# Questions tagged [exponential-sums]

The method of exponential sums is one of a few general methods enabling us to solve a wide range of miscellaneous problems from the theory of numbers and its applications. The strongest results have been obtained with the aid of this method. Therefore knowledge of the fundamentals of theory of exponential sums is necessary for studying modern number theory.

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### Integral over an exponential sum with squares

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### $L^p$ bound on a Weyl sum

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### Upper bound for an exponential sum involving characters of a finite field

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### Number of integer solutions to quadratic polynomial with integer coefficients

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### Estimating certain short Kloosterman sums

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### Converse of Gallagher identity

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### A finite alternating sum

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### An integral involving many exponential terms with quadratic exponents in the denominator

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### Way to express a number in its most compact sum of powers

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### Formula for exponential integral over a cone

**10**

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### A question regarding Bourgain's paper on $\Lambda(p)$-subsets

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### A good way to bound the following exponential sum over $\mathbb{Z}/q\mathbb{Z}$ involving linear forms?

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### Exponential sums with prime power modulus

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### Linear exponential sum with gcd

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### Upper bound for the integral over minor arcs of the exponential sum with prime omega function coefficients

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### Smoothed Weyl sum inequality

**7**

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### Conjecture about an exponential sum

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### Flow of zeros in the shifted exponential generating function?

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### Lower bound for some sums of roots of unity

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### Discrepancy bound of integer tensor product sequence?

**3**

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### Inequality for exponential sum in Dvoretzky 1972

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### Cancellation in this exponential sum?

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### The minimum of the maximum of a sequence of sinc functions

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### Coefficients $U_m(n,k)$ in the identity $n^{2m+1}=\sum\limits_{0\leq k \leq m}(-1)^{m-k}U_m(n,k)\cdot n^k$

**4**

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### Explicit numbers with square root cancellation in Weyl's exponential sum

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### Summing the infinite series $\sum_{k=0}^{\infty} \frac{x^k}{(k!)^2}$ [closed]

**3**

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### Proof of the identity $\left\lvert\sum_{j=0}^{p-1}e^{(2\pi i/p)(mj^2+nj)}\right\rvert=\sqrt p$ for $p$ odd prime

**2**

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### Uniform power-saving estimate for an exponential sum

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### Coefficients in the sum $\sum_{k=0}^{n-1}\sum_{j=0}^{m}A_{j,m}(n-k)^jk^j=n^{2m+1}, \ m=1,2,…$

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### Approximating $1_I$, $I\subset \lbrack 0,1\rbrack$, by trigonometric polynomials

**2**

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### Moments of certain exponential sum

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### Number of solutions for the inequality with square roots

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### Prerequisites to read Katz's Gauss Sums, Kloosterman Sums, and Monodromy Groups

**3**

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### How to estimate a mixed character sum $\sum_{h \in \mathbb{Z}/q \mathbb{Z}} \chi(f(h)) e(Ch/q)$?

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### An exponential sum like the Kloosterman sums

**5**

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### Does a Kloosterman sum composed with a rational function exhibit square root cancellation?

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### Does $\int_{0}^{\infty}e^{-xz}\sum_{n=0}^{\infty}a_{n}\frac{x^n}{n!}dx$ converge for $z>0$ with $a_{n} > n! $, for $ n>1$? [closed]

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### Exponential sum over polynomial values

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### How to evaluate this sum of roots of unity with condition to zero

**4**

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### On an observation which relates to the exponential sum $\sum_{n=1}^{[\sqrt{t/2\pi}]} n^{-\frac{1}{2}+it}$

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### Is this averaged exponential sum over primes small infinitely often?

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### An exponential sum estimate on small intervals

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### Upper bound for $ \sum_{p\leq x} a(p)p^{it} $

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### Bounding exponential sum of the form $\sum_{\mathbf{x} \in (\mathbb{Z}/q \mathbb{Z})^n } \chi_1(x_1)\cdots \chi_n (x_n) e(a F(\mathbf{x})/q)$

**2**

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### Hypotheses for exponent pairs

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### optimal estimate for generalized Kloosterman sum

**4**

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### estimate for a sum of products of Weil's sum

**11**

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### exponential sum over variety

**9**

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### Arguments of exponential sums

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**1**answer