On the theory of distribution of primes based on I. M. Vinogradov’s method of trigonometric sums

1979 ◽  
pp. 39-73
Author(s):  
A. F. Lavrik
Keyword(s):  
Distribution Of Primes ◽  
Trigonometric Sums

Trigonometric sums through Ramanujan’s theory of theta functions

2021 ◽  
Author(s):  
K. N. Harshitha ◽  
K. R. Vasuki ◽  
M. V. Yathirajsharma
Keyword(s):  
Theta Functions ◽  
Trigonometric Sums

scholarly journals Variants on Andrica’s Conjecture with and without the Riemann Hypothesis

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 289 ◽  
Author(s):  
Matt Visser
Keyword(s):  
Riemann Hypothesis ◽  
Distribution Of Primes

The gap between what we can explicitly prove regarding the distribution of primes and what we suspect regarding the distribution of primes is enormous. It is (reasonably) well-known that the Riemann hypothesis is not sufficient to prove Andrica’s conjecture: ∀n≥1, is p n + 1 - p n ≤ 1 ? However, can one at least get tolerably close? I shall first show that with a logarithmic modification, provided one assumes the Riemann hypothesis, one has p n + 1 /ln p n + 1 - p n /ln p n < 11/25; (n ≥ 1). Then, by considering more general mth roots, again assuming the Riemann hypothesis, I show that p n + 1 m - p n m < 44/(25 e[m < 2]); (n ≥ 3; m > 2). In counterpoint, if we limit ourselves to what we can currently prove unconditionally, then the only explicit Andrica-like results seem to be variants on the relatively weak results below: ln2 pn + 1 - ln2 pn < 9; ln3 pn + 1 - ln3 pn < 52; ln4 pn + 1 - ln4 pn < 991; (n ≥ 1). I shall also update the region on which Andrica’s conjecture is unconditionally verified.

Rational trigonometric sums on ?algebraic varieties?

1986 ◽  
Vol 39 (2) ◽  
pp. 89-97
Author(s):  
S. A. Stepanov
Keyword(s):  
Algebraic Varieties ◽  
Trigonometric Sums

MULTIPLE TRIGONOMETRIC SUMS

1976 ◽  
Vol 10 (1) ◽  
pp. 200-210 ◽  
Author(s):  
G I Arhipov ◽  
V N Čubarikov
Keyword(s):  
Trigonometric Sums
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