sums of length about $q^{3/2}$ of Hecke eigenvalues of automorphic forms on $\operatorname{SL}_{3}(\mathbf{Z})$ do not correlate with $q$-periodic functions with bounded Fourier transform. This generalizes the earlier results of Munshi and Holowinsky–Nelson, corresponding to multiplicative Dirichlet characters, and applies, in particular, to trace functions of small conductor modulo primes.
We determine the order of magnitude of $\mathbb{E}|\sum _{n\leqslant x}f(n)|^{2q}$, where $f(n)$ is a Steinhaus or Rademacher random multiplicative function, and $0\leqslant q\leqslant 1$. In the Steinhaus case, this is equivalent to determining the order of $\lim _{T\rightarrow \infty }\frac{1}{T}\int _{0}^{T}|\sum _{n\leqslant x}n^{-it}|^{2q}\,dt$.
In particular, we find that $\mathbb{E}|\sum _{n\leqslant x}f(n)|\asymp \sqrt{x}/(\log \log x)^{1/4}$. This proves a conjecture of Helson that one should have better than squareroot cancellation in the first moment and disproves counter-conjectures of various other authors. We deduce some consequences for the distribution and large deviations of $\sum _{n\leqslant x}f(n)$.
The proofs develop a connection between $\mathbb{E}|\sum _{n\leqslant x}f(n)|^{2q}$ and the $q$th moment of a critical, approximately Gaussian, multiplicative chaos and then establish the required estimates for that. We include some general introductory discussion about critical multiplicative chaos to help readers unfamiliar with that area.
In this paper we prove some one-level density results for the low-lying zeros of families of quadratic and quartic Hecke $L$-functions of the Gaussian field. As corollaries, we deduce that at least 94.27% and 5%, respectively, of the members of the quadratic family and the quartic family do not vanish at the central point.
We study the analogue of the Bombieri–Vinogradov theorem for
$\operatorname{SL}_{m}(\mathbb{Z})$
Hecke–Maass form
$F(z)$
. In particular, for
$\operatorname{SL}_{2}(\mathbb{Z})$
holomorphic or Maass Hecke eigenforms, symmetric-square lifts of holomorphic Hecke eigenforms on
$\operatorname{SL}_{2}(\mathbb{Z})$
, and
$\operatorname{SL}_{3}(\mathbb{Z})$
Maass Hecke eigenforms under the Ramanujan conjecture, the levels of distribution are all equal to
$1/2,$
which is as strong as the Bombieri–Vinogradov theorem. As an application, we study an automorphic version of Titchmarch’s divisor problem; namely for
$a\neq 0,$
where
$\unicode[STIX]{x1D70C}(n)$
are Fourier coefficients
$\unicode[STIX]{x1D706}_{f}(n)$
of a holomorphic Hecke eigenform
$f$
for
$\operatorname{SL}_{2}(\mathbb{Z})$
or Fourier coefficients
$A_{F}(n,1)$
of its symmetric-square lift
$F$
. Further, as a consequence, we get an asymptotic formula
where
$E_{1}(a)$
is a constant depending on
$a$
. Moreover, we also consider the asymptotic orthogonality of the Möbius function against the arithmetic function
$\unicode[STIX]{x1D70C}(n)d(n-a)$
.
where $[x]$ denotes the integral part of real $x$. The above summations were recently considered by Bordellès et al. [‘On a sum involving the Euler function’, Preprint, 2018, arXiv:1808.00188] and Wu [‘On a sum involving the Euler totient function’, Preprint, 2018, hal-01884018].
By combining classical techniques together with two novel asymptotic identities derived in recent work by Lenells and one of the authors, we analyse certain single sums of Riemann-zeta type. In addition, we analyse Euler-Zagier double exponential sums for particular values of Re{u} and Re{v} and for a variety of sets of summation, as well as particular cases of Mordell-Tornheim double sums.
In this note we give a characterization of $\ell ^{p}\times \cdots \times \ell ^{p}\rightarrow \ell ^{q}$ boundedness of maximal operators associated with multilinear convolution averages over spheres in $\mathbb{Z}^{n}$.
We establish Diophantine inequalities for the fractional parts of generalized polynomials, in particular for sequences
$\unicode[STIX]{x1D708}(n)=\lfloor n^{c}\rfloor +n^{k}$
with
$c>1$
a non-integral real number and
$k\in \mathbb{N}$
, as well as for
$\unicode[STIX]{x1D708}(p)$
where
$p$
runs through all prime numbers. This is related to classical work of Heilbronn and to recent results of Bergelson et al.
We generalize the work of Sarnak and Tsimerman to twisted sums of Kloosterman sums and thus give evidence towards the twisted Linnik–Selberg conjecture.
We obtain a non-trivial bound for cancellations between the Kloosterman sums modulo a large prime power with a prime argument running over very short intervals, which in turn is based on a new estimate on bilinear sums of Kloosterman sums. These results are analogues of those obtained by various authors for Kloosterman sums modulo a prime. However, the underlying technique is different and allows us to obtain non-trivial results starting from much shorter ranges.