The Estimation of Complete Exponential Sums

1985 ◽  
Vol 28 (4) ◽  
pp. 440-454 ◽  
Author(s):  
J. H. Loxton ◽  
R. C. Vaughan
Keyword(s):  
Exponential Sums ◽  
Exponential Sum ◽  
Integer Coefficients ◽  
Complex Zeros

AbstractThis paper proves a conjecture of Loxton and Smith about the size of the exponential sum S(f;q) formed by summing exp (2πif(x)/q) over x mod q, where f is a polynomial of degree n with integer coefficients. It is shown that |S(f;q)| ≤ Cfdn(q)qe/(e+1), where e is the maximum of the orders of the complex zeros of f'. An estimate is also obtained for Cf in terms of n, e and the different of f, and a number of examples are given to show that the estimate is best possible.

scholarly journals Linear Recurrences and Asymptotic Behavior of Exponential Sums of Symmetric Boolean Functions

10.37236/2004 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
Francis N. Castro ◽  
Luis A. Medina
Keyword(s):  
Asymptotic Behavior ◽  
Boolean Functions ◽  
Exponential Sums ◽  
Exponential Sum ◽  
Linear Recurrence ◽  
Symmetric Boolean Function ◽  
Linear Recurrences ◽  
Symmetric Boolean Functions ◽  
Integer Coefficients ◽  
Homogeneous Linear

In this paper we give an improvement of the degree of the homogeneous linear recurrence with integer coefficients that exponential sums of symmetric Boolean functions satisfy. This improvement is tight. We also compute the asymptotic behavior of symmetric Boolean functions and provide a formula that allows us to determine if a symmetric boolean function is asymptotically not balanced. In particular, when the degree of the symmetric function is a power of two, then the exponential sum is much smaller than $2^n$.

scholarly journals A note on two-term exponential sum and the reciprocal of the quartic Gauss sums

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Wenpeng Zhang ◽  
Xingxing Lv
Keyword(s):  
Exponential Sums ◽  
Exponential Sum ◽  
Gauss Sums ◽  
Power Mean ◽  
Hybrid Power ◽  
Analytic Methods ◽  
Hybrid Power Mean

AbstractThe main purpose of this article is by using the properties of the fourth character modulo a prime p and the analytic methods to study the calculating problem of a certain hybrid power mean involving the two-term exponential sums and the reciprocal of quartic Gauss sums, and to give some interesting calculating formulae of them.

A Multiple Exponential Sum to Modulus p2

1985 ◽  
Vol 28 (4) ◽  
pp. 394-396 ◽  
Author(s):  
D. R. Heath-Brown
Keyword(s):  
Algebraic Geometry ◽  
Exponential Sums ◽  
Exponential Sum ◽  
Total Degree ◽  
Image Position

AbstractFor suitable polynomials f(x) ∊ ℤ[x] in n variables, of total degree d, it is shown thatThis is, formally, a precise analogue of a theorem of Deligne [1] on exponential sums (mod p). However the proof uses no more than elementary algebraic geometry.

On a New Exponential Sum

2001 ◽  
Vol 44 (1) ◽  
pp. 87-92 ◽  
Author(s):  
Daniel Lieman ◽  
Igor Shparlinski
Keyword(s):  
Exponential Sums ◽  
Exponential Sum ◽  
Multiplicative Order

AbstractLet p be prime and let be of multiplicative order t modulo p. We consider exponential sums of the formand prove that for any ε > 0

On a variant of sum-product estimates and explicit exponential sum bounds in prime fields

2009 ◽  
Vol 146 (1) ◽  
pp. 1-21 ◽  
Author(s):  
J. BOURGAIN ◽  
M. Z. GARAEV
Keyword(s):  
Prime Order ◽  
Exponential Sums ◽  
Exponential Sum ◽  
Complex Numbers ◽  
Number Of Solutions ◽  
Prime Fields ◽  
Explicit Bounds ◽  
Multiplicative Subgroup ◽  
Image Position

AbstractLet Fp be the field of a prime order p and F*p be its multiplicative subgroup. In this paper we obtain a variant of sum-product estimates which in particular implies the bound for any subset A ⊂ Fp with 1 < |A| < p12/23. Then we apply our estimate to obtain explicit bounds for some exponential sums in Fp. We show that for any subsets X, Y, Z ⊂ F*p and any complex numbers αx, βy, γz with |αx| ≤ 1, |βy| ≤ 1, |γz| ≤ 1, the following bound holds: We apply this bound further to show that if H is a subgroup of F*p with |H| > p1/4, then Finally we show that if g is a generator of F*p then for any M < p the number of solutions of the equation is less than $M^{3-1/24+o(1)}\Bigl(1+(M^2/p)^{1/24}\Bigr).$. This implies that if p1/2 < M < p, then

Exponential sums involving q-digital functions

2019 ◽  
Vol 15 (06) ◽  
pp. 1143-1172
Author(s):  
Karam Aloui
Keyword(s):  
Real Number ◽  
Exponential Sums ◽  
Exponential Sum ◽  
Similar Estimate ◽  
Short Intervals ◽  
Sum Formula

We estimate the exponential sum [Formula: see text], where [Formula: see text] is a real number and [Formula: see text] are digital functions; in the spirit of the works of Kim and Berend–Kolesnik. A similar estimate along short intervals is also provided.

scholarly journals VINOGRADOV'S INTEGRAL AND BOUNDS FOR THE RIEMANN ZETA FUNCTION

2002 ◽  
Vol 85 (3) ◽  
pp. 565-633 ◽  
Author(s):  
KEVIN FORD
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Exponential Sums ◽  
Exponential Sum ◽  
Auxiliary Result ◽  
Mean Values ◽  
The Riemann Zeta Function ◽  
Secondary 11D72 ◽  
Explicit Bounds ◽  
Riemann Zeta

The main result is an upper bound for the Riemann zeta function in the critical strip: $\zeta(\sigma + it) \le A|t|^{B(1 - \sigma)^{3/2}} \log^{2/3} |t|$ with $A = 76.2$ and $B = 4.45$, valid for $\frac12 \le \sigma \le 1$ and $|t| \ge 3$. The previous best constant $B$ was 18.5. Tools include a variant of the Korobov–Vinogradov method of bounding exponential sums, an explicit version of T. D. Wooley's bounds for Vinogradov's integral, and explicit bounds for mean values of exponential sums over numbers without small prime factors, also using methods of Wooley. An auxiliary result is the exponential sum bound $S(N, t) \le 9.463 N^{1 - 1/(133.66\lambda^2)}$, where $N$ is a positive integer, $t$ is a real number, $\lambda = (\log t)/(\log N)$ and$S(N,t) = \max_{0 < u \le 1} \max_{N < R \le 2N} \left| \sum_{N < n \le R} (n + u)^{-it} \right|.$$2000 Mathematical Subject Classification: primary 11M06, 11N05, 11L15; secondary 11D72, 11M35.

scholarly journals On the fourth derivative test for exponential sums

2016 ◽  
Vol 28 (2) ◽  
Author(s):  
Olivier Robert
Keyword(s):  
Upper Bound ◽  
Exponential Sums ◽  
Exponential Sum ◽  
Fourth Derivative ◽  
Derivative Test

AbstractWe give an upper bound for the exponential sum ∑

MORE ON THE SUM-PRODUCT PHENOMENON IN PRIME FIELDS AND ITS APPLICATIONS

2005 ◽  
Vol 01 (01) ◽  
pp. 1-32 ◽  
Author(s):  
J. BOURGAIN
Keyword(s):  
Finite Fields ◽  
Exponential Sums ◽  
Rational Functions ◽  
Exponential Sum ◽  
Kloosterman Sums ◽  
Product Theorem ◽  
Short Intervals ◽  
Prime Fields ◽  
Product Sets ◽  
Mod P

In this paper we establish new estimates on sum-product sets and certain exponential sums in finite fields of prime order. Our first result is an extension of the sum-product theorem from [8] when sets of different sizes are involed. It is shown that if [Formula: see text] and pε < |B|, |C| < |A| < p1-ε, then |A + B| + |A · C| > pδ (ε)|A|. Next we exploit the Szemerédi–Trotter theorem in finite fields (also obtained in [8]) to derive several new facts on expanders and extractors. It is shown for instance that the function f(x,y) = x(x+y) from [Formula: see text] to [Formula: see text] satisfies |F(A,B)| > pβ for some β = β (α) > α whenever [Formula: see text] and |A| ~ |B|~ pα, 0 < α < 1. The exponential sum ∑x∈ A,y∈Bεp(axy+bx2y2), ab ≠ 0 ( mod p), may be estimated nontrivially for arbitrary sets [Formula: see text] satisfying |A|, |B| > pρ where ρ < 1/2 is some constant. From this, one obtains an explicit 2-source extractor (with exponential uniform distribution) if both sources have entropy ratio at last ρ. No such examples when ρ < 1/2 seemed known. These questions were largely motivated by recent works on pseudo-randomness such as [2] and [3]. Finally it is shown that if pε < |A| < p1-ε, then always |A + A|+|A-1 + A-1| > pδ(ε)|A|. This is the finite fields version of a problem considered in [11]. If A is an interval, there is a relation to estimates on incomplete Kloosterman sums. In the Appendix, we obtain an apparently new bound on bilinear Kloosterman sums over relatively short intervals (without the restrictions of Karatsuba's result [14]) which is of relevance to problems involving the divisor function (see [1]) and the distribution ( mod p) of certain rational functions on the primes (cf. [12]).

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