Exponential Sums and the Riemann Zeta Function IV

1993 ◽  
Vol s3-66 (1) ◽  
pp. 1-40 ◽  
Author(s):  
M. N. Huxley
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Exponential Sums ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

scholarly journals VINOGRADOV'S INTEGRAL AND BOUNDS FOR THE RIEMANN ZETA FUNCTION

2002 ◽  
Vol 85 (3) ◽  
pp. 565-633 ◽  
Author(s):  
KEVIN FORD
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Exponential Sums ◽  
Exponential Sum ◽  
Auxiliary Result ◽  
Mean Values ◽  
The Riemann Zeta Function ◽  
Secondary 11D72 ◽  
Explicit Bounds ◽  
Riemann Zeta

The main result is an upper bound for the Riemann zeta function in the critical strip: $\zeta(\sigma + it) \le A|t|^{B(1 - \sigma)^{3/2}} \log^{2/3} |t|$ with $A = 76.2$ and $B = 4.45$, valid for $\frac12 \le \sigma \le 1$ and $|t| \ge 3$. The previous best constant $B$ was 18.5. Tools include a variant of the Korobov–Vinogradov method of bounding exponential sums, an explicit version of T. D. Wooley's bounds for Vinogradov's integral, and explicit bounds for mean values of exponential sums over numbers without small prime factors, also using methods of Wooley. An auxiliary result is the exponential sum bound $S(N, t) \le 9.463 N^{1 - 1/(133.66\lambda^2)}$, where $N$ is a positive integer, $t$ is a real number, $\lambda = (\log t)/(\log N)$ and$S(N,t) = \max_{0 < u \le 1} \max_{N < R \le 2N} \left| \sum_{N < n \le R} (n + u)^{-it} \right|.$$2000 Mathematical Subject Classification: primary 11M06, 11N05, 11L15; secondary 11D72, 11M35.

scholarly journals The Vortex-like Behavior of the Riemann Zeta Function to the Right of the Critical Strip

2021 ◽  
Vol 77 (1) ◽  
Author(s):  
J. M. Sepulcre ◽  
T. Vidal
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Exponential Sums ◽  
Critical Strip ◽  
Real Numbers ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
The Right ◽  
Class Of Functions ◽  
Dense Set

AbstractBased on an equivalence relation that was established recently on exponential sums, in this paper we study the class of functions that are equivalent to the Riemann zeta function in the half-plane $$\{s\in {\mathbb {C}}:\mathrm{Re}\, s>1\}$$ { s ∈ C : Re s > 1 } . In connection with this class of functions, we first determine the value of the maximum abscissa from which the images of any function in it cannot take a prefixed argument. The main result shows that each of these functions experiments a vortex-like behavior in the sense that the main argument of its images varies indefinitely near the vertical line $$\mathrm{Re}\, s=1$$ Re s = 1 . In particular, regarding the Riemann zeta function $$\zeta (s)$$ ζ ( s ) , for every $$\sigma _0>1$$ σ 0 > 1 we can assure the existence of a relatively dense set of real numbers $$\{t_m\}_{m\ge 1}$$ { t m } m ≥ 1 such that the parametrized curve traced by the points $$(\mathrm{Re} (\zeta (\sigma +it_m)),\mathrm{Im}(\zeta (\sigma +it_m)))$$ ( Re ( ζ ( σ + i t m ) ) , Im ( ζ ( σ + i t m ) ) ) , with $$\sigma \in (1,\sigma _0)$$ σ ∈ ( 1 , σ 0 ) , makes a prefixed finite number of turns around the origin.

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