scholarly journals The symmetric-square L -function on the critical line

2014 ◽  
Vol 140 ◽  
pp. 196-214
Author(s):  
Qingfeng Sun
Author(s):  
RIZWANUR KHAN ◽  
MATTHEW P. YOUNG

Abstract We establish sharp bounds for the second moment of symmetric-square L-functions attached to Hecke Maass cusp forms $u_j$ with spectral parameter $t_j$ , where the second moment is a sum over $t_j$ in a short interval. At the central point $s=1/2$ of the L-function, our interval is smaller than previous known results. More specifically, for $\left \lvert t_j\right \rvert $ of size T, our interval is of size $T^{1/5}$ , whereas the previous best was $T^{1/3}$ , from work of Lam. A little higher up on the critical line, our second moment yields a subconvexity bound for the symmetric-square L-function. More specifically, we get subconvexity at $s=1/2+it$ provided $\left \lvert t_j\right \rvert ^{6/7+\delta }\le \lvert t\rvert \le (2-\delta )\left \lvert t_j\right \rvert $ for any fixed $\delta>0$ . Since $\lvert t\rvert $ can be taken significantly smaller than $\left \lvert t_j\right \rvert $ , this may be viewed as an approximation to the notorious subconvexity problem for the symmetric-square L-function in the spectral aspect at $s=1/2$ .


Synthese ◽  
2016 ◽  
Vol 197 (12) ◽  
pp. 5117-5136 ◽  
Author(s):  
J. Adam Carter

AbstractIn Chapter 3 of Judgment and Agency, Sosa (Judgment and Agency, 2015) explicates the concept of a fully apt performance. In the course of doing so, he draws from illustrative examples of practical performances and applies lessons drawn to the case of cognitive performances, and in particular, to the cognitive performance of judging. Sosa’s examples in the practical sphere are rich and instructive. But there is, I will argue, an interesting disanalogy between the practical and cognitive examples he relies on. Ultimately, I think the source of the disanalogy is a problematic picture of the cognitive performance of guessing and its connection to knowledge and defeat. Once this critical line of argument is advanced, an alternative picture of guessing, qua cognitive performance, is articulated, one which avoids the problems discussed, and yet remains compatible with Sosa’s broader framework.


Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1254
Author(s):  
Xue Han ◽  
Xiaofei Yan ◽  
Deyu Zhang

Let Pc(x)={p≤x|p,[pc]areprimes},c∈R+∖N and λsym2f(n) be the n-th Fourier coefficient associated with the symmetric square L-function L(s,sym2f). For any A>0, we prove that the mean value of λsym2f(n) over Pc(x) is ≪xlog−A−2x for almost all c∈ε,(5+3)/8−ε in the sense of Lebesgue measure. Furthermore, it holds for all c∈(0,1) under the Riemann Hypothesis. Furthermore, we obtain that asymptotic formula for λf2(n) over Pc(x) is ∑p,qprimep≤x,q=[pc]λf2(p)=xclog2x(1+o(1)), for almost all c∈ε,(5+3)/8−ε, where λf(n) is the normalized n-th Fourier coefficient associated with a holomorphic cusp form f for the full modular group.


2012 ◽  
Vol 85 (9) ◽  
Author(s):  
Paolo Cea ◽  
Leonardo Cosmai ◽  
Massimo D’Elia ◽  
Alessandro Papa ◽  
Francesco Sanfilippo

2010 ◽  
Vol 130 (9) ◽  
pp. 2078-2091 ◽  
Author(s):  
Neil Dummigan ◽  
Bernhard Heim
Keyword(s):  

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