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The typical size of character and zeta sums is $o(\sqrt{x})$
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abstract
We prove conjecturally sharp upper bounds for the Dirichlet character moments $\frac{1}{r-1} \sum_{\chi \; \text{mod} \; r} |\sum_{n \leq x} \chi(n)|^{2q}$, where $r$ is a large prime, $1 \leq x \leq r$, and $0 \leq q \leq 1$ is real. In particular, if both $x$ and $r/x$ tend to infinity with $r$ then $\frac{1}{r-1} \sum_{\chi \; \text{mod} \; r} |\sum_{n \leq x} \chi(n)| = o(\sqrt{x})$, and so the sums $\sum_{n \leq x} \chi(n)$ typically exhibit "better than squareroot cancellation". We prove analogous better than squareroot bounds for the moments $\frac{1}{T} \int_{0}^{T} |\sum_{n \leq x} n^{it}|^{2q} dt$ of zeta sums; of Dirichlet theta functions $\theta(1,\chi)$; and of the sums $\sum_{n \leq x} h(n) \chi(n)$, where $h(n)$ is any suitably bounded multiplicative function (for example the M\"{o}bius function $\mu(n)$). The proofs depend on similar better than squareroot cancellation phenomena for low moments of random multiplicative functions. An important ingredient is a reorganisation of the conditioning arguments from the random case, so that one only needs to "condition" on a small collection of fairly short prime number sums. The conditioned quantities arising can then be well approximated by twisted second moments, whose behaviour is the same for character and zeta sums as in the random case.
Forward citations
Cited by 6 Pith papers
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A sharp almost sure upper bound for partial sums of random multiplicative functions
For both Steinhaus and Rademacher random multiplicative functions, almost surely |sum_{n≤x} f(n)| ≪ sqrt(x)(log log x)^{1/4+ε}, matching Harper's lower bound.
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Large values of Dirichlet polynomials with multiplicative coefficients
New Omega results for maxima of Dirichlet polynomials with multiplicative coefficients in the range N ≤ sqrt(T) and in a transition regime, using resonance method and GCD sums.
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A few notes on the asymptotic behavior of Rademacher random multiplicative functions
Obtains unrestricted high-moment estimates and exponential tail bounds for sums of Rademacher multiplicative functions via martingales.
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Lower bounds for high moments of zeta sums
For every k>2, unconditionally, the average of |∑_{n≤x} n^{-it}|^{2k} over t∈[0,T] is ≫_k x^k (log L)^{(k-1)^2}, where L = min{x, T/x}.
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Large character sums with multiplicative coefficients
An Omega lower bound near sqrt(N) exp((sqrt(2)+o(1)) sqrt(log(q/N) log_3(q/N)/log_2(q/N))) is claimed for character sums weighted by completely multiplicative f, but the extra positivity condition in the statement for...
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Large values of character sums with multiplicative coefficients
The maximum over characters of |Σ_{n≤N} f(n)χ(n)|, for any multiplicative f with |f(n)| = 1, is at least √N exp((1+o(1))√(log(q/N)/log₂(q/N))) in the range exp((log q)^{1/2+δ}) ≤ N ≤ √q.
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