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Sign Changes of Partial Sums of Random Multiplicative Functions

Geis, Nicholas Stanley
http://orcid.org/0000-0002-4591-4565
http://rave.ohiolink.edu/etdc/view?acc_num=osu172132303546951

Abstract Details

2024, Doctor of Philosophy, Ohio State University, Mathematics.
Let $f$ be a Rademacher random multiplicative function and $$M_f(u):=\sum_{n \leq u} f(n)$$ be the partial sum of $f$. Let $V_f(x)$ denote the number of sign changes of $M_f(u)$ up to $x$. The primary goal of this work is to prove the first explicit almost-sure growth rate for $V_f(x)$ proven by the author in [14]. Namely, we show that for any $\alpha > 2$ \[ V_f(x) = \Omega \Big( (\log \log \log x)^{1/\alpha} \Big) \] as $x\to \infty$, almost surely. We accomplish this by constructing a family of intervals $[y_k, X_k]$, with both $y_k, X_k \to \infty$ as $k \to \infty$, that contain at least 1 sign change of $M_f(u)$ eventually, almost surely. This is the content of Chapter 5 and requires the main result of Chapter 3. A secondary goal of this work is to provide the necessary background for the first non-effective proof that $M_f(u)$ changes signs infinitely often almost surely by Aymone, Heap and Zhao [6]. This and related examples are the content of Chapter 4. Along the way, we also discuss the work of Wintner [32], who first introduced random multiplicative functions. We do this to correct a minor mistake in the original proofs. This is done in Chapter 2. Chapter 6 is concerned with a generalized problem on the sign changes of $M_f(u)$ when the sum is restricted to $y$-smooth numbers, i.e. positive integers $n$ such that if a prime $p \mid n$ then $p \leq y$. Although we do not answer the posed problem, we state results that are analogous to ones utilized in the proof of our main sign-counting result.
Ghaith Hiary (Advisor)
Wenzhi Luo (Committee Member)
James Cogdell (Committee Member)
129 p.
Mathematics
analytic number theory; probability theory; random multiplicative functions

Recommended Citations

Citations

  • Geis, N. S. (2024). Sign Changes of Partial Sums of Random Multiplicative Functions [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu172132303546951

    APA Style (7th edition)

  • Geis, Nicholas. Sign Changes of Partial Sums of Random Multiplicative Functions. 2024. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu172132303546951.

    MLA Style (8th edition)

  • Geis, Nicholas. "Sign Changes of Partial Sums of Random Multiplicative Functions." Doctoral dissertation, Ohio State University, 2024. http://rave.ohiolink.edu/etdc/view?acc_num=osu172132303546951

    Chicago Manual of Style (17th edition)

osu172132303546951
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This open access ETD is published by The Ohio State University and OhioLINK.