Extreme Values of |ζ(1 + It)|
نویسندگان
چکیده
Improving on a result of J.E. Littlewood, N. Levinson [3] showed that there are arbitrarily large t for which |ζ(1 + it)| ≥ e log2 t + O(1). (Throughout ζ(s) is the Riemann-zeta function, and logj denotes the j-th iterated logarithm, so that log1 n = logn and logj n = log(logj−1 n) for each j ≥ 2.) The best upper bound known is Vinogradov’s |ζ(1 + it)| (log t). Littlewood had shown that |ζ(1+it)| 2e log2 t assuming the Riemann Hypothesis, in fact by showing that the value of |ζ(1 + it)| could be closely approximated by its Euler product for primes up to log(2 + |t|) under this assumption. Under the further hypothesis that the Euler product up to log(2+ |t|) still serves as a good approximation, Littlewood conjectured that max|t|≤T |ζ(1 + it)| ∼ e log2 T , though later he wrote in [5] (in connection with a q-analogue): “there is perhaps no good reason for believing ... this hypothesis”. Our Theorem 1 evaluates the frequency with which such extreme values are attained; and if this density function were to persist to the end of the viable range then this implies the conjecture that
منابع مشابه
2 Andrew Granville And
Improving on a result of J.E. Littlewood, N. Levinson [3] showed that there are arbitrarily large t for which |ζ(1 + it)| ≥ e log2 t + O(1). (Throughout ζ(s) is the Riemann-zeta function, and logj denotes the j-th iterated logarithm, so that log1 n = logn and logj n = log(logj−1 n) for each j ≥ 2.) The best upper bound known is Vinogradov’s |ζ(1 + it)| ≪ (log t). Littlewood had shown that |ζ(1 ...
متن کاملOn the Distribution of Extreme Values of Zeta and L-functions in the Strip
Abstract. We study the distribution of large (and small) values of several families of L-functions on a line Re(s) = σ where 1/2 < σ < 1. We consider the Riemann zeta function ζ(s) in the t-aspect, Dirichlet L-functions in the q-aspect, and L-functions attached to primitive holomorphic cusp forms of weight 2 in the level aspect. For each family we show that the L-values can be very well modeled...
متن کاملJa n 20 08 THE TWO DIMENSIONAL DISTRIBUTION OF VALUES OF ζ ( 1 + it )
We prove several results on the distribution function of ζ(1 + it) in the complex plane, that is the joint distribution function of arg ζ(1 + it) and |ζ(1 + it)|. Similar results are also given for L(1, χ) (as χ varies over non-principal characters modulo a large prime q).
متن کاملTHE TWO DIMENSIONAL DISTRIBUTION OF VALUES OF ζ(1 + it)
We prove several results on the distribution function of ζ(1 + it) in the complex plane, that is the joint distribution function of arg ζ(1 + it) and |ζ(1 + it)|. Similar results are also given for L(1, χ) (as χ varies over non-principal characters modulo a large prime q).
متن کاملO ct 2 00 8 THE TWO DIMENSIONAL DISTRIBUTION OF VALUES OF ζ ( 1 + it )
We prove several results on the distribution function of ζ(1 + it) in the complex plane, that is the joint distribution function of arg ζ(1 + it) and |ζ(1 + it)|. Similar results are also given for L(1, χ) (as χ varies over non-principal characters modulo a large prime q).
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2006