1,720,995 research outputs found
On the minimal number of small elements generating finite prime fields
No full textDOI:
10.1017/S000497271700048
Shifted moments of l-functions and moments of theta functions
No full textAssuming the Riemann Hypothesis, Soundararajan [Ann. of Math.a (2) 170 (2009), 981-993] showed that fT0 l.1=2 C it/2κ T .log T /κk2+€ His method was used by Chandee [Q.A J. Math. 62 (2011), 545-572] to obtain upper bounds for shifted moments of the Riemann Zeta function. Building on these ideas of Chandee and Soundararajan, we obtain, conditionally, upper bounds for shifted moments of Dirichlet-functions which allow us to derive upper bounds for moments of theta functions
Character sums over squarefree and squarefull numbers
No full textWe give upper bounds for character sums over squarefree and squarefull numbers sharper than the prior known in the literature. As an application, we study the distribution of squarefull numbers in arithmetic progressions and sharpen (without restriction on the modulus) the recent results obtained by Liu and Zhang. © 2014 Springer Basel
Histoire sommaire des hospices civils de Versailles
No full textMunsch M. Histoire sommaire des hospices civils de Versailles. In: Revue d'histoire de la pharmacie, 26ᵉ année, n°104, 1938. pp. 436-441
Polynomial products modulo primes and applications
No full textFor any polynomial P(x) ∈ Z[x] , we study arithmetic dynamical systems generated by FP(n)=∏k≤nP(k)(modp),n≥ 1. We apply this to improve the lower bound on the number of distinct quadratic fields of the form Q(FP(n)) in short intervals M≤ n≤ M+ H previously due to Cilleruelo, Luca, Quirós and Shparlinski. As a second application, we estimate the average number of missing values of FP(n)(modp) for special families of polynomials, generalizing previous work of Banks, Garaev, Luca, Schinzel, Shparlinski and others
Distribution of factorials modulo p
No full textWe estimate the average number of residue classes missed by the sequence n! (mod p) for p ≤ x
Second Moment of Dirichlet L-Functions, Character Sums over Subgroups and Upper Bounds on Relative Class Numbers
No full textWe prove an asymptotic formula for the mean-square average of L-functions associated with subgroups of characters of sufficiently large size. Our proof relies on the study of certain character sums recently introduced by E. Elma, where p ≥ 3 is prime and d ≥ 1 is any odd divisor of p - 1. We obtain an asymptotic formula for which holds true for any odd divisor d of p - 1, thus removing E. Elma's restrictions on the size of d. This answers a question raised in Elma's paper. Our proof relies on both estimates on the frequency of large character sums and techniques from the theory of uniform distribution. As an application, in the range , we obtain a significant improvement over the trivial bound on the relative class numbers of the imaginary number fields of conductor and degree , where d ≥ 1 is odd
The second and fourth moments of theta functions at their central point
No full textLet χ range over the (p - 1)/2 even Dirichlet characters mod p ≥ 3, a prime. Let θ(x, χ) be the associated theta series. It is known that the square mean value of θ(1, χ) is asymptotic to p3/2/42 as p goes to infinity. We prove that the fourth mean value of θ(1, χ) is asymptotic to 3/16πp2logp as p goes to infinity. We give similar results for mean values of odd Dirichlet characters mod p. © 2012 Elsevier Inc
Upper and Lower Bounds for Higher Moments of Theta Functions
No full textWe obtain optimal lower bounds for moments of theta functions. On the other hand, we also get new upper bounds on individual theta values and moments of theta functions on average over primes. The upper bounds are based on bounds of character sums and, in particular, on a modification of some recent results of M. Z. Garaev
On smooth square-free numbers in arithmetic progressions
No full textBooker and Pomerance [Proc. Amer. Math. Soc. 145 (2017) 5035–5042] have shown that any residue class modulo a prime (Formula presented.) can be represented by a positive (Formula presented.) -smooth square-free integer (Formula presented.) with all prime factors up to (Formula presented.) and conjectured that in fact one can find such (Formula presented.) with (Formula presented.). Using bounds on double Kloosterman sums due to Garaev [Mat. Zametki 88 (2010) 365–373] we prove this conjecture in a stronger form (Formula presented.) and also consider more general versions of this question replacing (Formula presented.) -smoothness of (Formula presented.) by the stronger condition of (Formula presented.) -smoothness. Using bounds on multiplicative character sums and a sieve method, we also show that we can represent all residue classes by a positive square-free integer (Formula presented.) which is (Formula presented.) -smooth. Additionally, we obtain stronger results for almost all primes (Formula presented.)
- …
