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We investigate various mean value problems involving order three primitive Dirichlet characters. In particular, we obtain an asymptotic formula for the first moment of central values of the Dirichlet L-functions associated to this family, with a power saving in the error term. We also obtain a large-sieve type result for order three (and six) Dirichlet characters.
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Fix a smooth cubic form $F/mathbb{Q}$ in $6$ variables. For $N_F(X):=#{boldsymbol{x}in[-X,X]^6:F(boldsymbol{x})=0}$, the randomness prediction $N_F(X)=(c_text{HL}+o(1))cdot X^3$ as $Xtoinfty$ of Hardy-Littlewood may fail. Nonetheless, Hooley suggested a modified prediction accounting for special structured loci on the projective variety $V:= V(F)subseteqmathbb{P}^5_mathbb{Q}$. A weighted version of $N_F(X)$ essentially decomposes as a sum of adelic data over hyperplane sections $V_{boldsymbol{c}}subseteq V$, generically with nonzero discriminant $D(boldsymbol{c})$. Assuming standard conjectures for the Hasse-Weil $L$-functions $L(s,V_{boldsymbol{c}})$ over ${boldsymbol{c}inmathbb{Z}^6:D(boldsymbol{c}) eq0}$, Hooley proved the bound $N_F(X) = O_epsilon(X^{3+epsilon})$, essentially for any given diagonal $F$. Now assume (1) standard conjectures for each $L(s,V_{boldsymbol{c}})$, for certain tensor $L$-functions thereof, and for $L(s,V)$; (2) standard predictions (of Random Matrix Theory type) for the mean values of $1/L(s)$ and $1/L(s_1)L(s_2)$ over certain geometric families; (3) a quantitative form of the Square-free Sieve Conjecture for $D$; and (4) an effective bound on the local variation (in $boldsymbol{c}$) of the local factors $L_p(s,V_{boldsymbol{c}})$, in the spirit of Krasners lemma. Under (1)-(4), we establish (away from the Hessian of $F$) a weighted, localized version of Hooleys prediction for diagonal $F$ -- and hence the Hasse principle for $V/mathbb{Q}$. Still under (1)-(4), we conclude that asymptotically $100%$ of integers $a otin{4,5}bmod{9}$ lie in ${x^3+y^3+z^3:x,y,zinmathbb{Z}}$ -- and a positive fraction lie in ${x^3+y^3+z^3:x,y,zinmathbb{Z}_{geq0}}$.
In this paper, we consider how to express an Iwahori--Whittaker function through Demazure characters. Under some interesting combinatorial conditions, we obtain an explicit formula and thereby a generalization of the Casselman--Shalika formula. Under the same conditions, we compute the transition matrix between two natural bases for the space of Iwahori fixed vectors of an induced representation of a p-adic group; this generalizes a result of Bump--Nakasuji.
For a finite group $G$, let $K(G)$ denote the field generated over $mathbb{Q}$ by its character values. For $n>24$, G. R. Robinson and J. G. Thompson proved that $$K(A_n)=mathbb{Q}left ({ sqrt{p^*} : pleq n {text{ an odd prime with } p eq n-2}}right),$$ where $p^*:=(-1)^{frac{p-1}{2}}p$. Confirming a speculation of Thompson, we show that arbitrary suitable multiquadratic fields are similarly generated by the values of $A_n$-characters restricted to elements whose orders are only divisible by ramified primes. To be more precise, we say that a $pi$-number is a positive integer whose prime factors belong to a set of odd primes $pi:= {p_1, p_2,dots, p_t}$. Let $K_{pi}(A_n)$ be the field generated by the values of $A_n$-characters for even permutations whose orders are $pi$-numbers. If $tgeq 2$, then we determine a constant $N_{pi}$ with the property that for all $n> N_{pi}$, we have $$K_{pi}(A_n)=mathbb{Q}left(sqrt{p_1^*}, sqrt{p_2^*},dots, sqrt{p_t^*}right).$$
In this paper, we develop a computational approach for estimating the mean value of a quantity in the presence of uncertainty. We demonstrate that, under some mild assumptions, the upper and lower bounds of the mean value are efficiently computable via a sample reuse technique, of which the computational complexity is shown to posses a Poisson distribution.
Let Fq be a finite field with q=8 or q at least 16. Let S be a smooth cubic surface defined over Fq containing at least one rational line. We use a pigeonhole principle to prove that all the rational points on S are generated via tangent and secant operations from a single point.
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